once more unto the decomposition breach, dear friends!

NEWS.md

@@ -227,6 +227,21 @@ This section lists changes that do not have deprecation warnings.
   * `readuntil` now does *not* include the delimiter in its result, matching the
     behavior of `readline`. Pass `keep=true` to get the old behavior ([#25633]).
 
+  * `lu` methods now return decomposition objects such as `LU` rather than
+    tuples of arrays or tuples of numbers ([#26997], [#27159], [#27212]).
+
+  * `schur` methods now return decomposition objects such as `Schur` and
+    `GeneralizedSchur` rather than tuples of arrays ([#26997], [#27159], [#27212]).
+
+  * `lq` methods now return decomposition objects such as `LQ`
+    rather than tuples of arrays ([#26997], [#27159], [#27212]).
+
+  * `qr` methods now return decomposition objects such as `QR`, `QRPivoted`,
+    and `QRCompactWY` rather than tuples of arrays ([#26997], [#27159], [#27212]).
+
+  * `svd` methods now return decomposition objects such as `SVD` and
+    `GeneralizedSVD` rather than tuples of arrays or tuples of numbers ([#26997], [#27159], [#27212]).
+
   * `countlines` now always counts the last non-empty line even if it does not
     end with EOL, matching the behavior of `eachline` and `readlines` ([#25845]).
 
@@ -688,6 +703,31 @@ Deprecated or removed
   * The keyword `immutable` is fully deprecated to `struct`, and
     `type` is fully deprecated to `mutable struct` ([#19157], [#20418]).
 
+  * `lufact`, `schurfact`, `lqfact`, `qrfact`, `ldltfact`, `svdfact`,
+    `bkfact`, `hessfact`, `eigfact`, and `cholfact` have respectively been
+    deprecated to `lu`, `schur`, `lq`, `qr`, `ldlt`, `svd`, `bunchkaufman`,
+    `hessenberg`, `eigen`, and `cholesky` ([#26997], [#27159], [#27212]).
+
+  * `lufact!`, `schurfact!`, `lqfact!`, `qrfact!`, `ldltfact!`, `svdfact!`,
+    `bkfact!`, `hessfact!`, and `eigfact!` have respectively been deprecated to
+    `lu!`, `schur!`, `lq!`, `qr!`, `ldlt!`, `svd!`, `bunchkaufman!`,
+    `hessenberg!`, and `eigen!` ([#26997], [#27159], [#27212]).
+
+  * `eig(A[, args...])` has been deprecated in favor of `eigen(A[, args...])`.
+    Whereas the former returns a tuple of arrays, the latter returns an `Eigen` object.
+    So for a direct replacement, use `(eigen(A[, args...])...,)`. But going forward,
+    consider using the direct result of `eigen(A[, args...])` instead, either
+    destructured into its components (`vals, vecs = eigen(A[, args...])`) or
+    as an `Eigen` object (`X = eigen(A[, args...])`) ([#26997], [#27159], [#27212]).
+
+  * `eig(A::AbstractMatrix, B::AbstractMatrix)` and `eig(A::Number, B::Number)`
+    have been deprecated in favor of `eigen(A, B)`. Whereas the former each return
+    a tuple of arrays, the latter returns a `GeneralizedEigen` object. So for a direct
+    replacement, use `(eigen(A, B)...,)`. But going forward, consider using the
+    direct result of `eigen(A, B)` instead, either destructured into its components
+    (`vals, vecs = eigen(A, B)`), or as a `GeneralizedEigen` object
+    (`X = eigen(A, B)`) ([#26997], [#27159], [#27212]).
+
   * Indexing into multidimensional arrays with more than one index but fewer indices than there are
     dimensions is no longer permitted when those trailing dimensions have lengths greater than 1.
     Instead, reshape the array or add trailing indices so the dimensionality and number of indices

doc/src/manual/arrays.md

@@ -746,37 +746,37 @@ creating any temporaries, and by calling the appropriate LAPACK function with th
 dimension size and stride parameters.
 
 ```julia-repl
-julia> a = rand(10,10)
+julia> a = rand(10, 10)
 10×10 Array{Float64,2}:
- 0.561255   0.226678   0.203391  0.308912   …  0.750307  0.235023   0.217964
- 0.718915   0.537192   0.556946  0.996234      0.666232  0.509423   0.660788
- 0.493501   0.0565622  0.118392  0.493498      0.262048  0.940693   0.252965
- 0.0470779  0.736979   0.264822  0.228787      0.161441  0.897023   0.567641
- 0.343935   0.32327    0.795673  0.452242      0.468819  0.628507   0.511528
- 0.935597   0.991511   0.571297  0.74485    …  0.84589   0.178834   0.284413
- 0.160706   0.672252   0.133158  0.65554       0.371826  0.770628   0.0531208
- 0.306617   0.836126   0.301198  0.0224702     0.39344   0.0370205  0.536062
- 0.890947   0.168877   0.32002   0.486136      0.096078  0.172048   0.77672
- 0.507762   0.573567   0.220124  0.165816      0.211049  0.433277   0.539476
+ 0.517515  0.0348206  0.749042   0.0979679  …  0.75984     0.950481   0.579513
+ 0.901092  0.873479   0.134533   0.0697848     0.0586695   0.193254   0.726898
+ 0.976808  0.0901881  0.208332   0.920358      0.288535    0.705941   0.337137
+ 0.657127  0.0317896  0.772837   0.534457      0.0966037   0.700694   0.675999
+ 0.471777  0.144969   0.0718405  0.0827916     0.527233    0.173132   0.694304
+ 0.160872  0.455168   0.489254   0.827851   …  0.62226     0.0995456  0.946522
+ 0.291857  0.769492   0.68043    0.629461      0.727558    0.910796   0.834837
+ 0.775774  0.700731   0.700177   0.0126213     0.00822304  0.327502   0.955181
+ 0.9715    0.64354    0.848441   0.241474      0.591611    0.792573   0.194357
+ 0.646596  0.575456   0.0995212  0.038517      0.709233    0.477657   0.0507231
 
 julia> b = view(a, 2:2:8,2:2:4)
-4×2 SubArray{Float64,2,Array{Float64,2},Tuple{StepRange{Int64,Int64},StepRange{Int64,Int64}},false}:
- 0.537192  0.996234
- 0.736979  0.228787
- 0.991511  0.74485
- 0.836126  0.0224702
+4×2 view(::Array{Float64,2}, 2:2:8, 2:2:4) with eltype Float64:
+ 0.873479   0.0697848
+ 0.0317896  0.534457
+ 0.455168   0.827851
+ 0.700731   0.0126213
 
-julia> (q,r) = qr(b);
+julia> (q, r) = qr(b);
 
 julia> q
-4×2 Array{Float64,2}:
- -0.338809   0.78934
- -0.464815  -0.230274
- -0.625349   0.194538
- -0.527347  -0.534856
+4×4 LinearAlgebra.QRCompactWYQ{Float64,Array{Float64,2}}:
+ -0.722358    0.227524  -0.247784    -0.604181
+ -0.0262896  -0.575919  -0.804227     0.144377
+ -0.376419   -0.75072    0.540177    -0.0541979
+ -0.579497    0.230151  -0.00552346   0.781782
 
 julia> r
 2×2 Array{Float64,2}:
- -1.58553  -0.921517
-  0.0       0.866567
+ -1.20921  -0.383393
+  0.0      -0.910506
 ```

doc/src/manual/parallel-computing.md

@@ -457,7 +457,7 @@ we could compute the singular values of several large random matrices in paralle
 ```julia-repl
 julia> M = Matrix{Float64}[rand(1000,1000) for i = 1:10];
 
-julia> pmap(svd, M);
+julia> pmap(svdvals, M);
 ```
 
 Julia's [`pmap`](@ref) is designed for the case where each function call does a large amount
@@ -486,7 +486,7 @@ As an example, consider computing the singular values of matrices of different s
 ```julia-repl
 julia> M = Matrix{Float64}[rand(800,800), rand(600,600), rand(800,800), rand(600,600)];
 
-julia> pmap(svd, M);
+julia> pmap(svdvals, M);
 ```
 
 If one process handles both 800×800 matrices and another handles both 600×600 matrices, we will

doc/src/manual/performance-tips.md

@@ -544,7 +544,7 @@ function norm(A)
     if isa(A, Vector)
         return sqrt(real(dot(A,A)))
     elseif isa(A, Matrix)
-        return maximum(svd(A)[2])
+        return maximum(svdvals(A))
     else
         error("norm: invalid argument")
     end
@@ -555,7 +555,7 @@ This can be written more concisely and efficiently as:
 
 ```julia
 norm(x::Vector) = sqrt(real(dot(x,x)))
-norm(A::Matrix) = maximum(svd(A)[2])
+norm(A::Matrix) = maximum(svdvals(A))
 ```
 
 ## Write "type-stable" functions

stdlib/Distributed/test/distributed_exec.jl

@@ -576,8 +576,8 @@ end
 n = 10
 as = [rand(4,4) for i in 1:n]
 bs = deepcopy(as)
-cs = collect(Distributed.pgenerate(x->(sleep(rand()*0.1); svdfact(x)), bs))
-svdas = map(svdfact, as)
+cs = collect(Distributed.pgenerate(x->(sleep(rand()*0.1); svd(x)), bs))
+svdas = map(svd, as)
 for i in 1:n
     @test cs[i].U ≈ svdas[i].U
     @test cs[i].S ≈ svdas[i].S

stdlib/IterativeEigensolvers/src/IterativeEigensolvers.jl

@@ -78,7 +78,7 @@ function _eigs(A, B;
     sym = !iscmplx && issymmetric(A) && issymmetric(B)
     nevmax = sym ? n-1 : n-2
     if nevmax <= 0
-        throw(ArgumentError("input matrix A is too small. Use eigfact instead."))
+        throw(ArgumentError("input matrix A is too small. Use eigen instead."))
     end
     if nev > nevmax
         @warn "Adjusting nev from $nev to $nevmax"
@@ -317,10 +317,10 @@ function _svds(X; nsv::Int = 6, ritzvec::Bool = true, tol::Float64 = 0.0, maxite
         # left_sv  = sqrt(2) * ex[2][ 1:size(X,1),     ind ] .* sign.(ex[1][ind]')
         if size(X, 1) >= size(X, 2)
             V = ex[2]
-            U = qr(rmul!(X*V, Diagonal(inv.(svals))))[1]
+            U = Array(qr(rmul!(X*V, Diagonal(inv.(svals)))).Q)
         else
             U = ex[2]
-            V = qr(rmul!(X'U, Diagonal(inv.(svals))))[1]
+            V = Array(qr(rmul!(X'U, Diagonal(inv.(svals)))).Q)
         end
 
         # right_sv = sqrt(2) * ex[2][ size(X,1)+1:end, ind ]

stdlib/IterativeEigensolvers/test/runtests.jl

@@ -152,8 +152,8 @@ LinearAlgebra.A_mul_B!(rho2::StridedVector{T},Phi::CPM{T},rho::StridedVector{T})
 
 let
     # Generate random isometry
-    (Q,R) = qr(randn(100,50))
-    Q = reshape(Q,(50,2,50))
+    (Q, R) = qr(randn(100, 50))
+    Q = reshape(Array(Q), (50, 2, 50))
     # Construct trace-preserving completely positive map from this
     Phi = CPM(copy(Q))
     (d,v,nconv,numiter,numop,resid) = eigs(Phi,nev=1,which=:LM)
@@ -189,16 +189,16 @@ end
     S2 = svd(Array(A))
 
     ## singular values match:
-    @test S1[1].S ≈ S2[2][1:2]
+    @test S1[1].S ≈ S2.S[1:2]
     @testset "singular vectors" begin
         ## 1st left singular vector
         s1_left = sign(S1[1].U[3,1]) * S1[1].U[:,1]
-        s2_left = sign(S2[1][3,1]) * S2[1][:,1]
+        s2_left = sign(S2.U[3,1]) * S2.U[:,1]
         @test s1_left ≈ s2_left
 
         ## 1st right singular vector
         s1_right = sign(S1[1].V[3,1]) * S1[1].V[:,1]
-        s2_right = sign(S2[3][3,1]) * S2[3][:,1]
+        s2_right = sign(S2.V[3,1]) * S2.V[:,1]
         @test s1_right ≈ s2_right
     end
     # Issue number 10329
@@ -213,7 +213,7 @@ end
     end
     @testset "passing guess for Krylov vectors" begin
         S1 = svds(A, nsv = 2, v0=rand(eltype(A),size(A,2)))
-        @test S1[1].S ≈ S2[2][1:2]
+        @test S1[1].S ≈ S2.S[1:2]
     end
 
     @test_throws ArgumentError svds(A,nsv=0)
@@ -227,7 +227,7 @@ end
         for j in 2:i
             d[j] = d[1]
         end
-        A = qr(randn(rng, 20, 20))[1]*Diagonal(d)*qr(randn(rng, 20, 20))[1]
+        A = qr(randn(rng, 20, 20)).Q*Diagonal(d)*qr(randn(rng, 20, 20)).Q
         @testset "Number of singular values: $j" for j in 2:6
             # Default size of subspace
             F = svds(A, nsv = j, v0 = v0)
@@ -251,21 +251,21 @@ end
     S2 = svd(Array(A))
 
     ## singular values match:
-    @test S1[1].S ≈ S2[2][1:2]
+    @test S1[1].S ≈ S2.S[1:2]
     @testset "singular vectors" begin
         ## left singular vectors
         s1_left = abs.(S1[1].U[:,1:2])
-        s2_left = abs.(S2[1][:,1:2])
+        s2_left = abs.(S2.U[:,1:2])
         @test s1_left ≈ s2_left
 
         ## right singular vectors
         s1_right = abs.(S1[1].V[:,1:2])
-        s2_right = abs.(S2[3][:,1:2])
+        s2_right = abs.(S2.V[:,1:2])
         @test s1_right ≈ s2_right
     end
     @testset "passing guess for Krylov vectors" begin
         S1 = svds(A, nsv = 2, v0=rand(eltype(A),size(A,2)))
-        @test S1[1].S ≈ S2[2][1:2]
+        @test S1[1].S ≈ S2.S[1:2]
     end
 
     @test_throws ArgumentError svds(A,nsv=0)

stdlib/LinearAlgebra/docs/src/index.md

@@ -198,7 +198,7 @@ Legend:
 
 ### Matrix factorizations
 
-| Matrix type               | LAPACK | [`eig`](@ref) | [`eigvals`](@ref) | [`eigvecs`](@ref) | [`svd`](@ref) | [`svdvals`](@ref) |
+| Matrix type               | LAPACK | [`eigen`](@ref) | [`eigvals`](@ref) | [`eigvecs`](@ref) | [`svd`](@ref) | [`svdvals`](@ref) |
 |:------------------------- |:------ |:------------- |:----------------- |:----------------- |:------------- |:----------------- |
 | [`Symmetric`](@ref)       | SY     |               | ARI               |                   |               |                   |
 | [`Hermitian`](@ref)       | HE     |               | ARI               |                   |               |                   |
@@ -312,47 +312,40 @@ LinearAlgebra.LowerTriangular
 LinearAlgebra.UpperTriangular
 LinearAlgebra.UniformScaling
 LinearAlgebra.lu
-LinearAlgebra.lufact
-LinearAlgebra.lufact!
+LinearAlgebra.lu!
 LinearAlgebra.chol
-LinearAlgebra.cholfact
-LinearAlgebra.cholfact!
+LinearAlgebra.cholesky
+LinearAlgebra.cholesky!
 LinearAlgebra.lowrankupdate
 LinearAlgebra.lowrankdowndate
 LinearAlgebra.lowrankupdate!
 LinearAlgebra.lowrankdowndate!
-LinearAlgebra.ldltfact
-LinearAlgebra.ldltfact!
+LinearAlgebra.ldlt
+LinearAlgebra.ldlt!
 LinearAlgebra.qr
 LinearAlgebra.qr!
-LinearAlgebra.qrfact
-LinearAlgebra.qrfact!
 LinearAlgebra.QR
 LinearAlgebra.QRCompactWY
 LinearAlgebra.QRPivoted
-LinearAlgebra.lqfact!
-LinearAlgebra.lqfact
+LinearAlgebra.lq!
 LinearAlgebra.lq
-LinearAlgebra.bkfact
-LinearAlgebra.bkfact!
-LinearAlgebra.eig
+LinearAlgebra.bunchkaufman
+LinearAlgebra.bunchkaufman!
 LinearAlgebra.eigvals
 LinearAlgebra.eigvals!
 LinearAlgebra.eigmax
 LinearAlgebra.eigmin
 LinearAlgebra.eigvecs
-LinearAlgebra.eigfact
-LinearAlgebra.eigfact!
-LinearAlgebra.hessfact
-LinearAlgebra.hessfact!
-LinearAlgebra.schurfact
-LinearAlgebra.schurfact!
+LinearAlgebra.eigen
+LinearAlgebra.eigen!
+LinearAlgebra.hessenberg
+LinearAlgebra.hessenberg!
+LinearAlgebra.schur!
 LinearAlgebra.schur
 LinearAlgebra.ordschur
 LinearAlgebra.ordschur!
-LinearAlgebra.svdfact
-LinearAlgebra.svdfact!
 LinearAlgebra.svd
+LinearAlgebra.svd!
 LinearAlgebra.svdvals
 LinearAlgebra.svdvals!
 LinearAlgebra.Givens