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/* -- translated by f2c (version 20050501).
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You must link the resulting object file with libf2c:
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on Microsoft Windows system, link with libf2c.lib;
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on Linux or Unix systems, link with .../path/to/libf2c.a -lm
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or, if you install libf2c.a in a standard place, with -lf2c -lm
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-- in that order, at the end of the command line, as in
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Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,
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http://www.netlib.org/f2c/libf2c.zip
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#include "arpack_internal.h"
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/* Subroutine */ int igraphdlaln2_(logical *ltrans, integer *na, integer *nw,
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doublereal *smin, doublereal *ca, doublereal *a, integer *lda,
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doublereal *d1, doublereal *d2, doublereal *b, integer *ldb,
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doublereal *wr, doublereal *wi, doublereal *x, integer *ldx,
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doublereal *scale, doublereal *xnorm, integer *info)
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/* Initialized data */
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static logical zswap[4] = { FALSE_,FALSE_,TRUE_,TRUE_ };
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static logical rswap[4] = { FALSE_,TRUE_,FALSE_,TRUE_ };
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static integer ipivot[16] /* was [4][4] */ = { 1,2,3,4,2,1,4,3,3,4,1,2,
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/* System generated locals */
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integer a_dim1, a_offset, b_dim1, b_offset, x_dim1, x_offset;
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doublereal d__1, d__2, d__3, d__4, d__5, d__6;
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static doublereal equiv_0[4], equiv_1[4];
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static doublereal bi1, bi2, br1, br2, xi1, xi2, xr1, xr2, ci21, ci22,
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cr21, cr22, li21, csi, ui11, lr21, ui12, ui22;
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static doublereal csr, ur11, ur12, ur22;
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static doublereal bbnd, cmax, ui11r, ui12s, temp, ur11r, ur12s, u22abs;
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static doublereal bnorm, cnorm, smini;
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extern doublereal igraphdlamch_(char *);
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extern /* Subroutine */ int igraphdladiv_(doublereal *, doublereal *,
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doublereal *, doublereal *, doublereal *, doublereal *);
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static doublereal bignum, smlnum;
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/* -- LAPACK auxiliary routine (version 3.0) -- */
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/* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
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/* Courant Institute, Argonne National Lab, and Rice University */
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/* October 31, 1992 */
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/* .. Scalar Arguments .. */
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/* .. Array Arguments .. */
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/* DLALN2 solves a system of the form (ca A - w D ) X = s B */
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/* or (ca A' - w D) X = s B with possible scaling ("s") and */
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/* perturbation of A. (A' means A-transpose.) */
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/* A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA */
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/* real diagonal matrix, w is a real or complex value, and X and B are */
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/* NA x 1 matrices -- real if w is real, complex if w is complex. NA */
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/* If w is complex, X and B are represented as NA x 2 matrices, */
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/* the first column of each being the real part and the second */
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/* being the imaginary part. */
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/* "s" is a scaling factor (.LE. 1), computed by DLALN2, which is */
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/* so chosen that X can be computed without overflow. X is further */
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/* scaled if necessary to assure that norm(ca A - w D)*norm(X) is less */
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/* If both singular values of (ca A - w D) are less than SMIN, */
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/* SMIN*identity will be used instead of (ca A - w D). If only one */
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/* singular value is less than SMIN, one element of (ca A - w D) will be */
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/* perturbed enough to make the smallest singular value roughly SMIN. */
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/* If both singular values are at least SMIN, (ca A - w D) will not be */
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/* perturbed. In any case, the perturbation will be at most some small */
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/* multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values */
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/* are computed by infinity-norm approximations, and thus will only be */
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/* correct to a factor of 2 or so. */
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/* Note: all input quantities are assumed to be smaller than overflow */
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/* by a reasonable factor. (See BIGNUM.) */
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/* LTRANS (input) LOGICAL */
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/* =.TRUE.: A-transpose will be used. */
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/* =.FALSE.: A will be used (not transposed.) */
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/* NA (input) INTEGER */
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/* The size of the matrix A. It may (only) be 1 or 2. */
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/* NW (input) INTEGER */
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/* 1 if "w" is real, 2 if "w" is complex. It may only be 1 */
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/* SMIN (input) DOUBLE PRECISION */
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/* The desired lower bound on the singular values of A. This */
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/* should be a safe distance away from underflow or overflow, */
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/* say, between (underflow/machine precision) and (machine */
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/* precision * overflow ). (See BIGNUM and ULP.) */
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/* CA (input) DOUBLE PRECISION */
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/* The coefficient c, which A is multiplied by. */
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/* A (input) DOUBLE PRECISION array, dimension (LDA,NA) */
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/* The NA x NA matrix A. */
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/* LDA (input) INTEGER */
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/* The leading dimension of A. It must be at least NA. */
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/* D1 (input) DOUBLE PRECISION */
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/* The 1,1 element in the diagonal matrix D. */
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/* D2 (input) DOUBLE PRECISION */
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/* The 2,2 element in the diagonal matrix D. Not used if NW=1. */
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/* B (input) DOUBLE PRECISION array, dimension (LDB,NW) */
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/* The NA x NW matrix B (right-hand side). If NW=2 ("w" is */
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/* complex), column 1 contains the real part of B and column 2 */
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/* contains the imaginary part. */
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/* LDB (input) INTEGER */
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/* The leading dimension of B. It must be at least NA. */
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/* WR (input) DOUBLE PRECISION */
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/* The real part of the scalar "w". */
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/* WI (input) DOUBLE PRECISION */
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/* The imaginary part of the scalar "w". Not used if NW=1. */
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/* X (output) DOUBLE PRECISION array, dimension (LDX,NW) */
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/* The NA x NW matrix X (unknowns), as computed by DLALN2. */
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/* If NW=2 ("w" is complex), on exit, column 1 will contain */
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/* the real part of X and column 2 will contain the imaginary */
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/* LDX (input) INTEGER */
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/* The leading dimension of X. It must be at least NA. */
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/* SCALE (output) DOUBLE PRECISION */
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/* The scale factor that B must be multiplied by to insure */
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/* that overflow does not occur when computing X. Thus, */
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/* (ca A - w D) X will be SCALE*B, not B (ignoring */
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/* perturbations of A.) It will be at most 1. */
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/* XNORM (output) DOUBLE PRECISION */
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/* The infinity-norm of X, when X is regarded as an NA x NW */
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/* INFO (output) INTEGER */
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/* An error flag. It will be set to zero if no error occurs, */
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/* a negative number if an argument is in error, or a positive */
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/* number if ca A - w D had to be perturbed. */
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/* The possible values are: */
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/* = 0: No error occurred, and (ca A - w D) did not have to be */
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/* = 1: (ca A - w D) had to be perturbed to make its smallest */
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/* (or only) singular value greater than SMIN. */
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/* NOTE: In the interests of speed, this routine does not */
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/* check the inputs for errors. */
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/* ===================================================================== */
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/* .. Parameters .. */
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/* .. Local Scalars .. */
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/* .. Local Arrays .. */
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/* .. External Functions .. */
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/* .. External Subroutines .. */
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/* .. Intrinsic Functions .. */
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/* .. Equivalences .. */
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/* .. Data statements .. */
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/* Parameter adjustments */
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a_offset = 1 + a_dim1;
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b_offset = 1 + b_dim1;
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x_offset = 1 + x_dim1;
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/* .. Executable Statements .. */
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smlnum = 2. * igraphdlamch_("Safe minimum");
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bignum = 1. / smlnum;
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smini = max(*smin,smlnum);
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/* Don't check for input errors */
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/* Standard Initializations */
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/* 1 x 1 (i.e., scalar) system C X = B */
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/* Real 1x1 system. */
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csr = *ca * a[a_dim1 + 1] - *wr * *d1;
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/* If | C | < SMINI, use C = SMINI */
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/* Check scaling for X = B / C */
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bnorm = (d__1 = b[b_dim1 + 1], abs(d__1));
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if (cnorm < 1. && bnorm > 1.) {
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if (bnorm > bignum * cnorm) {
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x[x_dim1 + 1] = b[b_dim1 + 1] * *scale / csr;
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*xnorm = (d__1 = x[x_dim1 + 1], abs(d__1));
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/* Complex 1x1 system (w is complex) */
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csr = *ca * a[a_dim1 + 1] - *wr * *d1;
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cnorm = abs(csr) + abs(csi);
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/* If | C | < SMINI, use C = SMINI */
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/* Check scaling for X = B / C */
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bnorm = (d__1 = b[b_dim1 + 1], abs(d__1)) + (d__2 = b[(b_dim1 <<
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if (cnorm < 1. && bnorm > 1.) {
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if (bnorm > bignum * cnorm) {
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d__1 = *scale * b[b_dim1 + 1];
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d__2 = *scale * b[(b_dim1 << 1) + 1];
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igraphdladiv_(&d__1, &d__2, &csr, &csi, &x[x_dim1 + 1], &x[(x_dim1 << 1)
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*xnorm = (d__1 = x[x_dim1 + 1], abs(d__1)) + (d__2 = x[(x_dim1 <<
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/* Compute the real part of C = ca A - w D (or ca A' - w D ) */
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cr[0] = *ca * a[a_dim1 + 1] - *wr * *d1;
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cr[3] = *ca * a[(a_dim1 << 1) + 2] - *wr * *d2;
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cr[2] = *ca * a[a_dim1 + 2];
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cr[1] = *ca * a[(a_dim1 << 1) + 1];
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cr[1] = *ca * a[a_dim1 + 2];
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cr[2] = *ca * a[(a_dim1 << 1) + 1];
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/* Real 2x2 system (w is real) */
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/* Find the largest element in C */
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for (j = 1; j <= 4; ++j) {
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if ((d__1 = crv[j - 1], abs(d__1)) > cmax) {
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cmax = (d__1 = crv[j - 1], abs(d__1));
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/* If norm(C) < SMINI, use SMINI*identity. */
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d__3 = (d__1 = b[b_dim1 + 1], abs(d__1)), d__4 = (d__2 = b[
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b_dim1 + 2], abs(d__2));
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bnorm = max(d__3,d__4);
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if (smini < 1. && bnorm > 1.) {
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if (bnorm > bignum * smini) {
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temp = *scale / smini;
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x[x_dim1 + 1] = temp * b[b_dim1 + 1];
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x[x_dim1 + 2] = temp * b[b_dim1 + 2];
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*xnorm = temp * bnorm;
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/* Gaussian elimination with complete pivoting. */
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ur11 = crv[icmax - 1];
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cr21 = crv[ipivot[(icmax << 2) - 3] - 1];
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ur12 = crv[ipivot[(icmax << 2) - 2] - 1];
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cr22 = crv[ipivot[(icmax << 2) - 1] - 1];
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ur22 = cr22 - ur12 * lr21;
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/* If smaller pivot < SMINI, use SMINI */
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if (abs(ur22) < smini) {
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if (rswap[icmax - 1]) {
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d__2 = (d__1 = br1 * (ur22 * ur11r), abs(d__1)), d__3 = abs(br2);
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bbnd = max(d__2,d__3);
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if (bbnd > 1. && abs(ur22) < 1.) {
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if (bbnd >= bignum * abs(ur22)) {
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xr2 = br2 * *scale / ur22;
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xr1 = *scale * br1 * ur11r - xr2 * (ur11r * ur12);
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if (zswap[icmax - 1]) {
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d__1 = abs(xr1), d__2 = abs(xr2);
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*xnorm = max(d__1,d__2);
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/* Further scaling if norm(A) norm(X) > overflow */
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if (*xnorm > 1. && cmax > 1.) {
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if (*xnorm > bignum / cmax) {
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temp = cmax / bignum;
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x[x_dim1 + 1] = temp * x[x_dim1 + 1];
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x[x_dim1 + 2] = temp * x[x_dim1 + 2];
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*xnorm = temp * *xnorm;
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*scale = temp * *scale;
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/* Complex 2x2 system (w is complex) */
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/* Find the largest element in C */
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ci[0] = -(*wi) * *d1;
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ci[3] = -(*wi) * *d2;
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for (j = 1; j <= 4; ++j) {
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if ((d__1 = crv[j - 1], abs(d__1)) + (d__2 = civ[j - 1], abs(
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cmax = (d__1 = crv[j - 1], abs(d__1)) + (d__2 = civ[j - 1]
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/* If norm(C) < SMINI, use SMINI*identity. */
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d__5 = (d__1 = b[b_dim1 + 1], abs(d__1)) + (d__2 = b[(b_dim1
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<< 1) + 1], abs(d__2)), d__6 = (d__3 = b[b_dim1 + 2],
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abs(d__3)) + (d__4 = b[(b_dim1 << 1) + 2], abs(d__4));
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bnorm = max(d__5,d__6);
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if (smini < 1. && bnorm > 1.) {
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if (bnorm > bignum * smini) {
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temp = *scale / smini;
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x[x_dim1 + 1] = temp * b[b_dim1 + 1];
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x[x_dim1 + 2] = temp * b[b_dim1 + 2];
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x[(x_dim1 << 1) + 1] = temp * b[(b_dim1 << 1) + 1];
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x[(x_dim1 << 1) + 2] = temp * b[(b_dim1 << 1) + 2];
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*xnorm = temp * bnorm;
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/* Gaussian elimination with complete pivoting. */
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ur11 = crv[icmax - 1];
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ui11 = civ[icmax - 1];
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cr21 = crv[ipivot[(icmax << 2) - 3] - 1];
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ci21 = civ[ipivot[(icmax << 2) - 3] - 1];
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ur12 = crv[ipivot[(icmax << 2) - 2] - 1];
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ui12 = civ[ipivot[(icmax << 2) - 2] - 1];
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cr22 = crv[ipivot[(icmax << 2) - 1] - 1];
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ci22 = civ[ipivot[(icmax << 2) - 1] - 1];
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if (icmax == 1 || icmax == 4) {
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/* Code when off-diagonals of pivoted C are real */
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if (abs(ur11) > abs(ui11)) {
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/* Computing 2nd power */
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ur11r = 1. / (ur11 * (d__1 * d__1 + 1.));
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ui11r = -temp * ur11r;
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/* Computing 2nd power */
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ui11r = -1. / (ui11 * (d__1 * d__1 + 1.));
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ur11r = -temp * ui11r;
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ur12s = ur12 * ur11r;
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ui12s = ur12 * ui11r;
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ur22 = cr22 - ur12 * lr21;
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ui22 = ci22 - ur12 * li21;
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/* Code when diagonals of pivoted C are real */
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ur12s = ur12 * ur11r;
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ui12s = ui12 * ur11r;
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ur22 = cr22 - ur12 * lr21 + ui12 * li21;
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ui22 = -ur12 * li21 - ui12 * lr21;
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u22abs = abs(ur22) + abs(ui22);
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/* If smaller pivot < SMINI, use SMINI */
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if (u22abs < smini) {
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if (rswap[icmax - 1]) {
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bi2 = b[(b_dim1 << 1) + 1];
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bi1 = b[(b_dim1 << 1) + 2];
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bi1 = b[(b_dim1 << 1) + 1];
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bi2 = b[(b_dim1 << 1) + 2];
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br2 = br2 - lr21 * br1 + li21 * bi1;
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bi2 = bi2 - li21 * br1 - lr21 * bi1;
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d__1 = (abs(br1) + abs(bi1)) * (u22abs * (abs(ur11r) + abs(ui11r))
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), d__2 = abs(br2) + abs(bi2);
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bbnd = max(d__1,d__2);
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if (bbnd > 1. && u22abs < 1.) {
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if (bbnd >= bignum * u22abs) {
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igraphdladiv_(&br2, &bi2, &ur22, &ui22, &xr2, &xi2);
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xr1 = ur11r * br1 - ui11r * bi1 - ur12s * xr2 + ui12s * xi2;
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xi1 = ui11r * br1 + ur11r * bi1 - ui12s * xr2 - ur12s * xi2;
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if (zswap[icmax - 1]) {
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x[(x_dim1 << 1) + 1] = xi2;
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x[(x_dim1 << 1) + 2] = xi1;
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x[(x_dim1 << 1) + 1] = xi1;
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x[(x_dim1 << 1) + 2] = xi2;
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d__1 = abs(xr1) + abs(xi1), d__2 = abs(xr2) + abs(xi2);
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*xnorm = max(d__1,d__2);
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/* Further scaling if norm(A) norm(X) > overflow */
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if (*xnorm > 1. && cmax > 1.) {
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if (*xnorm > bignum / cmax) {
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temp = cmax / bignum;
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x[x_dim1 + 1] = temp * x[x_dim1 + 1];
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x[x_dim1 + 2] = temp * x[x_dim1 + 2];
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x[(x_dim1 << 1) + 1] = temp * x[(x_dim1 << 1) + 1];
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x[(x_dim1 << 1) + 2] = temp * x[(x_dim1 << 1) + 2];
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*xnorm = temp * *xnorm;
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*scale = temp * *scale;
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} /* igraphdlaln2_ */
added
removed
src/lapack/dlaln2.c
