From 62534e55438210b4da43d2ada6c6ed311cbcf99a Mon Sep 17 00:00:00 2001 From: Dongdong Kong Date: Mon, 13 May 2024 10:58:03 +0800 Subject: [PATCH] mv references and formula to BandMats --- Project.toml | 2 +- data/test_CA-NS6.jl | 16 + docs/formula/main_whit.jl | 76 --- docs/formula/whit3_hat.jl | 94 --- .../whittaker_Cholesky_weinert2007.ipynb | 347 ----------- docs/formula/whittaker_Cholesky_whit3.ipynb | 565 ------------------ docs/formula/whittaker_LU_Eilers.ipynb | 216 ------- docs/formula/whittaker_LU_Eilers_d3.ipynb | 278 --------- ...nalytical Chemistry_A Perfect Smoother.pdf | Bin 127349 -> 0 bytes ...ih, 2010, Tridiagonal Matrix Algorithm.pdf | Bin 99449 -> 0 bytes docs/references/smooth.m | 71 --- docs/references/tsmooth.m | 62 -- docs/references/weinert2007.pdf | Bin 583301 -> 0 bytes src/Smooth/Whittaker/WHIT.jl | 15 +- src/Smooth/Whittaker/main_whittaker.jl | 3 + test/test-smooth_whit.jl | 14 +- 16 files changed, 40 insertions(+), 1719 deletions(-) delete mode 100644 docs/formula/main_whit.jl delete mode 100644 docs/formula/whit3_hat.jl delete mode 100644 docs/formula/whittaker_Cholesky_weinert2007.ipynb delete mode 100644 docs/formula/whittaker_Cholesky_whit3.ipynb delete mode 100644 docs/formula/whittaker_LU_Eilers.ipynb delete mode 100644 docs/formula/whittaker_LU_Eilers_d3.ipynb delete mode 100644 docs/references/Eilers_2003_Analytical Chemistry_A Perfect Smoother.pdf delete mode 100644 docs/references/Salih, 2010, Tridiagonal Matrix Algorithm.pdf delete mode 100644 docs/references/smooth.m delete mode 100644 docs/references/tsmooth.m delete mode 100644 docs/references/weinert2007.pdf diff --git a/Project.toml b/Project.toml index f1eb2b9..f467dd3 100644 --- a/Project.toml +++ b/Project.toml @@ -31,6 +31,7 @@ VegCurveFitPlotExt = "Plots" CategoricalArrays = "0.10" DataFrames = "1.6" DelimitedFiles = "1.9" +nlminb_jll = "0.1.1" NaNStatistics = "0.6" Parameters = "0.12" RecipesBase = "1.3" @@ -38,7 +39,6 @@ Reexport = "1" StaticArrays = "1.9" UnPack = "1" julia = "1.9" -nlminb_jll = "0.1.1" [extras] DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" diff --git a/data/test_CA-NS6.jl b/data/test_CA-NS6.jl index 53e2f39..0fe80c3 100644 --- a/data/test_CA-NS6.jl +++ b/data/test_CA-NS6.jl @@ -4,6 +4,22 @@ using Test y, w, t = deserialize("data/phenofit_CA-NS6") +n = length(y) +x = 1:n + +include_cve = true +@time z, cve = whit3(y, w; lambda, include_cve) +@time z1, cve1 = whit3(y, w, x; lambda, include_cve) + +using Plots +plot() +plot!(z, label="old") +plot!(z1, label="test") + +@time z2, cve2 = WHIT(y, w; d=3, lambda=2, include_cve) +@time z2, cve2 = WHIT(y, w, x; d=3, lambda=2, include_cve) + +# x = diff(t) lambda = 2.0 include_cve = false diff --git a/docs/formula/main_whit.jl b/docs/formula/main_whit.jl deleted file mode 100644 index a98a441..0000000 --- a/docs/formula/main_whit.jl +++ /dev/null @@ -1,76 +0,0 @@ -using LinearAlgebra, SparseArrays -using Symbolics -import Symbolics: scalarize, variables - -@variables λ - -speye(n) = SparseArrays.sparse(I, n, n) - -function Base.diff(x::SparseMatrixCSC, d::Integer=1) - D = x[2:end, :] .- x[1:end-1, :] - d >= 2 ? diff(D, d - 1) : D -end - -function ddmat(x::AbstractVector, d::Integer=2) - m = length(x) - if d == 0 - return speye(m) - else - # dx = x[(d+1):m] - x[1:(m-d)] # bug may here - return diff(ddmat(x, d - 1)) - end -end - -function LU_decompose(A₁) - n = size(A₁, 1) - T = typeof(A₁) - L = T(diagm(ones(n))) - - ## 徒手LU分解 - for i = 1:n-1 - r1 = A₁[i, :] - # U[i, :] = r1 - for j = i+1:n - f = A₁[j, i] / A₁[i, i] - L[j, i] = f - A₁[j, :] .= A₁[j, :] .- (f * r1) - # 为啥要引入U，这样已经求解完成了 - # L[:, 1] = A₁[j, :] - end - # println("i = $i") - # display(A₁) - end - (; L, U=A₁) -end - -function diag_m(x) - n = length(x) - M = Matrix{Num}(undef, n, n) - for i = 1:n, j = 1:n - if i == j - M[i, j] = x[i] - else - M[i, j] = 0.0 - end - end - M -end - -# 代数余子式; algebraic complement -function complement(A::AbstractArray, i=1, j=1; verbose=false) - i, j = j, i - m, n = size(A) - _A = A[setdiff(1:m, i), setdiff(1:n, j)] - verbose && display(_A) - (-1)^(i + j) * det(_A) -end - -function complement(A::AbstractArray) - R = similar(A) - fill!(R, 0) - m, n = size(A) - for i = 1:m, j = 1:n - R[i, j] = complement(A, i, j) # 代数余子式需要进行一次转置，才能得到A* = C' - end - R -end diff --git a/docs/formula/whit3_hat.jl b/docs/formula/whit3_hat.jl deleted file mode 100644 index d70e65e..0000000 --- a/docs/formula/whit3_hat.jl +++ /dev/null @@ -1,94 +0,0 @@ -# 一种`带状矩阵对角阵`的快速算法 -# 用于`Whittaker smoother`求解 -# -# Dongdong Kong, CUG, 2024-05-07 - -function def_U(c, e, f) - n = length(c) - U = Matrix{Num}(undef, n, n) - fill!(U, 0) - for i = 1:n - U[i, i] = 1 - i + 1 <= n && (U[i, i+1] = c[i]) - i + 2 <= n && (U[i, i+2] = e[i]) - i + 3 <= n && (U[i, i+3] = f[i]) - end - U -end - -function def_U(c, e) - n = length(c) - U = Matrix{Num}(undef, n, n) - fill!(U, 0) - for i = 1:n - U[i, i] = 1 - i + 1 <= n && (U[i, i+1] = c[i]) - i + 2 <= n && (U[i, i+2] = e[i]) - end - U -end - -""" -- `U`: [ - 1 c₁ e₁ f₁ 0 - 0 1 c₂ e₂ f₂ - 0 0 1 c₃ e₃ - 0 0 0 1 c₄ - 0 0 0 0 1 -] -""" -function cal_diag(U2, d; m=3) - # S2: [n, m+1] - # U2: [n, m] - n = length(d) - S = variables(:S, 1:n, 1:m+1) # m=2,3个临时变量已足够 - fill!(S, 0) - S[n, 1] = 1 / d[n] - - for i = n-1:-1:1 - S[i, 1] = 1 / d[i] - for l = 1:min(m, n - i) - S[i, 1+l] = 0 - for k = 1:min(n - i, m) - # if k <= l - # S[i, 1+l] -= U[i, i+k] * S[i+k, l-k+1] - # else - # S[i, 1+l] -= U[i, i+k] * S[i+l, k-l+1] - # end - _i, _j = k <= l ? (i + k, l - k + 1) : (i + l, k - l + 1) - S[i, 1+l] -= U2[i, k] * S[_i, _j] - end - S[i, 1] -= U2[i, l] * S[i, 1+l] - end - end - S[:, 1] -end - -function cal_diag_full(U, d; m=3) - # B = (U' * D * U) # Hutchinson 1985, Eq. 3.1 - # B^(-1) = B * U^(-1)' + (1 - U) * B^(-1) # Hutchinson 1985, Eq. 3.3 - n = length(d) - b = variables(:b, 1:n, 1:n) # elements of B^-1, 对称矩阵，因此只求一半的元素即可 - b[n, n] = 1 / d[n] - # b[n-1, n] = -U[n-1,n]*b[n,n] - # b[n-1, n-1] = d[n-1] - U[n-1,n]*b[n-1,n] - - for i = n-1:-1:1 - b[i, i] = 1 / d[i] - - for l = 1:min(m, n - i) - b[i, i+l] = 0 - for k = 1:min(n - i, m) - _i, _j = k <= l ? (i + k, i + l) : (i + l, i + k) - b[i, i+l] -= U[i, i+k] * b[_i, _j] - # if k <= l - # b[i, i+l] -= U[i, i+k] * b[i+k, i+l] - # else - # b[i, i+l] -= U[i, i+k] * b[i+l, i+k] - # end - end - b[i, i] -= U[i, i+l] * b[i, i+l] - end - end - diag(b) -end diff --git a/docs/formula/whittaker_Cholesky_weinert2007.ipynb b/docs/formula/whittaker_Cholesky_weinert2007.ipynb deleted file mode 100644 index 19d1074..0000000 --- a/docs/formula/whittaker_Cholesky_weinert2007.ipynb +++ /dev/null @@ -1,347 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 2, - "id": "92ea33dd-6fc9-4938-aa19-26b7156068b0", - "metadata": {}, - "outputs": [], - "source": [ - "using LinearAlgebra, SparseArrays\n", - "using Symbolics\n", - "\n", - "@variables λ e_1 f_1 e_2 f_2 e_3 f_3 e_4 f_4 e_5 f_5 f_6;" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "id": "a44f8bad-41c5-460c-b44e-c1f534c92ee1", - "metadata": {}, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "L\n" - ] - }, - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{cccccc}\n", - "1 & 0 & 0 & 0 & 0 & 0 \\\\\n", - " - e_{1} & 1 & 0 & 0 & 0 & 0 \\\\\n", - "f_{1} & - e_{2} & 1 & 0 & 0 & 0 \\\\\n", - "0 & f_{2} & - e_{3} & 1 & 0 & 0 \\\\\n", - "0 & 0 & f_{3} & - e_{4} & 1 & 0 \\\\\n", - "0 & 0 & 0 & f_{4} & - e_{5} & 1 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "6×6 Matrix{Num}:\n", - " 1 0 0 0 0 0\n", - " -e_1 1 0 0 0 0\n", - " f_1 -e_2 1 0 0 0\n", - " 0 f_2 -e_3 1 0 0\n", - " 0 0 f_3 -e_4 1 0\n", - " 0 0 0 f_4 -e_5 1" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{cccccc}\n", - "\\frac{1}{f_{1}} & 0 & 0 & 0 & 0 & 0 \\\\\n", - "0 & \\frac{1}{f_{2}} & 0 & 0 & 0 & 0 \\\\\n", - "0 & 0 & \\frac{1}{f_{3}} & 0 & 0 & 0 \\\\\n", - "0 & 0 & 0 & \\frac{1}{f_{4}} & 0 & 0 \\\\\n", - "0 & 0 & 0 & 0 & \\frac{1}{f_{5}} & 0 \\\\\n", - "0 & 0 & 0 & 0 & 0 & \\frac{1}{f_{6}} \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "6×6 Matrix{Num}:\n", - " 1 / f_1 0 0 0 0 0\n", - " 0 1 / f_2 0 0 0 0\n", - " 0 0 1 / f_3 0 0 0\n", - " 0 0 0 1 / f_4 0 0\n", - " 0 0 0 0 1 / f_5 0\n", - " 0 0 0 0 0 1 / f_6" - ] - }, - "execution_count": 3, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "L =[\n", - " 1 0 0 0 0 0\n", - " -e_1 1 0 0 0 0\n", - " f_1 -e_2 1 0 0 0\n", - " 0 f_2 -e_3 1 0 0\n", - " 0 0 f_3 -e_4 1 0\n", - " 0 0 0 f_4 -e_5 1\n", - " ]\n", - "\n", - "println(\"L\"); display(L)\n", - "D = diagm([f_1, f_2, f_3, f_4, f_5, f_6])^-1" - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "id": "6cba1139-83f8-40eb-bf2e-69bdf19d3356", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{cc}\n", - "\\frac{\\theta_{1}}{f_{1}} & w_{1} \\\\\n", - "\\frac{ - e_{1} \\theta_{1}}{f_{1}} + \\frac{\\theta_{2}}{f_{2}} & w_{2} \\\\\n", - "\\frac{ - e_{2} \\theta_{2}}{f_{2}} + \\frac{\\theta_{3}}{f_{3}} + \\theta_{1} & w_{3} \\\\\n", - "\\frac{ - e_{3} \\theta_{3}}{f_{3}} + \\frac{\\theta_{4}}{f_{4}} + \\theta_{2} & w_{4} \\\\\n", - "\\frac{\\theta_{5}}{f_{5}} + \\frac{ - e_{4} \\theta_{4}}{f_{4}} + \\theta_{3} & w_{5} \\\\\n", - "\\frac{ - e_{5} \\theta_{5}}{f_{5}} + \\frac{\\theta_{6}}{f_{6}} + \\theta_{4} & w_{6} \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "6×2 Matrix{Num}:\n", - " θ_1 / f_1 w_1\n", - " (-e_1*θ_1) / f_1 + θ_2 / f_2 w_2\n", - " (-e_2*θ_2) / f_2 + θ_3 / f_3 + θ_1 w_3\n", - " (-e_3*θ_3) / f_3 + θ_4 / f_4 + θ_2 w_4\n", - " θ_5 / f_5 + (-e_4*θ_4) / f_4 + θ_3 w_5\n", - " (-e_5*θ_5) / f_5 + θ_6 / f_6 + θ_4 w_6" - ] - }, - "execution_count": 5, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "@variables θ_1 θ_2 θ_3 θ_4 θ_5 θ_6\n", - "@variables w_1 w_2 w_3 w_4 w_5 w_6\n", - "θ = [θ_1 θ_2 θ_3 θ_4 θ_5 θ_6]'\n", - "w = [w_1 w_2 w_3 w_4 w_5 w_6]'\n", - "[L * D * θ w]" - ] - }, - { - "cell_type": "markdown", - "id": "cb7e310a-21d0-4443-b780-fc6d7d6ddabb", - "metadata": {}, - "source": [ - "θ1 = f1 * w1\n", - "θ2 = f2(w2 + θ1e1/f1)\n", - "θ3 = f3(w3 + θ2e2/f2 - θ1 )\n" - ] - }, - { - "cell_type": "code", - "execution_count": 5, - "id": "a9244286-51ca-4ddd-be35-733cae30a07f", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{cccccc}\n", - "\\frac{1}{f_{1}} & \\frac{ - e_{1}}{f_{1}} & 1 & 0 & 0 & 0 \\\\\n", - "\\frac{ - e_{1}}{f_{1}} & \\frac{e_{1}^{2}}{f_{1}} + \\frac{1}{f_{2}} & - e_{1} + \\frac{ - e_{2}}{f_{2}} & 1 & 0 & 0 \\\\\n", - "1 & - e_{1} + \\frac{ - e_{2}}{f_{2}} & f_{1} + \\frac{1}{f_{3}} + \\frac{e_{2}^{2}}{f_{2}} & - e_{2} + \\frac{ - e_{3}}{f_{3}} & 1 & 0 \\\\\n", - "0 & 1 & - e_{2} + \\frac{ - e_{3}}{f_{3}} & f_{2} + \\frac{e_{3}^{2}}{f_{3}} + \\frac{1}{f_{4}} & - e_{3} + \\frac{ - e_{4}}{f_{4}} & 1 \\\\\n", - "0 & 0 & 1 & - e_{3} + \\frac{ - e_{4}}{f_{4}} & f_{3} + \\frac{e_{4}^{2}}{f_{4}} + \\frac{1}{f_{5}} & - e_{4} + \\frac{ - e_{5}}{f_{5}} \\\\\n", - "0 & 0 & 0 & 1 & - e_{4} + \\frac{ - e_{5}}{f_{5}} & f_{4} + \\frac{e_{5}^{2}}{f_{5}} + \\frac{1}{f_{6}} \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "6×6 Matrix{Num}:\n", - " 1 / f_1 (-e_1) / f_1 … 0\n", - " (-e_1) / f_1 (e_1^2) / f_1 + 1 / f_2 0\n", - " 1 -e_1 + (-e_2) / f_2 0\n", - " 0 1 1\n", - " 0 0 -e_4 + (-e_5) / f_5\n", - " 0 0 … f_4 + (e_5^2) / f_5 + 1 / f_6" - ] - }, - "execution_count": 5, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "A2 = L * D * L'" - ] - }, - { - "cell_type": "markdown", - "id": "36b9b1c0-01a8-4972-8a54-d94dcce51f17", - "metadata": {}, - "source": [ - "## 先求出e, f, 然后根据w, 确定theta\n", - "```julia\n", - "let u_i = a_1 - e_i-1 # u_i 没必要存在\n", - "\n", - "-e1/f1 = a1\n", - "e1 = a1 * f1\n", - "f2 = 1/(a0 - e1^2/f1)\n", - "\n", - "-e2/f2 = e1 - a1\n", - "e2 = -f2(-a1 + e1) = f2(a1 - e1) = f2 * u2\n", - "\n", - "f3 = 1/(a0 -e2^2/f2 - f1) (e2^2/f2 = u2 * e2,)\n", - "\n", - "-e3/f3 = -a1 + e2 \n", - "e3 = f3 * (a1 - e2)\n", - "```" - ] - }, - { - "cell_type": "code", - "execution_count": 12, - "id": "21f525ef-8f71-4520-a205-46c0541e65c0", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{cccccc}\n", - "6 + \\frac{2}{3 \\lambda} & -4 + \\frac{1}{6} \\lambda & 1 & 0 & 0 & 0 \\\\\n", - "-4 + \\frac{1}{6} \\lambda & 6 + \\frac{2}{3} \\lambda & -4 + \\frac{1}{6} \\lambda & 1 & 0 & 0 \\\\\n", - "1 & -4 + \\frac{1}{6} \\lambda & 6 + \\frac{2}{3} \\lambda & -4 + \\frac{1}{6} \\lambda & 1 & 0 \\\\\n", - "0 & 1 & -4 + \\frac{1}{6} \\lambda & 6 + \\frac{2}{3} \\lambda & -4 + \\frac{1}{6} \\lambda & 1 \\\\\n", - "0 & 0 & 1 & -4 + \\frac{1}{6} \\lambda & 6 + \\frac{2}{3} \\lambda & -4 + \\frac{1}{6} \\lambda \\\\\n", - "0 & 0 & 0 & 1 & -4 + \\frac{1}{6} \\lambda & 6 + \\frac{2}{3} \\lambda \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "6×6 Matrix{Num}:\n", - " 6 + 2 / (3λ) -4 + (1//6)*λ 1 … 0 0\n", - " -4 + (1//6)*λ 6 + (2//3)*λ -4 + (1//6)*λ 0 0\n", - " 1 -4 + (1//6)*λ 6 + (2//3)*λ 1 0\n", - " 0 1 -4 + (1//6)*λ -4 + (1//6)*λ 1\n", - " 0 0 1 6 + (2//3)*λ -4 + (1//6)*λ\n", - " 0 0 0 … -4 + (1//6)*λ 6 + (2//3)*λ" - ] - }, - "execution_count": 12, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "A = [\n", - " 2/3λ+6 λ/6-4 1 0 0 0\n", - " λ/6-4 2λ/3+6 λ/6-4 1 0 0\n", - " 1 λ/6-4 2λ/3+6 λ/6-4 1 0\n", - " 0 1 λ/6-4 2λ/3+6 λ/6-4 1\n", - " 0 0 1 λ/6-4 2λ/3+6 λ/6-4\n", - " 0 0 0 1 λ/6-4 2λ/3+6\n", - "]" - ] - }, - { - "cell_type": "markdown", - "id": "ff3b301e-c3b4-4e3f-b6cc-a6735fcdbd29", - "metadata": {}, - "source": [ - "## second step\n", - "$$\n", - "L^T c = \\theta\n", - "$$" - ] - }, - { - "cell_type": "code", - "execution_count": 23, - "id": "fc25cae6-0089-461f-954a-1e65ddac8b6c", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{cc}\n", - "c_{1} - c_{2} e_{1} + c_{3} f_{1} & \\theta_{1} \\\\\n", - "c_{2} - c_{3} e_{2} + c_{4} f_{2} & \\theta_{2} \\\\\n", - "c_{3} - c_{4} e_{3} + c_{5} f_{3} & \\theta_{3} \\\\\n", - "c_{4} - c_{5} e_{4} + c_{6} f_{4} & \\theta_{4} \\\\\n", - "c_{5} - c_{6} e_{5} & \\theta_{5} \\\\\n", - "c_{6} & \\theta_{6} \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "6×2 Matrix{Num}:\n", - " c_1 - c_2*e_1 + c_3*f_1 θ_1\n", - " c_2 - c_3*e_2 + c_4*f_2 θ_2\n", - " c_3 - c_4*e_3 + c_5*f_3 θ_3\n", - " c_4 - c_5*e_4 + c_6*f_4 θ_4\n", - " c_5 - c_6*e_5 θ_5\n", - " c_6 θ_6" - ] - }, - "execution_count": 23, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "@variables c_1 c_2 c_3 c_4 c_5 c_6\n", - "c = [c_1 c_2 c_3 c_4 c_5 c_6]'\n", - "\n", - "[L' * c θ]\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Julia 1.10.3", - "language": "julia", - "name": "julia-1.10" - }, - "language_info": { - "file_extension": ".jl", - "mimetype": "application/julia", - "name": "julia", - "version": "1.10.3" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/docs/formula/whittaker_Cholesky_whit3.ipynb b/docs/formula/whittaker_Cholesky_whit3.ipynb deleted file mode 100644 index f3ce362..0000000 --- a/docs/formula/whittaker_Cholesky_whit3.ipynb +++ /dev/null @@ -1,565 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 64, - "id": "cd3c38ac-780c-4e99-9e4f-7c26c3d1cf6d", - "metadata": {}, - "outputs": [], - "source": [ - "# 这里是whittaker 3阶向量求解过程\n", - "include(\"main_whit.jl\")\n", - "\n", - "using SymbolicUtils" - ] - }, - { - "cell_type": "code", - "execution_count": 74, - "id": "a2c429bc-be39-41e5-a7a0-31ce5e4e0439", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{ccc}\n", - "c_1 & e_1 & f_1 \\\\\n", - "c_2 & e_2 & f_2 \\\\\n", - "c_3 & e_3 & f_3 \\\\\n", - "c_4 & e_4 & f_4 \\\\\n", - "c_5 & e_5 & f_5 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "5×3 Matrix{Num}:\n", - " c[1] e[1] f[1]\n", - " c[2] e[2] f[2]\n", - " c[3] e[3] f[3]\n", - " c[4] e[4] f[4]\n", - " c[5] e[5] f[5]" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "include(\"whit3_hat.jl\")\n", - "\n", - "n = 5\n", - "@variables λ \n", - "\n", - "@variables d[1:n]\n", - "D = diag_m(d)\n", - "\n", - "@variables c[1:n]\n", - "@variables e[1:n]\n", - "@variables f[1:n]\n", - "\n", - "z = variables(:z, 1:n)\n", - "w = variables(:w, 1:n)\n", - "y = variables(:y, 1:n)\n", - "θ = variables(:θ, 1:n)\n", - "\n", - "U = def_U(c, e, f)\n", - "U2 = hcat(c, e, f)\n", - "# U = def_U(c, e)\n", - "# L = U'\n", - "# x = inv(L)\n", - "# H = x' * inv(D) * x\n", - "# H\n", - "# diag(H)" - ] - }, - { - "cell_type": "code", - "execution_count": 60, - "id": "35d27b6c", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{ccccc}\n", - "d_1 & 0 & 0 & 0 & 0 \\\\\n", - " - c_1 d_2 & d_2 & 0 & 0 & 0 \\\\\n", - "\\left( - e_1 + c_1 c_2 \\right) d_3 & - c_2 d_3 & d_3 & 0 & 0 \\\\\n", - "\\left( - f_1 + c_3 e_1 - c_1 \\left( - e_2 + c_2 c_3 \\right) \\right) d_4 & \\left( - e_2 + c_2 c_3 \\right) d_4 & - c_3 d_4 & d_4 & 0 \\\\\n", - "\\left( c_4 f_1 - \\left( - e_3 + c_3 c_4 \\right) e_1 + c_1 \\left( f_2 - c_4 e_2 + c_2 \\left( - e_3 + c_3 c_4 \\right) \\right) \\right) d_5 & \\left( - f_2 + c_4 e_2 - c_2 \\left( - e_3 + c_3 c_4 \\right) \\right) d_5 & \\left( - e_3 + c_3 c_4 \\right) d_5 & - c_4 d_5 & d_5 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "5×5 Matrix{Num}:\n", - " d[1] … 0.0 0.0\n", - " -c[1]*d[2] 0.0 0.0\n", - " (-e[1] + c[1]*c[2])*d[3] 0.0 0.0\n", - " (-f[1] + c[3]*e[1] - c[1]*(-e[2] + c[2]*c[3]))*d[4] d[4] 0.0\n", - " (c[4]*f[1] - (-e[3] + c[3]*c[4])*e[1] + c[1]*(f[2] - c[4]*e[2] + c[2]*(-e[3] + c[3]*c[4])))*d[5] -c[4]*d[5] d[5]" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "D * (U^-1)'" - ] - }, - { - "cell_type": "code", - "execution_count": 76, - "id": "3f4e1d0e", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{c}\n", - "0 \\\\\n", - "0 \\\\\n", - "0 \\\\\n", - "0 \\\\\n", - "0 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "5-element Vector{Num}:\n", - " 0\n", - " 0\n", - " 0\n", - " 0\n", - " 0" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "# 成功推导\n", - "s = cal_diag(U2, d; m=3)\n", - "s_full = cal_diag_full(U, d; m=3)\n", - "s - s_full # \n", - "# s\n" - ] - }, - { - "cell_type": "code", - "execution_count": 113, - "id": "b835c8ad", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{ccccc}\n", - "1 & 0 & 0 & 0 & 0 \\\\\n", - " - c_1 & 1 & 0 & 0 & 0 \\\\\n", - " - e_1 + c_1 c_2 & - c_2 & 1 & 0 & 0 \\\\\n", - "c_3 e_1 - c_1 \\left( - e_2 + c_2 c_3 \\right) & - e_2 + c_2 c_3 & - c_3 & 1 & 0 \\\\\n", - " - \\left( - e_3 + c_3 c_4 \\right) e_1 + c_1 \\left( - c_4 e_2 + c_2 \\left( - e_3 + c_3 c_4 \\right) \\right) & c_4 e_2 - c_2 \\left( - e_3 + c_3 c_4 \\right) & - e_3 + c_3 c_4 & - c_4 & 1 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "5×5 Matrix{Num}:\n", - " 1 … 0 0 0\n", - " -c[1] 0 0 0\n", - " -e[1] + c[1]*c[2] 1 0 0\n", - " c[3]*e[1] - c[1]*(-e[2] + c[2]*c[3]) -c[3] 1 0\n", - " -(-e[3] + c[3]*c[4])*e[1] + c[1]*(-c[4]*e[2] + c[2]*(-e[3] + c[3]*c[4])) -e[3] + c[3]*c[4] -c[4] 1" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "complement(L)" - ] - }, - { - "cell_type": "code", - "execution_count": 124, - "id": "37149752", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{c}\n", - "\\frac{a_{7}ˏ_1^{2}}{d_7} + \\frac{a_{2}ˏ_1^{2}}{d_2} + \\frac{a_{4}ˏ_1^{2}}{d_4} + \\frac{a_{3}ˏ_1^{2}}{d_3} + \\frac{1}{d_1} + \\frac{a_{6}ˏ_1^{2}}{d_6} + \\frac{a_{5}ˏ_1^{2}}{d_5} \\\\\n", - "\\frac{a_{5}ˏ_2^{2}}{d_5} + \\frac{a_{6}ˏ_2^{2}}{d_6} + \\frac{a_{4}ˏ_2^{2}}{d_4} + \\frac{1}{d_2} + \\frac{a_{7}ˏ_2^{2}}{d_7} + \\frac{a_{3}ˏ_2^{2}}{d_3} \\\\\n", - "\\frac{1}{d_3} + \\frac{a_{5}ˏ_3^{2}}{d_5} + \\frac{a_{6}ˏ_3^{2}}{d_6} + \\frac{a_{7}ˏ_3^{2}}{d_7} + \\frac{a_{4}ˏ_3^{2}}{d_4} \\\\\n", - "\\frac{a_{6}ˏ_4^{2}}{d_6} + \\frac{a_{5}ˏ_4^{2}}{d_5} + \\frac{1}{d_4} + \\frac{a_{7}ˏ_4^{2}}{d_7} \\\\\n", - "\\frac{1}{d_5} + \\frac{a_{7}ˏ_5^{2}}{d_7} + \\frac{a_{6}ˏ_5^{2}}{d_6} \\\\\n", - "\\frac{1}{d_6} + \\frac{a_{7}ˏ_6^{2}}{d_7} \\\\\n", - "\\frac{1}{d_7} \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "7-element Vector{Num}:\n", - " (a₇ˏ₁^2) / d[7] + (a₂ˏ₁^2) / d[2] + (a₄ˏ₁^2) / d[4] + (a₃ˏ₁^2) / d[3] + 1 / d[1] + (a₆ˏ₁^2) / d[6] + (a₅ˏ₁^2) / d[5]\n", - " (a₅ˏ₂^2) / d[5] + (a₆ˏ₂^2) / d[6] + (a₄ˏ₂^2) / d[4] + 1 / d[2] + (a₇ˏ₂^2) / d[7] + (a₃ˏ₂^2) / d[3]\n", - " 1 / d[3] + (a₅ˏ₃^2) / d[5] + (a₆ˏ₃^2) / d[6] + (a₇ˏ₃^2) / d[7] + (a₄ˏ₃^2) / d[4]\n", - " (a₆ˏ₄^2) / d[6] + (a₅ˏ₄^2) / d[5] + 1 / d[4] + (a₇ˏ₄^2) / d[7]\n", - " 1 / d[5] + (a₇ˏ₅^2) / d[7] + (a₆ˏ₅^2) / d[6]\n", - " 1 / d[6] + (a₇ˏ₆^2) / d[7]\n", - " 1 / d[7]" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{c}\n", - "\\frac{a_{7}ˏ_1^{2}}{d_7} + \\frac{a_{2}ˏ_1^{2}}{d_2} + \\frac{a_{4}ˏ_1^{2}}{d_4} + \\frac{a_{3}ˏ_1^{2}}{d_3} + \\frac{1}{d_1} + \\frac{a_{6}ˏ_1^{2}}{d_6} + \\frac{a_{5}ˏ_1^{2}}{d_5} \\\\\n", - "\\frac{a_{5}ˏ_2^{2}}{d_5} + \\frac{a_{6}ˏ_2^{2}}{d_6} + \\frac{a_{4}ˏ_2^{2}}{d_4} + \\frac{1}{d_2} + \\frac{a_{7}ˏ_2^{2}}{d_7} + \\frac{a_{3}ˏ_2^{2}}{d_3} \\\\\n", - "\\frac{1}{d_3} + \\frac{a_{5}ˏ_3^{2}}{d_5} + \\frac{a_{6}ˏ_3^{2}}{d_6} + \\frac{a_{7}ˏ_3^{2}}{d_7} + \\frac{a_{4}ˏ_3^{2}}{d_4} \\\\\n", - "\\frac{a_{6}ˏ_4^{2}}{d_6} + \\frac{a_{5}ˏ_4^{2}}{d_5} + \\frac{1}{d_4} + \\frac{a_{7}ˏ_4^{2}}{d_7} \\\\\n", - "\\frac{1}{d_5} + \\frac{a_{7}ˏ_5^{2}}{d_7} + \\frac{a_{6}ˏ_5^{2}}{d_6} \\\\\n", - "\\frac{1}{d_6} + \\frac{a_{7}ˏ_6^{2}}{d_7} \\\\\n", - "\\frac{1}{d_7} \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "7-element Vector{Num}:\n", - " (a₇ˏ₁^2) / d[7] + (a₂ˏ₁^2) / d[2] + (a₄ˏ₁^2) / d[4] + (a₃ˏ₁^2) / d[3] + 1 / d[1] + (a₆ˏ₁^2) / d[6] + (a₅ˏ₁^2) / d[5]\n", - " (a₅ˏ₂^2) / d[5] + (a₆ˏ₂^2) / d[6] + (a₄ˏ₂^2) / d[4] + 1 / d[2] + (a₇ˏ₂^2) / d[7] + (a₃ˏ₂^2) / d[3]\n", - " 1 / d[3] + (a₅ˏ₃^2) / d[5] + (a₆ˏ₃^2) / d[6] + (a₇ˏ₃^2) / d[7] + (a₄ˏ₃^2) / d[4]\n", - " (a₆ˏ₄^2) / d[6] + (a₅ˏ₄^2) / d[5] + 1 / d[4] + (a₇ˏ₄^2) / d[7]\n", - " 1 / d[5] + (a₇ˏ₅^2) / d[7] + (a₆ˏ₅^2) / d[6]\n", - " 1 / d[6] + (a₇ˏ₆^2) / d[7]\n", - " 1 / d[7]" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "## 定义一个三角矩阵\n", - "n = 7\n", - "\n", - "@variables d[1:n]\n", - "h = variables(:h, 1:n)\n", - "# D = diag_m(d)\n", - "\n", - "A = variables(:a, 1:n, 1:n)\n", - "for i = 1:n, j=i:n\n", - " A[i, j] = i == j ? 1 : 0\n", - "end\n", - "A\n", - "# (L L')^-1 = L^-1' * L^-1\n", - "H = A' * inv(D) * A\n", - "display(diag(H))\n", - "\n", - "# 现在找到了解法，然后是如何把结果填进去\n", - "for i = 1:n\n", - " h[i] = 0\n", - " len = n - i + 1\n", - " for j = n:-1:n-len+1\n", - " h[i] += A[j, i]^2 / d[j]\n", - " end\n", - "end\n", - "h\n", - "# h - diag(H)" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "55d2472d", - "metadata": {}, - "outputs": [], - "source": [ - "# # Compute diagonal of inverse\n", - "s0[n] = 1 / d[n]\n", - "s0[n-1] = 1 / d[n-1] + c[n-1]^2 * s0[n]\n", - "s1[n-1] = -c[n-1] * s0[n]\n", - "\n", - "@inbounds @fastmath for i = n-2:-1:1\n", - " s1[i] = -c[i] * s0[i+1] - e[i] * s1[i+1]\n", - " s2[i] = -c[i] * s1[i+1] - e[i] * s0[i+2]\n", - "\n", - " # s0[i] = 1 / d[i] -\n", - " # c[i] * (-c[i] * s0[i+1] - e[i] * s1[i+1]) -\n", - " # e[i] * (-c[i] * s1[i+1] - e[i] * s0[i+2])\n", - " s0[i] = 1 / d[i] + c[i]^2 * s0[i+1] + 2c[i] * e[i] * s1[i+1] + e[i]^2 * s0[i+2]\n", - "end" - ] - }, - { - "cell_type": "code", - "execution_count": 50, - "id": "2ea4def7", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{c}\n", - " - \\left( - \\left( \\frac{c_4 e_3}{d_5} - c_3 \\left( \\frac{1}{d_4} + \\frac{c_4^{2}}{d_5} \\right) \\right) e_2 - c_2 \\left( \\frac{1}{d_3} - e_3 \\left( \\frac{ - e_3}{d_5} + \\frac{c_3 c_4}{d_5} \\right) - c_3 \\left( \\frac{c_4 e_3}{d_5} - c_3 \\left( \\frac{1}{d_4} + \\frac{c_4^{2}}{d_5} \\right) \\right) \\right) \\right) e_1 - c_1 \\left( \\frac{1}{d_2} - \\left( - e_2 \\left( \\frac{1}{d_4} + \\frac{c_4^{2}}{d_5} \\right) - c_2 \\left( \\frac{c_4 e_3}{d_5} - c_3 \\left( \\frac{1}{d_4} + \\frac{c_4^{2}}{d_5} \\right) \\right) \\right) e_2 - c_2 \\left( - \\left( \\frac{c_4 e_3}{d_5} - c_3 \\left( \\frac{1}{d_4} + \\frac{c_4^{2}}{d_5} \\right) \\right) e_2 - c_2 \\left( \\frac{1}{d_3} - e_3 \\left( \\frac{ - e_3}{d_5} + \\frac{c_3 c_4}{d_5} \\right) - c_3 \\left( \\frac{c_4 e_3}{d_5} - c_3 \\left( \\frac{1}{d_4} + \\frac{c_4^{2}}{d_5} \\right) \\right) \\right) \\right) \\right) \\\\\n", - " - \\left( \\frac{c_4 e_3}{d_5} - c_3 \\left( \\frac{1}{d_4} + \\frac{c_4^{2}}{d_5} \\right) \\right) e_2 - c_2 \\left( \\frac{1}{d_3} - e_3 \\left( \\frac{ - e_3}{d_5} + \\frac{c_3 c_4}{d_5} \\right) - c_3 \\left( \\frac{c_4 e_3}{d_5} - c_3 \\left( \\frac{1}{d_4} + \\frac{c_4^{2}}{d_5} \\right) \\right) \\right) \\\\\n", - "\\frac{c_4 e_3}{d_5} - c_3 \\left( \\frac{1}{d_4} + \\frac{c_4^{2}}{d_5} \\right) \\\\\n", - "\\frac{ - c_4}{d_5} \\\\\n", - "s1_5 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "5-element Vector{Num}:\n", - " -(-((c₄*e₃) / d₅ - c₃*(1 / d₄ + (c₄^2) / d₅))*e₂ - c₂*(1 / d₃ - e₃*((-e₃) / d₅ + (c₃*c₄) / d₅) - c₃*((c₄*e₃) / d₅ - c₃*(1 / d₄ + (c₄^2) / d₅))))*e₁ - c₁*(1 / d₂ - (-e₂*(1 / d₄ + (c₄^2) / d₅) - c₂*((c₄*e₃) / d₅ - c₃*(1 / d₄ + (c₄^2) / d₅)))*e₂ - c₂*(-((c₄*e₃) / d₅ - c₃*(1 / d₄ + (c₄^2) / d₅))*e₂ - c₂*(1 / d₃ - e₃*((-e₃) / d₅ + (c₃*c₄) / d₅) - c₃*((c₄*e₃) / d₅ - c₃*(1 / d₄ + (c₄^2) / d₅)))))\n", - " -((c₄*e₃) / d₅ - c₃*(1 / d₄ + (c₄^2) / d₅))*e₂ - c₂*(1 / d₃ - e₃*((-e₃) / d₅ + (c₃*c₄) / d₅) - c₃*((c₄*e₃) / d₅ - c₃*(1 / d₄ + (c₄^2) / d₅)))\n", - " (c₄*e₃) / d₅ - c₃*(1 / d₄ + (c₄^2) / d₅)\n", - " (-c₄) / d₅\n", - " s1₅" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "d = variables(:d, 1:n)\n", - "c = variables(:c, 1:n)\n", - "e = variables(:e, 1:n)\n", - "f = variables(:f, 1:n)\n", - "\n", - "# h = variables(:h, 1:n)\n", - "s0 = variables(:s0, 1:n)\n", - "s1 = variables(:s1, 1:n)\n", - "s2 = variables(:s2, 1:n)\n", - "\n", - "s0[n] = 1 / d[n]\n", - "s0[n-1] = 1 / d[n-1] + c[n-1]^2 * s0[n]\n", - "s1[n-1] = -c[n-1] * s0[n]\n", - "\n", - "@inbounds @fastmath for i = n-2:-1:1\n", - " s1[i] = -c[i] * s0[i+1] - e[i] * s1[i+1]\n", - " s2[i] = -c[i] * s1[i+1] - e[i] * s0[i+2]\n", - " s0[i] = 1 / d[i] - c[i] * s1[i] - e[i] * s2[i]\n", - "end\n", - "\n", - "s1" - ] - }, - { - "cell_type": "code", - "execution_count": 45, - "id": "978e9891-eeff-419f-9a0a-1532275e7b42", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{c}\n", - "c_4 f_1 - \\left( - e_3 + c_3 c_4 \\right) e_1 - c_1 \\left( - f_2 + c_4 e_2 - c_2 \\left( - e_3 + c_3 c_4 \\right) \\right) \\\\\n", - " - f_2 + c_4 e_2 - c_2 \\left( - e_3 + c_3 c_4 \\right) \\\\\n", - " - e_3 + c_3 c_4 \\\\\n", - " - c_4 \\\\\n", - "1 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "5-element Vector{Num}:\n", - " c₄*f₁ - (-e₃ + c₃*c₄)*e₁ - c₁*(-f₂ + c₄*e₂ - c₂*(-e₃ + c₃*c₄))\n", - " -f₂ + c₄*e₂ - c₂*(-e₃ + c₃*c₄)\n", - " -e₃ + c₃*c₄\n", - " -c₄\n", - " 1" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "## 设计一套求解的算法\n", - "# @variables h[1:n]\n", - "h = variables(:h, 1:n)\n", - "s = variables(:s, 1:n)\n", - "\n", - "h[n] = 1\n", - "h[n-1] = -c[n-1]\n", - "h[n-2] = -e[n-2] * h[n] - c[n-2] * h[n-1]\n", - "@inbounds for i = n-3:-1:1\n", - " h[i] = -f[i] * h[i+3] - e[i] * h[i+2] - c[i] * h[i+1]\n", - "end\n", - "h\n", - "\n", - "# function cal_diag!(s::V, h::V, d::V) where {V<:AbstractVector{<:Real}}\n", - "# n = length(d)\n", - "s[n] = h[1]^2 / d[n]\n", - "s[n-1] = h[2]^2 / d[n] + h[1] / d[n-1]\n", - "\n", - "@inbounds for k = n:-1:1\n", - " s[k] = 0.0\n", - " for (i, j) in zip(k:n, n:-1:k)\n", - " s[k] += h[i]^2 / d[j]\n", - " end\n", - "end\n", - "# s\n", - "h\n", - "# s - diag(H)\n", - "# s\n", - "# end" - ] - }, - { - "cell_type": "code", - "execution_count": 37, - "id": "95234236-46a3-4346-9576-c27ff284868f", - "metadata": {}, - "outputs": [], - "source": [ - "# H = inv(L * D * L')\n", - "# diag(H)" - ] - }, - { - "cell_type": "code", - "execution_count": 17, - "id": "e34b6b09-f606-4d4c-9629-2ba664330115", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{c}\n", - "y_1 \\\\\n", - "y_2 - c_1 y_1 \\\\\n", - "y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\\\\n", - "y_4 - f_1 y_1 - \\left( y_2 - c_1 y_1 \\right) e_2 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) c_3 \\\\\n", - "y_5 - \\left( y_2 - c_1 y_1 \\right) f_2 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) e_3 - \\left( y_4 - f_1 y_1 - \\left( y_2 - c_1 y_1 \\right) e_2 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) c_3 \\right) c_4 \\\\\n", - "y_6 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) f_3 - \\left( y_4 - f_1 y_1 - \\left( y_2 - c_1 y_1 \\right) e_2 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) c_3 \\right) e_4 - \\left( y_5 - \\left( y_2 - c_1 y_1 \\right) f_2 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) e_3 - \\left( y_4 - f_1 y_1 - \\left( y_2 - c_1 y_1 \\right) e_2 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) c_3 \\right) c_4 \\right) c_5 \\\\\n", - "y_7 - \\left( y_4 - f_1 y_1 - \\left( y_2 - c_1 y_1 \\right) e_2 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) c_3 \\right) f_4 - \\left( y_5 - \\left( y_2 - c_1 y_1 \\right) f_2 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) e_3 - \\left( y_4 - f_1 y_1 - \\left( y_2 - c_1 y_1 \\right) e_2 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) c_3 \\right) c_4 \\right) e_5 - \\left( y_6 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) f_3 - \\left( y_4 - f_1 y_1 - \\left( y_2 - c_1 y_1 \\right) e_2 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) c_3 \\right) e_4 - \\left( y_5 - \\left( y_2 - c_1 y_1 \\right) f_2 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) e_3 - \\left( y_4 - f_1 y_1 - \\left( y_2 - c_1 y_1 \\right) e_2 - \\left( y_3 - e_1 y_1 - \\left( y_2 - c_1 y_1 \\right) c_2 \\right) c_3 \\right) c_4 \\right) c_5 \\right) c_6 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "7-element Vector{Num}:\n", - " y₁\n", - " y₂ - c[1]*y₁\n", - " y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2]\n", - " y₄ - f[1]*y₁ - (y₂ - c[1]*y₁)*e[2] - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*c[3]\n", - " y₅ - (y₂ - c[1]*y₁)*f[2] - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*e[3] - (y₄ - f[1]*y₁ - (y₂ - c[1]*y₁)*e[2] - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*c[3])*c[4]\n", - " y₆ - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*f[3] - (y₄ - f[1]*y₁ - (y₂ - c[1]*y₁)*e[2] - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*c[3])*e[4] - (y₅ - (y₂ - c[1]*y₁)*f[2] - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*e[3] - (y₄ - f[1]*y₁ - (y₂ - c[1]*y₁)*e[2] - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*c[3])*c[4])*c[5]\n", - " y₇ - (y₄ - f[1]*y₁ - (y₂ - c[1]*y₁)*e[2] - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*c[3])*f[4] - (y₅ - (y₂ - c[1]*y₁)*f[2] - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*e[3] - (y₄ - f[1]*y₁ - (y₂ - c[1]*y₁)*e[2] - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*c[3])*c[4])*e[5] - (y₆ - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*f[3] - (y₄ - f[1]*y₁ - (y₂ - c[1]*y₁)*e[2] - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*c[3])*e[4] - (y₅ - (y₂ - c[1]*y₁)*f[2] - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*e[3] - (y₄ - f[1]*y₁ - (y₂ - c[1]*y₁)*e[2] - (y₃ - e[1]*y₁ - (y₂ - c[1]*y₁)*c[2])*c[3])*c[4])*c[5])*c[6]" - ] - }, - "execution_count": 17, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "L * θ\n", - "L \\ y" - ] - }, - { - "cell_type": "code", - "execution_count": 19, - "id": "1c25770d-29f2-483c-aa36-36b01fa50ef3", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{cc}\n", - "\\left( z_1 + c_1 z_2 + e_1 z_3 + f_1 z_4 \\right) d_1 & \\theta_1 \\\\\n", - "\\left( z_2 + c_2 z_3 + e_2 z_4 + f_2 z_5 \\right) d_2 & \\theta_2 \\\\\n", - "\\left( z_3 + c_3 z_4 + e_3 z_5 + f_3 z_6 \\right) d_3 & \\theta_3 \\\\\n", - "\\left( z_4 + c_4 z_5 + e_4 z_6 + f_4 z_7 \\right) d_4 & \\theta_4 \\\\\n", - "\\left( z_5 + c_5 z_6 + e_5 z_7 \\right) d_5 & \\theta_5 \\\\\n", - "\\left( z_6 + c_6 z_7 \\right) d_6 & \\theta_6 \\\\\n", - "d_7 z_7 & \\theta_7 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "7×2 Matrix{Num}:\n", - " (z₁ + c[1]*z₂ + e[1]*z₃ + f[1]*z₄)*d[1] θ₁\n", - " (z₂ + c[2]*z₃ + e[2]*z₄ + f[2]*z₅)*d[2] θ₂\n", - " (z₃ + c[3]*z₄ + e[3]*z₅ + f[3]*z₆)*d[3] θ₃\n", - " (z₄ + c[4]*z₅ + e[4]*z₆ + f[4]*z₇)*d[4] θ₄\n", - " (z₅ + c[5]*z₆ + e[5]*z₇)*d[5] θ₅\n", - " (z₆ + c[6]*z₇)*d[6] θ₆\n", - " d[7]*z₇ θ₇" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "display([D * L' * z θ])\n", - "# z' = (L' * z) \\ θ\n" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Julia 1.10.3", - "language": "julia", - "name": "julia-1.10" - }, - "language_info": { - "file_extension": ".jl", - "mimetype": "application/julia", - "name": "julia", - "version": "1.10.3" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/docs/formula/whittaker_LU_Eilers.ipynb b/docs/formula/whittaker_LU_Eilers.ipynb deleted file mode 100644 index 6b45e9e..0000000 --- a/docs/formula/whittaker_LU_Eilers.ipynb +++ /dev/null @@ -1,216 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 2, - "id": "8717e83f-8793-45a4-8cc0-228e60917acd", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "cofactor (generic function with 3 methods)" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "include(\"main_whit.jl\")" - ] - }, - { - "cell_type": "code", - "execution_count": 3, - "id": "0b3d0446-5bf2-494d-9a96-efa5b7ee07ad", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{cccccc}\n", - "w_1 + \\lambda & - 2 \\lambda & \\lambda & 0 & 0 & 0 \\\\\n", - " - 2 \\lambda & w_2 + 5 \\lambda & - 4 \\lambda & \\lambda & 0 & 0 \\\\\n", - "\\lambda & - 4 \\lambda & w_3 + 6 \\lambda & - 4 \\lambda & \\lambda & 0 \\\\\n", - "0 & \\lambda & - 4 \\lambda & w_4 + 6 \\lambda & - 4 \\lambda & \\lambda \\\\\n", - "0 & 0 & \\lambda & - 4 \\lambda & w_5 + 5 \\lambda & - 2 \\lambda \\\\\n", - "0 & 0 & 0 & \\lambda & - 2 \\lambda & w_6 + \\lambda \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "6×6 Matrix{Num}:\n", - " w₁ + λ -2λ λ 0.0 0.0 0.0\n", - " -2λ w₂ + 5λ -4λ λ 0.0 0.0\n", - " λ -4λ w₃ + 6λ -4λ λ 0.0\n", - " 0.0 λ -4λ w₄ + 6λ -4λ λ\n", - " 0.0 0.0 λ -4λ w₅ + 5λ -2λ\n", - " 0.0 0.0 0.0 λ -2λ w₆ + λ" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{cccccc}\n", - "1 & 0 & 0 & 0 & 0 & 0 \\\\\n", - "\\frac{ - 2 \\lambda}{w_1 + \\lambda} & 1 & 0 & 0 & 0 & 0 \\\\\n", - "\\frac{\\lambda}{w_1 + \\lambda} & \\frac{\\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} & 1 & 0 & 0 & 0 \\\\\n", - "0 & \\frac{\\lambda}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} & \\frac{\\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} - 4 \\lambda}{w_3 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right)^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda} & 1 & 0 & 0 \\\\\n", - "0 & 0 & \\frac{\\lambda}{w_3 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right)^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda} & \\frac{\\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{w_3 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right)^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda} - 4 \\lambda}{w_4 + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{w_3 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right)^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + 6 \\lambda} & 1 & 0 \\\\\n", - "0 & 0 & 0 & \\frac{\\lambda}{w_4 + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{w_3 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right)^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + 6 \\lambda} & \\frac{\\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{w_3 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right)^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{w_4 + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{w_3 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right)^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + 6 \\lambda} - 2 \\lambda}{w_5 + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{w_3 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right)^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{w_4 + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{w_3 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right)^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_3 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_1 + \\lambda} - 4 \\lambda \\right)^{2}}{w_2 + \\frac{ - 4 \\lambda^{2}}{w_1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda} + 5 \\lambda} & 1 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "6×6 Matrix{Num}:\n", - " 1 … 0\n", - " (-2λ) / (w₁ + λ) 0\n", - " λ / (w₁ + λ) 0\n", - " 0.0 0\n", - " 0.0 0\n", - " 0.0 … 1" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "n = 6\n", - "w = variables(:w, 1:n)\n", - "D = ddmat(1:n, 2)\n", - "# M = diagm(ones(n))\n", - "M = diag_m(w[1:n]) + λ * D' * D\n", - "display(M)\n", - "# M = diagm(W) + λ * D' * D\n", - "\n", - "U = deepcopy(M)\n", - "L = typeof(U)(diagm(ones(n)))\n", - "\n", - "r = lu(M)\n", - "## lu采用for循环的形式，\n", - "r.L" - ] - }, - { - "cell_type": "code", - "execution_count": 7, - "id": "51947888-a1e3-4ade-9df4-dc4caeb2add2", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{cccccc}\n", - "w_{1} + \\lambda & - 2 \\lambda & \\lambda & 0 & 0 & 0 \\\\\n", - "0 & w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda & \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda & \\lambda & 0 & 0 \\\\\n", - "0 & 0 & w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda & \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right) \\lambda}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} - 4 \\lambda & \\lambda & 0 \\\\\n", - "0 & 0 & 0 & w_{4} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right) \\lambda}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda & \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right) \\lambda}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} - 4 \\lambda & \\lambda \\\\\n", - "0 & 0 & 0 & 0 & w_{5} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right) \\lambda}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{w_{4} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right) \\lambda}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} + 5 \\lambda & \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right) \\lambda}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{w_{4} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right) \\lambda}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} - 2 \\lambda \\\\\n", - "0 & 0 & 0 & 0 & 0 & w_{6} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right) \\lambda}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{w_{4} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right) \\lambda}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} - 2 \\lambda \\right)^{2}}{w_{5} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right) \\lambda}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{w_{4} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right) \\lambda}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{w_{4} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right) \\lambda}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{w_{3} + \\frac{ - \\lambda^{2}}{w_{1} + \\lambda} + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{w_{1} + \\lambda} - 4 \\lambda \\right)^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} + \\frac{ - \\lambda^{2}}{w_{2} + \\frac{ - 4 \\lambda^{2}}{w_{1} + \\lambda} + 5 \\lambda} + 6 \\lambda} + \\lambda \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "6×6 Matrix{Num}:\n", - " w_1 + λ … 0.0\n", - " 0 0.0\n", - " 0 0.0\n", - " 0 λ\n", - " 0 (-((-((-((2(λ^2)) / (w_1 + λ) - 4λ)*λ) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) - 4λ)*λ) / (w_3 + (-(λ^2)) / (w_1 + λ) + (-(((2(λ^2)) / (w_1 + λ) - 4λ)^2)) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) + 6λ) - 4λ)*λ) / (w_4 + (-(((-((2(λ^2)) / (w_1 + λ) - 4λ)*λ) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) - 4λ)^2)) / (w_3 + (-(λ^2)) / (w_1 + λ) + (-(((2(λ^2)) / (w_1 + λ) - 4λ)^2)) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) + 6λ) + (-(λ^2)) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) + 6λ) - 2λ\n", - " 0 … w_6 + (-(((-((-((-((2(λ^2)) / (w_1 + λ) - 4λ)*λ) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) - 4λ)*λ) / (w_3 + (-(λ^2)) / (w_1 + λ) + (-(((2(λ^2)) / (w_1 + λ) - 4λ)^2)) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) + 6λ) - 4λ)*λ) / (w_4 + (-(((-((2(λ^2)) / (w_1 + λ) - 4λ)*λ) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) - 4λ)^2)) / (w_3 + (-(λ^2)) / (w_1 + λ) + (-(((2(λ^2)) / (w_1 + λ) - 4λ)^2)) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) + 6λ) + (-(λ^2)) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) + 6λ) - 2λ)^2)) / (w_5 + (-(((-((-((2(λ^2)) / (w_1 + λ) - 4λ)*λ) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) - 4λ)*λ) / (w_3 + (-(λ^2)) / (w_1 + λ) + (-(((2(λ^2)) / (w_1 + λ) - 4λ)^2)) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) + 6λ) - 4λ)^2)) / (w_4 + (-(((-((2(λ^2)) / (w_1 + λ) - 4λ)*λ) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) - 4λ)^2)) / (w_3 + (-(λ^2)) / (w_1 + λ) + (-(((2(λ^2)) / (w_1 + λ) - 4λ)^2)) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) + 6λ) + (-(λ^2)) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) + 6λ) + (-(λ^2)) / (w_3 + (-(λ^2)) / (w_1 + λ) + (-(((2(λ^2)) / (w_1 + λ) - 4λ)^2)) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) + 6λ) + 5λ) + (-(λ^2)) / (w_4 + (-(((-((2(λ^2)) / (w_1 + λ) - 4λ)*λ) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) - 4λ)^2)) / (w_3 + (-(λ^2)) / (w_1 + λ) + (-(((2(λ^2)) / (w_1 + λ) - 4λ)^2)) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) + 6λ) + (-(λ^2)) / (w_2 + (-4(λ^2)) / (w_1 + λ) + 5λ) + 6λ) + λ" - ] - }, - "execution_count": 7, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "r.U" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "c80fd687-a0d6-4f73-8e6d-cb4158065161", - "metadata": {}, - "outputs": [], - "source": [] - }, - { - "cell_type": "code", - "execution_count": 11, - "id": "709c66c6-8ba1-40d5-8d5b-7021c64e0cb7", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{cccccccc}\n", - "1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", - "\\frac{ - 2 \\lambda}{1 + \\lambda} & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", - "\\frac{\\lambda}{1 + \\lambda} & \\frac{\\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n", - "0 & \\frac{\\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} & \\frac{\\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} & 1 & 0 & 0 & 0 & 0 \\\\\n", - "0 & 0 & \\frac{\\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} & \\frac{\\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} & 1 & 0 & 0 & 0 \\\\\n", - "0 & 0 & 0 & \\frac{\\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} & \\frac{\\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} - 4 \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} & 1 & 0 & 0 \\\\\n", - "0 & 0 & 0 & 0 & \\frac{\\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} & \\frac{\\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} - 4 \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} & 1 & 0 \\\\\n", - "0 & 0 & 0 & 0 & 0 & \\frac{\\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} & \\frac{\\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} - 2 \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\lambda^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right) \\lambda}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - \\left( \\frac{2 \\lambda^{2}}{1 + \\lambda} - 4 \\lambda \\right)^{2}}{1 + \\frac{ - 4 \\lambda^{2}}{1 + \\lambda} + 5 \\lambda} + \\frac{ - \\lambda^{2}}{1 + \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} + 6 \\lambda} + 5 \\lambda} & 1 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "8×8 Matrix{Num}:\n", - " 1 … 0\n", - " (-2λ) / (1.0 + λ) 0\n", - " λ / (1.0 + λ) 0\n", - " 0.0 0\n", - " 0.0 0\n", - " 0.0 … 0\n", - " 0.0 0\n", - " 0.0 1" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Julia 1.10.3", - "language": "julia", - "name": "julia-1.10" - }, - "language_info": { - "file_extension": ".jl", - "mimetype": "application/julia", - "name": "julia", - "version": "1.10.3" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/docs/formula/whittaker_LU_Eilers_d3.ipynb b/docs/formula/whittaker_LU_Eilers_d3.ipynb deleted file mode 100644 index 6d16c9b..0000000 --- a/docs/formula/whittaker_LU_Eilers_d3.ipynb +++ /dev/null @@ -1,278 +0,0 @@ -{ - "cells": [ - { - "cell_type": "code", - "execution_count": 2, - "id": "8717e83f-8793-45a4-8cc0-228e60917acd", - "metadata": {}, - "outputs": [ - { - "data": { - "text/plain": [ - "cofactor (generic function with 3 methods)" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "# 使用LDL'分解，求解Whittaker smoother (d=3)\n", - "# 根据该代码，归纳总结出whit3的向量公式\n", - "# 采用a1, a2，混合代入，或许能得到更简洁的公式\n", - "# Dongdong Kong, CUG, 2024-05-06\n", - "include(\"main_whit.jl\")" - ] - }, - { - "cell_type": "code", - "execution_count": 4, - "id": "0b3d0446-5bf2-494d-9a96-efa5b7ee07ad", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{ccccccc}\n", - "w_1 + \\lambda & - 3 \\lambda & 3 \\lambda & - \\lambda & 0 & 0 & 0 \\\\\n", - " - 3 \\lambda & w_2 + 10 \\lambda & - 12 \\lambda & 6 \\lambda & - \\lambda & 0 & 0 \\\\\n", - "3 \\lambda & - 12 \\lambda & w_3 + 19 \\lambda & - 15 \\lambda & 6 \\lambda & - \\lambda & 0 \\\\\n", - " - \\lambda & 6 \\lambda & - 15 \\lambda & w_4 + 20 \\lambda & - 15 \\lambda & 6 \\lambda & - \\lambda \\\\\n", - "0 & - \\lambda & 6 \\lambda & - 15 \\lambda & w_5 + 19 \\lambda & - 12 \\lambda & 3 \\lambda \\\\\n", - "0 & 0 & - \\lambda & 6 \\lambda & - 12 \\lambda & w_6 + 10 \\lambda & - 3 \\lambda \\\\\n", - "0 & 0 & 0 & - \\lambda & 3 \\lambda & - 3 \\lambda & w_7 + \\lambda \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "7×7 Matrix{Num}:\n", - " w₁ + λ -3λ 3λ -λ 0.0 0.0 0.0\n", - " -3λ w₂ + 10λ -12λ 6λ -λ 0.0 0.0\n", - " 3λ -12λ w₃ + 19λ -15λ 6λ -λ 0.0\n", - " -λ 6λ -15λ w₄ + 20λ -15λ 6λ -λ\n", - " 0.0 -λ 6λ -15λ w₅ + 19λ -12λ 3λ\n", - " 0.0 0.0 -λ 6λ -12λ w₆ + 10λ -3λ\n", - " 0.0 0.0 0.0 -λ 3λ -3λ w₇ + λ" - ] - }, - "metadata": {}, - "output_type": "display_data" - }, - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{ccccccc}\n", - "1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", - "\\frac{ - 3 \\lambda}{w_1 + \\lambda} & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n", - "\\frac{3 \\lambda}{w_1 + \\lambda} & \\frac{\\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} & 1 & 0 & 0 & 0 & 0 \\\\\n", - "\\frac{ - \\lambda}{w_1 + \\lambda} & \\frac{\\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} & \\frac{\\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} & 1 & 0 & 0 & 0 \\\\\n", - "0 & \\frac{ - \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} & \\frac{\\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} & \\frac{\\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} & 1 & 0 & 0 \\\\\n", - "0 & 0 & \\frac{ - \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} & \\frac{\\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + 6 \\lambda}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} & \\frac{\\frac{\\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + 6 \\lambda \\right)}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} - 12 \\lambda}{w_5 + \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + \\frac{ - \\lambda^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} & 1 & 0 \\\\\n", - "0 & 0 & 0 & \\frac{ - \\lambda}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} & \\frac{\\frac{\\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right) \\lambda}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + 3 \\lambda}{w_5 + \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + \\frac{ - \\lambda^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} & \\frac{\\frac{ - \\left( \\frac{\\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + 6 \\lambda \\right)}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} - 12 \\lambda \\right) \\left( \\frac{\\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right) \\lambda}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + 3 \\lambda \\right)}{w_5 + \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + \\frac{ - \\lambda^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{\\left( \\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + 6 \\lambda \\right) \\lambda}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} - 3 \\lambda}{w_6 + \\frac{ - \\left( \\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + 6 \\lambda \\right)^{2}}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + \\frac{ - \\lambda^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\left( \\frac{\\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + 6 \\lambda \\right)}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} - 12 \\lambda \\right)^{2}}{w_5 + \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_4 + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} - 15 \\lambda \\right)^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right)^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + \\frac{ - \\lambda^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + 10 \\lambda} + 19 \\lambda} + 10 \\lambda} & 1 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "7×7 Matrix{Num}:\n", - " 1 … 0\n", - " (-3λ) / (w₁ + λ) 0\n", - " (3λ) / (w₁ + λ) 0\n", - " (-λ) / (w₁ + λ) 0\n", - " 0.0 0\n", - " 0.0 … 0\n", - " 0.0 1" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "n = 7\n", - "\n", - "w = variables(:w, 1:n)\n", - "D = ddmat(1:n, 3)\n", - "\n", - "# M = diagm(ones(n))\n", - "M = diag_m(w[1:n]) + λ * D' * D\n", - "display(M)\n", - "# M = diagm(W) + λ * D' * D\n", - "# inv(M)\n", - "# U = deepcopy(M)\n", - "# L = typeof(U)(diagm(ones(n)));\n", - "\n", - "r = lu(M)\n", - "r.L\n", - "# diag(r.U)[1:n]" - ] - }, - { - "cell_type": "markdown", - "id": "8af76dfd-c7ed-440c-8ce8-982289ebfa50", - "metadata": {}, - "source": [ - "## 2. 一种更简明的公式推导方法" - ] - }, - { - "cell_type": "code", - "execution_count": 57, - "id": "51947888-a1e3-4ade-9df4-dc4caeb2add2", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{ccccccc}\n", - "1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", - "\\frac{ - 3 \\lambda}{d_1} & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n", - "\\frac{3 \\lambda}{w_1 + \\lambda} & \\frac{\\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda}{d_2} & 1 & 0 & 0 & 0 & 0 \\\\\n", - "\\frac{ - \\lambda}{w_1 + \\lambda} & \\frac{\\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} & \\frac{\\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda}{d_3} & 1 & 0 & 0 & 0 \\\\\n", - "0 & \\frac{ - \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} & \\frac{\\frac{\\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 6 \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} & \\frac{\\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda}{d_4} & 1 & 0 & 0 \\\\\n", - "0 & 0 & \\frac{ - \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} & \\frac{\\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + 6 \\lambda}{w_4 + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{d_3} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} & \\frac{\\frac{ - \\left( \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{d_2} + 6 \\lambda \\right)}{d_3} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + 6 \\lambda \\right)}{w_4 + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{d_3} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + \\frac{\\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{d_2} + 6 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} - 12 \\lambda}{d_5} & 1 & 0 \\\\\n", - "0 & 0 & 0 & \\frac{ - \\lambda}{w_4 + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{d_3} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} & \\frac{\\frac{\\left( \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{d_2} + 6 \\lambda \\right)}{d_3} - 15 \\lambda \\right) \\lambda}{w_4 + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{d_3} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + 3 \\lambda}{w_5 + \\frac{ - \\left( \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{d_2} + 6 \\lambda \\right)}{d_3} - 15 \\lambda \\right) \\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right)}{d_4} + \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{d_2} + 6 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 19 \\lambda} & \\frac{\\frac{\\left( \\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\lambda}{d_3} + 6 \\lambda \\right) \\lambda}{w_4 + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{d_3} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + \\frac{ - \\left( \\frac{\\left( \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{d_2} + 6 \\lambda \\right)}{d_3} - 15 \\lambda \\right) \\lambda}{w_4 + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{d_3} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + 3 \\lambda \\right) \\left( \\frac{\\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 6 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\lambda}{d_3} + 6 \\lambda \\right)}{d_4} - 12 \\lambda \\right)}{w_5 + \\frac{ - \\left( \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{d_2} + 6 \\lambda \\right)}{d_3} - 15 \\lambda \\right) \\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right)}{d_4} + \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{d_2} + 6 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 19 \\lambda} - 3 \\lambda}{d_6} & 1 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "7×7 Matrix{Num}:\n", - " 1.0 … 0.0\n", - " (-3λ) / d₁ 0.0\n", - " (3λ) / (w₁ + λ) 0.0\n", - " (-λ) / (w₁ + λ) 0.0\n", - " 0.0 0.0\n", - " 0.0 … 0.0\n", - " 0.0 1.0" - ] - }, - "metadata": {}, - "output_type": "display_data" - } - ], - "source": [ - "d = variables(:d, 1:n)\n", - "c = variables(:c, 1:n)\n", - "e = variables(:e, 1:n)\n", - "f = variables(:f, 1:n)\n", - "\n", - "U = deepcopy(M)\n", - "L = typeof(U)(diagm(ones(n)))\n", - "\n", - "## Elier采用的是LU分解 \n", - "# 仅运行三次，猜测公式的形式\n", - "for i = 1:n-1\n", - " r1 = U[i, :]\n", - " j = i+1\n", - " c[i] = U[j, i] / d[i]\n", - " L[j, i] = c[i] # ci, ei\n", - " U[j, :] .= U[j, :] .- (c[i] * r1)\n", - " d[i] = U[i, i]\n", - " \n", - " j = i+2\n", - " j > n && continue\n", - " e[i] = U[j, i] / U[i, i]\n", - " L[j, i] = e[i] # ci, ei\n", - " U[j, :] .= U[j, :] .- (e[i] * r1)\n", - "\n", - " j = i+3\n", - " j > n && continue\n", - " f[i] = U[j, i] / U[i, i]\n", - " L[j, i] = f[i] # ci, ei\n", - " U[j, :] .= U[j, :] .- (f[i] * r1)\n", - " # for j = i+1:min(i+2, n)\n", - " # f = U[j, i] / U[i, i]\n", - " # L[j, i] = f # ci, ei\n", - " # U[j, :] .= U[j, :] .- (f * r1)\n", - " # end\n", - "end\n", - "\n", - "## Elier采用的是LU分解 \n", - "display(L)\n", - "# display(U)\n" - ] - }, - { - "cell_type": "code", - "execution_count": 58, - "id": "663aaf1a-5ccb-495f-95d6-35228826cbc4", - "metadata": {}, - "outputs": [ - { - "data": { - "text/latex": [ - "$$ \\begin{equation}\n", - "\\left[\n", - "\\begin{array}{c}\n", - "w_1 + \\lambda \\\\\n", - "w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda \\\\\n", - "w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda \\\\\n", - "w_4 + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{d_3} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda \\\\\n", - "w_5 + \\frac{ - \\left( \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{d_2} + 6 \\lambda \\right)}{d_3} - 15 \\lambda \\right) \\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right)}{d_4} + \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{d_2} + 6 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + \\frac{ - \\lambda^{2}}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 19 \\lambda \\\\\n", - "w_6 + \\frac{ - \\left( \\frac{\\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 6 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + \\frac{ - \\left( \\frac{ - \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + 6 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\lambda}{d_3} + 6 \\lambda \\right)}{d_4} - 12 \\lambda \\right) \\left( \\frac{ - \\left( \\frac{\\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\lambda}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{d_2} + 6 \\lambda \\right)}{d_3} - 15 \\lambda \\right) \\left( \\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + 6 \\lambda \\right)}{w_4 + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{d_3} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + \\frac{\\left( \\frac{\\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\lambda}{d_2} + 6 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} - 12 \\lambda \\right)}{d_5} + \\frac{ - \\lambda^{2}}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + \\frac{ - \\left( \\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\lambda}{d_3} + 6 \\lambda \\right) \\left( \\frac{\\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right) \\lambda}{w_3 + \\frac{ - 9 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{d_2} + 19 \\lambda} + 6 \\lambda \\right)}{w_4 + \\frac{ - \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{9 \\lambda^{2}}{d_1} - 12 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} - 15 \\lambda \\right) \\left( \\frac{3 \\lambda^{2}}{w_1 + \\lambda} + \\frac{ - \\left( \\frac{9 \\lambda^{2}}{w_1 + \\lambda} - 12 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{d_2} - 15 \\lambda \\right)}{d_3} + \\frac{ - \\left( \\frac{ - 3 \\lambda^{2}}{w_1 + \\lambda} + 6 \\lambda \\right) \\left( \\frac{ - 3 \\lambda^{2}}{d_1} + 6 \\lambda \\right)}{w_2 + \\frac{ - 9 \\lambda^{2}}{d_1} + 10 \\lambda} + \\frac{ - \\lambda^{2}}{w_1 + \\lambda} + 20 \\lambda} + 10 \\lambda \\\\\n", - "d_7 \\\\\n", - "\\end{array}\n", - "\\right]\n", - "\\end{equation}\n", - " $$" - ], - "text/plain": [ - "7-element Vector{Num}:\n", - " w₁ + λ\n", - " w₂ + (-9(λ^2)) / d₁ + 10λ\n", - " w₃ + (-9(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((9(λ^2)) / d₁ - 12λ)) / d₂ + 19λ\n", - " w₄ + (-((3(λ^2)) / (w₁ + λ) + (-((-3(λ^2)) / (w₁ + λ) + 6λ)*((9(λ^2)) / d₁ - 12λ)) / (w₂ + (-9(λ^2)) / d₁ + 10λ) - 15λ)*((3(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((-3(λ^2)) / d₁ + 6λ)) / d₂ - 15λ)) / d₃ + (-((-3(λ^2)) / (w₁ + λ) + 6λ)*((-3(λ^2)) / d₁ + 6λ)) / (w₂ + (-9(λ^2)) / d₁ + 10λ) + (-(λ^2)) / (w₁ + λ) + 20λ\n", - " w₅ + (-((((-3(λ^2)) / (w₁ + λ) + 6λ)*λ) / (w₂ + (-9(λ^2)) / d₁ + 10λ) + (-((3(λ^2)) / (w₁ + λ) + (-((-3(λ^2)) / (w₁ + λ) + 6λ)*((9(λ^2)) / d₁ - 12λ)) / (w₂ + (-9(λ^2)) / d₁ + 10λ) - 15λ)*((((9(λ^2)) / (w₁ + λ) - 12λ)*λ) / d₂ + 6λ)) / d₃ - 15λ)*((-((((9(λ^2)) / d₁ - 12λ)*λ) / (w₂ + (-9(λ^2)) / d₁ + 10λ) + 6λ)*((3(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((-3(λ^2)) / d₁ + 6λ)) / d₂ - 15λ)) / (w₃ + (-9(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((9(λ^2)) / d₁ - 12λ)) / d₂ + 19λ) + (((-3(λ^2)) / d₁ + 6λ)*λ) / (w₂ + (-9(λ^2)) / d₁ + 10λ) - 15λ)) / d₄ + (-((((9(λ^2)) / d₁ - 12λ)*λ) / (w₂ + (-9(λ^2)) / d₁ + 10λ) + 6λ)*((((9(λ^2)) / (w₁ + λ) - 12λ)*λ) / d₂ + 6λ)) / (w₃ + (-9(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((9(λ^2)) / d₁ - 12λ)) / d₂ + 19λ) + (-(λ^2)) / (w₂ + (-9(λ^2)) / d₁ + 10λ) + 19λ\n", - " w₆ + (-((((((9(λ^2)) / d₁ - 12λ)*λ) / (w₂ + (-9(λ^2)) / d₁ + 10λ) + 6λ)*λ) / (w₃ + (-9(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((9(λ^2)) / d₁ - 12λ)) / d₂ + 19λ) + (-((-((((9(λ^2)) / d₁ - 12λ)*λ) / (w₂ + (-9(λ^2)) / d₁ + 10λ) + 6λ)*((3(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((-3(λ^2)) / d₁ + 6λ)) / d₂ - 15λ)) / (w₃ + (-9(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((9(λ^2)) / d₁ - 12λ)) / d₂ + 19λ) + (((-3(λ^2)) / d₁ + 6λ)*λ) / (w₂ + (-9(λ^2)) / d₁ + 10λ) - 15λ)*((((3(λ^2)) / (w₁ + λ) + (-((-3(λ^2)) / (w₁ + λ) + 6λ)*((9(λ^2)) / d₁ - 12λ)) / (w₂ + (-9(λ^2)) / d₁ + 10λ) - 15λ)*λ) / d₃ + 6λ)) / d₄ - 12λ)*((-((((-3(λ^2)) / (w₁ + λ) + 6λ)*λ) / (w₂ + (-9(λ^2)) / d₁ + 10λ) + (-((3(λ^2)) / (w₁ + λ) + (-((-3(λ^2)) / (w₁ + λ) + 6λ)*((9(λ^2)) / d₁ - 12λ)) / (w₂ + (-9(λ^2)) / d₁ + 10λ) - 15λ)*((((9(λ^2)) / (w₁ + λ) - 12λ)*λ) / d₂ + 6λ)) / d₃ - 15λ)*((((3(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((-3(λ^2)) / d₁ + 6λ)) / d₂ - 15λ)*λ) / (w₃ + (-9(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((9(λ^2)) / d₁ - 12λ)) / d₂ + 19λ) + 6λ)) / (w₄ + (-((3(λ^2)) / (w₁ + λ) + (-((-3(λ^2)) / (w₁ + λ) + 6λ)*((9(λ^2)) / d₁ - 12λ)) / (w₂ + (-9(λ^2)) / d₁ + 10λ) - 15λ)*((3(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((-3(λ^2)) / d₁ + 6λ)) / d₂ - 15λ)) / d₃ + (-((-3(λ^2)) / (w₁ + λ) + 6λ)*((-3(λ^2)) / d₁ + 6λ)) / (w₂ + (-9(λ^2)) / d₁ + 10λ) + (-(λ^2)) / (w₁ + λ) + 20λ) + (((((9(λ^2)) / (w₁ + λ) - 12λ)*λ) / d₂ + 6λ)*λ) / (w₃ + (-9(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((9(λ^2)) / d₁ - 12λ)) / d₂ + 19λ) - 12λ)) / d₅ + (-(λ^2)) / (w₃ + (-9(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((9(λ^2)) / d₁ - 12λ)) / d₂ + 19λ) + (-((((3(λ^2)) / (w₁ + λ) + (-((-3(λ^2)) / (w₁ + λ) + 6λ)*((9(λ^2)) / d₁ - 12λ)) / (w₂ + (-9(λ^2)) / d₁ + 10λ) - 15λ)*λ) / d₃ + 6λ)*((((3(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((-3(λ^2)) / d₁ + 6λ)) / d₂ - 15λ)*λ) / (w₃ + (-9(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((9(λ^2)) / d₁ - 12λ)) / d₂ + 19λ) + 6λ)) / (w₄ + (-((3(λ^2)) / (w₁ + λ) + (-((-3(λ^2)) / (w₁ + λ) + 6λ)*((9(λ^2)) / d₁ - 12λ)) / (w₂ + (-9(λ^2)) / d₁ + 10λ) - 15λ)*((3(λ^2)) / (w₁ + λ) + (-((9(λ^2)) / (w₁ + λ) - 12λ)*((-3(λ^2)) / d₁ + 6λ)) / d₂ - 15λ)) / d₃ + (-((-3(λ^2)) / (w₁ + λ) + 6λ)*((-3(λ^2)) / d₁ + 6λ)) / (w₂ + (-9(λ^2)) / d₁ + 10λ) + (-(λ^2)) / (w₁ + λ) + 20λ) + 10λ\n", - " d₇" - ] - }, - "execution_count": 58, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "d" - ] - }, - { - "cell_type": "code", - "execution_count": null, - "id": "c80fd687-a0d6-4f73-8e6d-cb4158065161", - "metadata": {}, - "outputs": [], - "source": [ - "# U = deepcopy(M)\n", - "# L,U = LU_decompose(U)\n", - "# L" - ] - } - ], - "metadata": { - "kernelspec": { - "display_name": "Julia 1.10.3", - "language": "julia", - "name": "julia-1.10" - }, - "language_info": { - "file_extension": ".jl", - "mimetype": "application/julia", - "name": "julia", - "version": "1.10.3" - } - }, - "nbformat": 4, - "nbformat_minor": 5 -} diff --git a/docs/references/Eilers_2003_Analytical Chemistry_A Perfect Smoother.pdf b/docs/references/Eilers_2003_Analytical Chemistry_A Perfect Smoother.pdf deleted file mode 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E@kQaXunhWcJu67{_z%KRw);CxdwR7g@Zln$9`5Ff=K7CTbBpqQO{FO-sc z2kJ@bcZjGqFv%2Upp;Cpev!UUrfpCnMU-Yd)f|6_fTML1d#zDzgxXq42JpwJHj?SF zf-DWBWG{2|3&qaD3GK2Nrc_zI1lTBw0(?EzYRmT*;dImmS#JWwW9jdz63PNCj`wLJLA1r z*>9E=Pmg)nMu@UiWO9w|wq187sFV4ZBNPg}PX0Y0S!9%UIWTR%-s+iUtR?-I!|hHK IlK*o2A3u|V;s5{u diff --git a/src/Smooth/Whittaker/WHIT.jl b/src/Smooth/Whittaker/WHIT.jl index 7f0a881..57f77f6 100644 --- a/src/Smooth/Whittaker/WHIT.jl +++ b/src/Smooth/Whittaker/WHIT.jl @@ -1,22 +1,29 @@ using SparseArrays -speye(n) = SparseArrays.sparse(I, n, n) +function Base.diff(x::AbstractMatrix, d::Integer=1) + D = x[2:end, :] .- x[1:end-1, :] + d >= 2 ? diffn(D, d - 1) : D +end function Base.diff(x::SparseMatrixCSC, d::Integer=1) D = x[2:end, :] .- x[1:end-1, :] d >= 2 ? diff(D, d - 1) : D end +# ddmat(x::AbstractVector, d::Integer=2) = diff(spdiagm(x), d) + +# 相比于原版进行了微调 function ddmat(x::AbstractVector, d::Integer=2) n = length(x) if d == 0 - return speye(n) + return sparse(I, n, n) else - return diff(ddmat(x, d - 1)) + dx = ( x[d+1:n] - x[1:(n-d)] ) ./ d + V = spdiagm( 1 ./ dx) + return V * diff(ddmat(x, d - 1)) end end - function WHIT(y::AbstractVector, w::AbstractVector; kw...) WHIT(y, w, 1:length(y); kw...) end diff --git a/src/Smooth/Whittaker/main_whittaker.jl b/src/Smooth/Whittaker/main_whittaker.jl index b61e8ac..93864d8 100644 --- a/src/Smooth/Whittaker/main_whittaker.jl +++ b/src/Smooth/Whittaker/main_whittaker.jl @@ -4,12 +4,15 @@ using Statistics # include("smooth_whittaker/whittaker2.jl") # include("smooth_whittaker/lambda_cv.jl") # include("smooth_whittaker/smooth_whit_v01.jl") + include("lambda_init.jl") include("lambda_vcurve.jl") include("whit2.jl") include("whit2_Frasso2015.jl") include("whit3.jl") +include("whit3_x.jl") + include("WHIT.jl") # include("smooth_whit.jl") # include("smooth_whit_GEE.jl") diff --git a/test/test-smooth_whit.jl b/test/test-smooth_whit.jl index 991ef85..7f78b20 100644 --- a/test/test-smooth_whit.jl +++ b/test/test-smooth_whit.jl @@ -1,5 +1,6 @@ using Test using UnPack +using VegCurveFit @testset "whittaker smoother" begin y = [5.0, 8, 9, 10, 12, 10, 15, 10, 9, 19, 19, 17, 13, 14, 18, 19, 18, 12, 18, @@ -18,7 +19,7 @@ using UnPack lamb_cv = lambda_cv(y, w, is_plot=true) lamb_vcurve = lambda_vcurve(y, w, is_plot=true) - + z1, cve_cv = whit2(y, w, lambda=lamb_cv) z2, cve_vcurve = whit2(y, w, lambda=lamb_vcurve) @test cve_cv < cve @@ -31,20 +32,23 @@ end d = deserialize("../data/Tumbarumba_EVI2") @unpack y, t, w = d - z, cve1 = whit2_cv(y, w, lambda=2) - z, cve2 = whit2(y, w, lambda=2) + z1, cve1 = whit2_cv(y, w, lambda=2) + z2, cve2 = whit2(y, w, lambda=2) @test cve1 ≈ cve2 @test_nowarn r = smooth_whit(y, w) # whit3 + x = 1:length(y) z1, cve1 = whit3(y, w; lambda=2) - z2, cve2 = WHIT(y, w; lambda=2, d=3) + z2, cve2 = WHIT(y, w, x; lambda=2, d=3) + @test cve1 ≈ cve2 @test maximum(z1 - z2) <= 1e-10 # whit2 z1, cve1 = whit2(y, w; lambda=2) - z2, cve2 = WHIT(y, w; lambda=2, d=2) + z2, cve2 = WHIT(y, w, x; lambda=2, d=2) + @test cve1 ≈ cve2 @test maximum(z1 - z2) <= 1e-10 end