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/* ../../../dependencies/lapack/src/slaln2.f -- translated by f2c (version 20061008).
You must link the resulting object file with libf2c:
on Microsoft Windows system, link with libf2c.lib;
on Linux or Unix systems, link with .../path/to/libf2c.a -lm
or, if you install libf2c.a in a standard place, with -lf2c -lm
-- in that order, at the end of the command line, as in
cc *.o -lf2c -lm
Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
http://www.netlib.org/f2c/libf2c.zip
*/
#include "f2c.h"
/* Subroutine */ int slaln2_(logical *ltrans, integer *na, integer *nw, real *
smin, real *ca, real *a, integer *lda, real *d1, real *d2, real *b,
integer *ldb, real *wr, real *wi, real *x, integer *ldx, real *scale,
real *xnorm, integer *info)
{
/* Initialized data */
static logical cswap[4] = { FALSE_,FALSE_,TRUE_,TRUE_ };
static logical rswap[4] = { FALSE_,TRUE_,FALSE_,TRUE_ };
static integer ipivot[16] /* was [4][4] */ = { 1,2,3,4,2,1,4,3,3,4,1,2,
4,3,2,1 };
/* System generated locals */
integer a_dim1, a_offset, b_dim1, b_offset, x_dim1, x_offset;
real r__1, r__2, r__3, r__4, r__5, r__6;
static real equiv_0[4], equiv_1[4];
/* Local variables */
static integer j;
#define ci (equiv_0)
#define cr (equiv_1)
static real bi1, bi2, br1, br2, xi1, xi2, xr1, xr2, ci21, ci22, cr21,
cr22, li21, csi, ui11, lr21, ui12, ui22;
#define civ (equiv_0)
static real csr, ur11, ur12, ur22;
#define crv (equiv_1)
static real bbnd, cmax, ui11r, ui12s, temp, ur11r, ur12s, u22abs;
static integer icmax;
static real bnorm, cnorm, smini;
extern doublereal slamch_(char *, ftnlen);
static real bignum;
extern /* Subroutine */ int sladiv_(real *, real *, real *, real *, real *
, real *);
static real smlnum;
/* -- LAPACK auxiliary routine (version 3.0) -- */
/* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
/* Courant Institute, Argonne National Lab, and Rice University */
/* October 31, 1992 */
/* .. Scalar Arguments .. */
/* .. */
/* .. Array Arguments .. */
/* .. */
/* Purpose */
/* ======= */
/* SLALN2 solves a system of the form (ca A - w D ) X = s B */
/* or (ca A' - w D) X = s B with possible scaling ("s") and */
/* perturbation of A. (A' means A-transpose.) */
/* A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA */
/* real diagonal matrix, w is a real or complex value, and X and B are */
/* NA x 1 matrices -- real if w is real, complex if w is complex. NA */
/* may be 1 or 2. */
/* If w is complex, X and B are represented as NA x 2 matrices, */
/* the first column of each being the real part and the second */
/* being the imaginary part. */
/* "s" is a scaling factor (.LE. 1), computed by SLALN2, which is */
/* so chosen that X can be computed without overflow. X is further */
/* scaled if necessary to assure that norm(ca A - w D)*norm(X) is less */
/* than overflow. */
/* If both singular values of (ca A - w D) are less than SMIN, */
/* SMIN*identity will be used instead of (ca A - w D). If only one */
/* singular value is less than SMIN, one element of (ca A - w D) will be */
/* perturbed enough to make the smallest singular value roughly SMIN. */
/* If both singular values are at least SMIN, (ca A - w D) will not be */
/* perturbed. In any case, the perturbation will be at most some small */
/* multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values */
/* are computed by infinity-norm approximations, and thus will only be */
/* correct to a factor of 2 or so. */
/* Note: all input quantities are assumed to be smaller than overflow */
/* by a reasonable factor. (See BIGNUM.) */
/* Arguments */
/* ========== */
/* LTRANS (input) LOGICAL */
/* =.TRUE.: A-transpose will be used. */
/* =.FALSE.: A will be used (not transposed.) */
/* NA (input) INTEGER */
/* The size of the matrix A. It may (only) be 1 or 2. */
/* NW (input) INTEGER */
/* 1 if "w" is real, 2 if "w" is complex. It may only be 1 */
/* or 2. */
/* SMIN (input) REAL */
/* The desired lower bound on the singular values of A. This */
/* should be a safe distance away from underflow or overflow, */
/* say, between (underflow/machine precision) and (machine */
/* precision * overflow ). (See BIGNUM and ULP.) */
/* CA (input) REAL */
/* The coefficient c, which A is multiplied by. */
/* A (input) REAL array, dimension (LDA,NA) */
/* The NA x NA matrix A. */
/* LDA (input) INTEGER */
/* The leading dimension of A. It must be at least NA. */
/* D1 (input) REAL */
/* The 1,1 element in the diagonal matrix D. */
/* D2 (input) REAL */
/* The 2,2 element in the diagonal matrix D. Not used if NW=1. */
/* B (input) REAL array, dimension (LDB,NW) */
/* The NA x NW matrix B (right-hand side). If NW=2 ("w" is */
/* complex), column 1 contains the real part of B and column 2 */
/* contains the imaginary part. */
/* LDB (input) INTEGER */
/* The leading dimension of B. It must be at least NA. */
/* WR (input) REAL */
/* The real part of the scalar "w". */
/* WI (input) REAL */
/* The imaginary part of the scalar "w". Not used if NW=1. */
/* X (output) REAL array, dimension (LDX,NW) */
/* The NA x NW matrix X (unknowns), as computed by SLALN2. */
/* If NW=2 ("w" is complex), on exit, column 1 will contain */
/* the real part of X and column 2 will contain the imaginary */
/* part. */
/* LDX (input) INTEGER */
/* The leading dimension of X. It must be at least NA. */
/* SCALE (output) REAL */
/* The scale factor that B must be multiplied by to insure */
/* that overflow does not occur when computing X. Thus, */
/* (ca A - w D) X will be SCALE*B, not B (ignoring */
/* perturbations of A.) It will be at most 1. */
/* XNORM (output) REAL */
/* The infinity-norm of X, when X is regarded as an NA x NW */
/* real matrix. */
/* INFO (output) INTEGER */
/* An error flag. It will be set to zero if no error occurs, */
/* a negative number if an argument is in error, or a positive */
/* number if ca A - w D had to be perturbed. */
/* The possible values are: */
/* = 0: No error occurred, and (ca A - w D) did not have to be */
/* perturbed. */
/* = 1: (ca A - w D) had to be perturbed to make its smallest */
/* (or only) singular value greater than SMIN. */
/* NOTE: In the interests of speed, this routine does not */
/* check the inputs for errors. */
/* ===================================================================== */
/* .. Parameters .. */
/* .. */
/* .. Local Scalars .. */
/* .. */
/* .. Local Arrays .. */
/* .. */
/* .. External Functions .. */
/* .. */
/* .. External Subroutines .. */
/* .. */
/* .. Intrinsic Functions .. */
/* .. */
/* .. Equivalences .. */
/* .. */
/* .. Data statements .. */
/* Parameter adjustments */
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
b_dim1 = *ldb;
b_offset = 1 + b_dim1;
b -= b_offset;
x_dim1 = *ldx;
x_offset = 1 + x_dim1;
x -= x_offset;
/* Function Body */
/* .. */
/* .. Executable Statements .. */
/* Compute BIGNUM */
smlnum = 2.f * slamch_("Safe minimum", (ftnlen)12);
bignum = 1.f / smlnum;
smini = dmax(*smin,smlnum);
/* Don't check for input errors */
*info = 0;
/* Standard Initializations */
*scale = 1.f;
if (*na == 1) {
/* 1 x 1 (i.e., scalar) system C X = B */
if (*nw == 1) {
/* Real 1x1 system. */
/* C = ca A - w D */
csr = *ca * a[a_dim1 + 1] - *wr * *d1;
cnorm = dabs(csr);
/* If | C | < SMINI, use C = SMINI */
if (cnorm < smini) {
csr = smini;
cnorm = smini;
*info = 1;
}
/* Check scaling for X = B / C */
bnorm = (r__1 = b[b_dim1 + 1], dabs(r__1));
if (cnorm < 1.f && bnorm > 1.f) {
if (bnorm > bignum * cnorm) {
*scale = 1.f / bnorm;
}
}
/* Compute X */
x[x_dim1 + 1] = b[b_dim1 + 1] * *scale / csr;
*xnorm = (r__1 = x[x_dim1 + 1], dabs(r__1));
} else {
/* Complex 1x1 system (w is complex) */
/* C = ca A - w D */
csr = *ca * a[a_dim1 + 1] - *wr * *d1;
csi = -(*wi) * *d1;
cnorm = dabs(csr) + dabs(csi);
/* If | C | < SMINI, use C = SMINI */
if (cnorm < smini) {
csr = smini;
csi = 0.f;
cnorm = smini;
*info = 1;
}
/* Check scaling for X = B / C */
bnorm = (r__1 = b[b_dim1 + 1], dabs(r__1)) + (r__2 = b[(b_dim1 <<
1) + 1], dabs(r__2));
if (cnorm < 1.f && bnorm > 1.f) {
if (bnorm > bignum * cnorm) {
*scale = 1.f / bnorm;
}
}
/* Compute X */
r__1 = *scale * b[b_dim1 + 1];
r__2 = *scale * b[(b_dim1 << 1) + 1];
sladiv_(&r__1, &r__2, &csr, &csi, &x[x_dim1 + 1], &x[(x_dim1 << 1)
+ 1]);
*xnorm = (r__1 = x[x_dim1 + 1], dabs(r__1)) + (r__2 = x[(x_dim1 <<
1) + 1], dabs(r__2));
}
} else {
/* 2x2 System */
/* Compute the real part of C = ca A - w D (or ca A' - w D ) */
cr[0] = *ca * a[a_dim1 + 1] - *wr * *d1;
cr[3] = *ca * a[(a_dim1 << 1) + 2] - *wr * *d2;
if (*ltrans) {
cr[2] = *ca * a[a_dim1 + 2];
cr[1] = *ca * a[(a_dim1 << 1) + 1];
} else {
cr[1] = *ca * a[a_dim1 + 2];
cr[2] = *ca * a[(a_dim1 << 1) + 1];
}
if (*nw == 1) {
/* Real 2x2 system (w is real) */
/* Find the largest element in C */
cmax = 0.f;
icmax = 0;
for (j = 1; j <= 4; ++j) {
if ((r__1 = crv[j - 1], dabs(r__1)) > cmax) {
cmax = (r__1 = crv[j - 1], dabs(r__1));
icmax = j;
}
/* L10: */
}
/* If norm(C) < SMINI, use SMINI*identity. */
if (cmax < smini) {
/* Computing MAX */
r__3 = (r__1 = b[b_dim1 + 1], dabs(r__1)), r__4 = (r__2 = b[
b_dim1 + 2], dabs(r__2));
bnorm = dmax(r__3,r__4);
if (smini < 1.f && bnorm > 1.f) {
if (bnorm > bignum * smini) {
*scale = 1.f / bnorm;
}
}
temp = *scale / smini;
x[x_dim1 + 1] = temp * b[b_dim1 + 1];
x[x_dim1 + 2] = temp * b[b_dim1 + 2];
*xnorm = temp * bnorm;
*info = 1;
return 0;
}
/* Gaussian elimination with complete pivoting. */
ur11 = crv[icmax - 1];
cr21 = crv[ipivot[(icmax << 2) - 3] - 1];
ur12 = crv[ipivot[(icmax << 2) - 2] - 1];
cr22 = crv[ipivot[(icmax << 2) - 1] - 1];
ur11r = 1.f / ur11;
lr21 = ur11r * cr21;
ur22 = cr22 - ur12 * lr21;
/* If smaller pivot < SMINI, use SMINI */
if (dabs(ur22) < smini) {
ur22 = smini;
*info = 1;
}
if (rswap[icmax - 1]) {
br1 = b[b_dim1 + 2];
br2 = b[b_dim1 + 1];
} else {
br1 = b[b_dim1 + 1];
br2 = b[b_dim1 + 2];
}
br2 -= lr21 * br1;
/* Computing MAX */
r__2 = (r__1 = br1 * (ur22 * ur11r), dabs(r__1)), r__3 = dabs(br2)
;
bbnd = dmax(r__2,r__3);
if (bbnd > 1.f && dabs(ur22) < 1.f) {
if (bbnd >= bignum * dabs(ur22)) {
*scale = 1.f / bbnd;
}
}
xr2 = br2 * *scale / ur22;
xr1 = *scale * br1 * ur11r - xr2 * (ur11r * ur12);
if (cswap[icmax - 1]) {
x[x_dim1 + 1] = xr2;
x[x_dim1 + 2] = xr1;
} else {
x[x_dim1 + 1] = xr1;
x[x_dim1 + 2] = xr2;
}
/* Computing MAX */
r__1 = dabs(xr1), r__2 = dabs(xr2);
*xnorm = dmax(r__1,r__2);
/* Further scaling if norm(A) norm(X) > overflow */
if (*xnorm > 1.f && cmax > 1.f) {
if (*xnorm > bignum / cmax) {
temp = cmax / bignum;
x[x_dim1 + 1] = temp * x[x_dim1 + 1];
x[x_dim1 + 2] = temp * x[x_dim1 + 2];
*xnorm = temp * *xnorm;
*scale = temp * *scale;
}
}
} else {
/* Complex 2x2 system (w is complex) */
/* Find the largest element in C */
ci[0] = -(*wi) * *d1;
ci[1] = 0.f;
ci[2] = 0.f;
ci[3] = -(*wi) * *d2;
cmax = 0.f;
icmax = 0;
for (j = 1; j <= 4; ++j) {
if ((r__1 = crv[j - 1], dabs(r__1)) + (r__2 = civ[j - 1],
dabs(r__2)) > cmax) {
cmax = (r__1 = crv[j - 1], dabs(r__1)) + (r__2 = civ[j -
1], dabs(r__2));
icmax = j;
}
/* L20: */
}
/* If norm(C) < SMINI, use SMINI*identity. */
if (cmax < smini) {
/* Computing MAX */
r__5 = (r__1 = b[b_dim1 + 1], dabs(r__1)) + (r__2 = b[(b_dim1
<< 1) + 1], dabs(r__2)), r__6 = (r__3 = b[b_dim1 + 2],
dabs(r__3)) + (r__4 = b[(b_dim1 << 1) + 2], dabs(
r__4));
bnorm = dmax(r__5,r__6);
if (smini < 1.f && bnorm > 1.f) {
if (bnorm > bignum * smini) {
*scale = 1.f / bnorm;
}
}
temp = *scale / smini;
x[x_dim1 + 1] = temp * b[b_dim1 + 1];
x[x_dim1 + 2] = temp * b[b_dim1 + 2];
x[(x_dim1 << 1) + 1] = temp * b[(b_dim1 << 1) + 1];
x[(x_dim1 << 1) + 2] = temp * b[(b_dim1 << 1) + 2];
*xnorm = temp * bnorm;
*info = 1;
return 0;
}
/* Gaussian elimination with complete pivoting. */
ur11 = crv[icmax - 1];
ui11 = civ[icmax - 1];
cr21 = crv[ipivot[(icmax << 2) - 3] - 1];
ci21 = civ[ipivot[(icmax << 2) - 3] - 1];
ur12 = crv[ipivot[(icmax << 2) - 2] - 1];
ui12 = civ[ipivot[(icmax << 2) - 2] - 1];
cr22 = crv[ipivot[(icmax << 2) - 1] - 1];
ci22 = civ[ipivot[(icmax << 2) - 1] - 1];
if (icmax == 1 || icmax == 4) {
/* Code when off-diagonals of pivoted C are real */
if (dabs(ur11) > dabs(ui11)) {
temp = ui11 / ur11;
/* Computing 2nd power */
r__1 = temp;
ur11r = 1.f / (ur11 * (r__1 * r__1 + 1.f));
ui11r = -temp * ur11r;
} else {
temp = ur11 / ui11;
/* Computing 2nd power */
r__1 = temp;
ui11r = -1.f / (ui11 * (r__1 * r__1 + 1.f));
ur11r = -temp * ui11r;
}
lr21 = cr21 * ur11r;
li21 = cr21 * ui11r;
ur12s = ur12 * ur11r;
ui12s = ur12 * ui11r;
ur22 = cr22 - ur12 * lr21;
ui22 = ci22 - ur12 * li21;
} else {
/* Code when diagonals of pivoted C are real */
ur11r = 1.f / ur11;
ui11r = 0.f;
lr21 = cr21 * ur11r;
li21 = ci21 * ur11r;
ur12s = ur12 * ur11r;
ui12s = ui12 * ur11r;
ur22 = cr22 - ur12 * lr21 + ui12 * li21;
ui22 = -ur12 * li21 - ui12 * lr21;
}
u22abs = dabs(ur22) + dabs(ui22);
/* If smaller pivot < SMINI, use SMINI */
if (u22abs < smini) {
ur22 = smini;
ui22 = 0.f;
*info = 1;
}
if (rswap[icmax - 1]) {
br2 = b[b_dim1 + 1];
br1 = b[b_dim1 + 2];
bi2 = b[(b_dim1 << 1) + 1];
bi1 = b[(b_dim1 << 1) + 2];
} else {
br1 = b[b_dim1 + 1];
br2 = b[b_dim1 + 2];
bi1 = b[(b_dim1 << 1) + 1];
bi2 = b[(b_dim1 << 1) + 2];
}
br2 = br2 - lr21 * br1 + li21 * bi1;
bi2 = bi2 - li21 * br1 - lr21 * bi1;
/* Computing MAX */
r__1 = (dabs(br1) + dabs(bi1)) * (u22abs * (dabs(ur11r) + dabs(
ui11r))), r__2 = dabs(br2) + dabs(bi2);
bbnd = dmax(r__1,r__2);
if (bbnd > 1.f && u22abs < 1.f) {
if (bbnd >= bignum * u22abs) {
*scale = 1.f / bbnd;
br1 = *scale * br1;
bi1 = *scale * bi1;
br2 = *scale * br2;
bi2 = *scale * bi2;
}
}
sladiv_(&br2, &bi2, &ur22, &ui22, &xr2, &xi2);
xr1 = ur11r * br1 - ui11r * bi1 - ur12s * xr2 + ui12s * xi2;
xi1 = ui11r * br1 + ur11r * bi1 - ui12s * xr2 - ur12s * xi2;
if (cswap[icmax - 1]) {
x[x_dim1 + 1] = xr2;
x[x_dim1 + 2] = xr1;
x[(x_dim1 << 1) + 1] = xi2;
x[(x_dim1 << 1) + 2] = xi1;
} else {
x[x_dim1 + 1] = xr1;
x[x_dim1 + 2] = xr2;
x[(x_dim1 << 1) + 1] = xi1;
x[(x_dim1 << 1) + 2] = xi2;
}
/* Computing MAX */
r__1 = dabs(xr1) + dabs(xi1), r__2 = dabs(xr2) + dabs(xi2);
*xnorm = dmax(r__1,r__2);
/* Further scaling if norm(A) norm(X) > overflow */
if (*xnorm > 1.f && cmax > 1.f) {
if (*xnorm > bignum / cmax) {
temp = cmax / bignum;
x[x_dim1 + 1] = temp * x[x_dim1 + 1];
x[x_dim1 + 2] = temp * x[x_dim1 + 2];
x[(x_dim1 << 1) + 1] = temp * x[(x_dim1 << 1) + 1];
x[(x_dim1 << 1) + 2] = temp * x[(x_dim1 << 1) + 2];
*xnorm = temp * *xnorm;
*scale = temp * *scale;
}
}
}
}
return 0;
/* End of SLALN2 */
} /* slaln2_ */
#undef crv
#undef civ
#undef cr
#undef ci
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