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qrsolv_.c
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qrsolv_.c
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/* qrsolv.f -- translated by f2c (version 20020621).
You must link the resulting object file with the libraries:
-lf2c -lm (in that order)
*/
#include "minpack.h"
#include <math.h>
#include "minpackP.h"
__minpack_attr__
void __minpack_func__(qrsolv)(const int *n, real *r__, const int *ldr,
const int *ipvt, const real *diag, const real *qtb, real *x,
real *sdiag, real *wa)
{
/* Initialized data */
#define p5 ((real).5)
#define p25 ((real).25)
/* System generated locals */
int r_dim1, r_offset, i__1, i__2, i__3;
real d__1, d__2;
/* Local variables */
int i__, j, k, l, jp1, kp1;
real tan__, cos__, sin__, sum, temp, cotan;
int nsing;
real qtbpj;
/* ********** */
/* subroutine qrsolv */
/* given an m by n matrix a, an n by n diagonal matrix d, */
/* and an m-vector b, the problem is to determine an x which */
/* solves the system */
/* a*x = b , d*x = 0 , */
/* in the least squares sense. */
/* this subroutine completes the solution of the problem */
/* if it is provided with the necessary information from the */
/* qr factorization, with column pivoting, of a. that is, if */
/* a*p = q*r, where p is a permutation matrix, q has orthogonal */
/* columns, and r is an upper triangular matrix with diagonal */
/* elements of nonincreasing magnitude, then qrsolv expects */
/* the full upper triangle of r, the permutation matrix p, */
/* and the first n components of (q transpose)*b. the system */
/* a*x = b, d*x = 0, is then equivalent to */
/* t t */
/* r*z = q *b , p *d*p*z = 0 , */
/* where x = p*z. if this system does not have full rank, */
/* then a least squares solution is obtained. on output qrsolv */
/* also provides an upper triangular matrix s such that */
/* t t t */
/* p *(a *a + d*d)*p = s *s . */
/* s is computed within qrsolv and may be of separate interest. */
/* the subroutine statement is */
/* subroutine qrsolv(n,r,ldr,ipvt,diag,qtb,x,sdiag,wa) */
/* where */
/* n is a positive integer input variable set to the order of r. */
/* r is an n by n array. on input the full upper triangle */
/* must contain the full upper triangle of the matrix r. */
/* on output the full upper triangle is unaltered, and the */
/* strict lower triangle contains the strict upper triangle */
/* (transposed) of the upper triangular matrix s. */
/* ldr is a positive integer input variable not less than n */
/* which specifies the leading dimension of the array r. */
/* ipvt is an integer input array of length n which defines the */
/* permutation matrix p such that a*p = q*r. column j of p */
/* is column ipvt(j) of the identity matrix. */
/* diag is an input array of length n which must contain the */
/* diagonal elements of the matrix d. */
/* qtb is an input array of length n which must contain the first */
/* n elements of the vector (q transpose)*b. */
/* x is an output array of length n which contains the least */
/* squares solution of the system a*x = b, d*x = 0. */
/* sdiag is an output array of length n which contains the */
/* diagonal elements of the upper triangular matrix s. */
/* wa is a work array of length n. */
/* subprograms called */
/* fortran-supplied ... dabs,dsqrt */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more */
/* ********** */
/* Parameter adjustments */
--wa;
--sdiag;
--x;
--qtb;
--diag;
--ipvt;
r_dim1 = *ldr;
r_offset = 1 + r_dim1 * 1;
r__ -= r_offset;
/* Function Body */
/* copy r and (q transpose)*b to preserve input and initialize s. */
/* in particular, save the diagonal elements of r in x. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
i__2 = *n;
for (i__ = j; i__ <= i__2; ++i__) {
r__[i__ + j * r_dim1] = r__[j + i__ * r_dim1];
/* L10: */
}
x[j] = r__[j + j * r_dim1];
wa[j] = qtb[j];
/* L20: */
}
/* eliminate the diagonal matrix d using a givens rotation. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* prepare the row of d to be eliminated, locating the */
/* diagonal element using p from the qr factorization. */
l = ipvt[j];
if (diag[l] == 0.) {
goto L90;
}
i__2 = *n;
for (k = j; k <= i__2; ++k) {
sdiag[k] = 0.;
/* L30: */
}
sdiag[j] = diag[l];
/* the transformations to eliminate the row of d */
/* modify only a single element of (q transpose)*b */
/* beyond the first n, which is initially zero. */
qtbpj = 0.;
i__2 = *n;
for (k = j; k <= i__2; ++k) {
/* determine a givens rotation which eliminates the */
/* appropriate element in the current row of d. */
if (sdiag[k] == 0.) {
goto L70;
}
if ((d__1 = r__[k + k * r_dim1], abs(d__1)) >= (d__2 = sdiag[k],
abs(d__2))) {
goto L40;
}
cotan = r__[k + k * r_dim1] / sdiag[k];
/* Computing 2nd power */
d__1 = cotan;
sin__ = p5 / sqrt(p25 + p25 * (d__1 * d__1));
cos__ = sin__ * cotan;
goto L50;
L40:
tan__ = sdiag[k] / r__[k + k * r_dim1];
/* Computing 2nd power */
d__1 = tan__;
cos__ = p5 / sqrt(p25 + p25 * (d__1 * d__1));
sin__ = cos__ * tan__;
L50:
/* compute the modified diagonal element of r and */
/* the modified element of ((q transpose)*b,0). */
r__[k + k * r_dim1] = cos__ * r__[k + k * r_dim1] + sin__ * sdiag[
k];
temp = cos__ * wa[k] + sin__ * qtbpj;
qtbpj = -sin__ * wa[k] + cos__ * qtbpj;
wa[k] = temp;
/* accumulate the tranformation in the row of s. */
kp1 = k + 1;
if (*n < kp1) {
goto L70;
}
i__3 = *n;
for (i__ = kp1; i__ <= i__3; ++i__) {
temp = cos__ * r__[i__ + k * r_dim1] + sin__ * sdiag[i__];
sdiag[i__] = -sin__ * r__[i__ + k * r_dim1] + cos__ * sdiag[
i__];
r__[i__ + k * r_dim1] = temp;
/* L60: */
}
L70:
/* L80: */
;
}
L90:
/* store the diagonal element of s and restore */
/* the corresponding diagonal element of r. */
sdiag[j] = r__[j + j * r_dim1];
r__[j + j * r_dim1] = x[j];
/* L100: */
}
/* solve the triangular system for z. if the system is */
/* singular, then obtain a least squares solution. */
nsing = *n;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (sdiag[j] == 0. && nsing == *n) {
nsing = j - 1;
}
if (nsing < *n) {
wa[j] = 0.;
}
/* L110: */
}
if (nsing < 1) {
goto L150;
}
i__1 = nsing;
for (k = 1; k <= i__1; ++k) {
j = nsing - k + 1;
sum = 0.;
jp1 = j + 1;
if (nsing < jp1) {
goto L130;
}
i__2 = nsing;
for (i__ = jp1; i__ <= i__2; ++i__) {
sum += r__[i__ + j * r_dim1] * wa[i__];
/* L120: */
}
L130:
wa[j] = (wa[j] - sum) / sdiag[j];
/* L140: */
}
L150:
/* permute the components of z back to components of x. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
l = ipvt[j];
x[l] = wa[j];
/* L160: */
}
return;
/* last card of subroutine qrsolv. */
} /* qrsolv_ */