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matrix.c
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matrix.c
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/*****************************************************************************************\
* *
* Matrix inversion, determinants, and linear equations via LU-decomposition *
* *
* Author: Gene Myers *
* Date : April 2007 *
* Mod : June 2008 -- Added TDG's and Cubic Spline to enable snakes and curves *
* Dec 2008 -- Refined TDG's and cubic splines to Decompose/Solve paradigm *
* *
\*****************************************************************************************/
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include <float.h>
#include "gene_core.h"
#include "matrix.h"
#define TINY 1.0e-20
/****************************************************************************************
* *
* LU-FACTORIZATION SYSTEM SOLVER *
* *
****************************************************************************************/
// M is a square double matrix where the row index moves the fastest.
// LU_Decompose takes M and produces an LU factorization of M that
// can then be used to rapidly solve the system for given right hand sides
// and to compute M's determinant. The return value is NULL if the matrix
// is nonsingular. If the matrix appears unstable (had to use a very nearly
// zero pivot) then the integer pointed at by stable will be zero, and
// non-zero otherwise. M is subsumed and effectively destroyed by the routine.
LU_Factor *LU_Decompose(Double_Matrix *M, int *stable)
{ LU_Factor *F;
int n, i, j;
int *p, sign;
double *v;
double *avec[1001], **a;
n = M->n;
if (n > 1000)
a = Malloc(sizeof(double)*n,"Allocating LU Factor work space");
else
a = avec;
F = Malloc(sizeof(LU_Factor),"Allocating LU Factor");
p = Malloc((sizeof(int) + sizeof(double))*n,"Allocating LU Factor");
if (a == NULL || F == NULL || p == NULL)
exit (1);
v = (double *) (p+n);
p[0] = 0;
a[0] = M->m;
for (i = 1; i < n; i++)
{ a[i] = a[i-1] + n;
p[i] = i;
}
*stable = 1;
sign = 1;
for (i = 0; i < n; i++) // Find the scale factors for each row in v.
{ double b, f, *r;
r = a[i];
b = 0.;
for (j = 0; j < n; j++)
{ f = fabs(r[j]);
if (f > b)
b = f;
}
if (b == 0.0)
{ free(p);
free(F);
if (n > 1000)
free(a);
return (NULL);
}
v[i] = 1./b;
}
for (j = 0; j < n; j++) // For each column
{ double b, s, *r;
int k, w;
for (i = 0; i < j; i++) // Determine U
{ r = a[i];
s = r[j];
for (k = 0; k < i; k++)
s -= r[k]*a[k][j];
r[j] = s;
}
b = -1.;
w = j;
for (i = j; i < n; i++) // Determine L without dividing by pivot, in order to
{ r = a[i]; // determine who the pivot should be.
s = r[j];
for (k = 0; k < j; k++)
s -= r[k]*a[k][j];
r[j] = s;
s = v[i]*fabs(s); // Update best pivot seen thus far
if (s > b)
{ b = s;
w = i;
}
}
if (w != j) // Pivot if necessary
{ r = a[w];
a[w] = a[j];
a[j] = r;
k = p[w];
p[w] = p[j];
p[j] = k;
sign = -sign;
v[w] = v[j];
}
if (fabs(a[j][j]) < TINY) // Complete column of L by dividing by selected pivot
{ if (a[j][j] < 0.)
a[j][j] = -TINY;
else
a[j][j] = TINY;
*stable = 0;
}
b = 1./a[j][j];
for (i = j+1; i < n; i++)
a[i][j] *= b;
}
#ifdef DEBUG_LU
{ int i, j;
printf("\nLU Decomposition\n");
for (i = 0; i < n; i++)
{ printf(" %2d: ",p[i]);
for (j = 0; j < n; j++)
printf(" %8g",a[i][j]);
printf("\n");
}
}
#endif
if (n > 1000)
free(a);
F->sign = sign;
F->perm = p;
F->lu_mat = M;
return (F);
}
// Display LU factorization F to specified file
void Show_LU_Product(FILE *file, LU_Factor *F)
{ int n, i, j, k;
int *p;
double u, **a, *d;
n = F->lu_mat->n;
d = F->lu_mat->m;
p = F->perm;
a = (double **) (p+n);
for (i = 0; i < n; i++)
a[i] = d + p[i]*n;
fprintf(file,"\nLU Product:\n");
for (i = 0; i < n; i++)
{ for (j = 0; j < i; j++)
{ u = 0.;
for (k = 0; k <= j; k++)
u += a[i][k] * a[k][j];
fprintf(file," %g",u);
}
for (j = i; j < n; j++)
{ u = a[i][j];
for (k = 0; k < i; k++)
u += a[i][k] * a[k][j];
fprintf(file," %g",u);
}
fprintf(file,"\n");
}
}
// Given rhs vector B and LU-factorization F, solve the system of equations
// and return the result in B.
// To invert M = L*U given the LU-decomposition, simply call LU_Solve with
// b = [ 0^k-1 1 0^n-k] to get the k'th column of the inverse matrix.
Double_Vector *LU_Solve(Double_Vector *B, LU_Factor *F)
{ double *x;
int n, i, j;
int *p;
double *a, *b, s, *r;
n = F->lu_mat->n;
a = F->lu_mat->m;
p = F->perm;
b = B->m;
x = (double *) (p+n);
for (i = 0; i < n; i++)
{ r = a + p[i]*n;
s = b[p[i]];
for (j = 0; j < i; j++)
s -= r[j] * x[j];
x[i] = s;
}
for (i = n; i-- > 0; )
{ r = a + p[i]*n;
s = x[i];
for (j = i+1; j < n; j++)
s -= r[j] * b[j];
b[i] = s/r[i];
}
return (B);
}
// Transpose a matrix M in-place and as a convenience return a pointer to it
Double_Matrix *Transpose_Matrix(Double_Matrix *M)
{ int n;
double *a;
int p, q;
int i, j;
n = M->n;
a = M->m;
p = 0;
for (j = 0; j < n; j++) // Transpose the result
{ q = j;
for (i = 0; i < j; i++)
{ double x = a[p];
a[p++] = a[q];
a[q] = x;
q += n;
}
p += (n-j);
}
return (M);
}
// Generate the right inverse of the matrix that gave rise to the LU factorization f.
// That is for matrix A, return matrix A^-1 s.t. A * A^-1 = I. If transpose is non-zero
// then the transpose of the right inverse is returned.
Double_Matrix *LU_Invert(LU_Factor *F, int transpose)
{ int n, i, j;
Double_Matrix *M, G;
double *m, *g;
n = F->lu_mat->n;
M = Malloc(sizeof(Double_Matrix),"Allocating matrix");
m = Malloc(sizeof(double)*n*n,"Allocating matrix");
if (M == NULL || m == NULL)
exit (1);
M->n = n;
M->m = m;
G.n = n;
g = m;
for (i = 0; i < n; i++) // Find the inverse of each column in the
{ G.m = g;
for (j = 0; j < n; j++)
g[j] = 0.;
g[i] = 1.;
LU_Solve(&G,F);
g += n;
}
if (!transpose)
Transpose_Matrix(M);
return (M);
}
// Given an LU-factorization F, return the value of the determinant of the
// original matrix.
double LU_Determinant(LU_Factor *F)
{ int i, n;
int *p;
double *a, det;
n = F->lu_mat->n;
a = F->lu_mat->m;
p = F->perm;
det = F->sign;
for (i = 0; i < n; i++)
det *= a[p[i]*n+i];
return (det);
}