dlaln2.c

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/* dlaln2.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"
#include "blaswrap.h"

/* Subroutine */ int dlaln2_(logical *ltrans, integer *na, integer *nw, 
        doublereal *smin, doublereal *ca, doublereal *a, integer *lda, 
        doublereal *d1, doublereal *d2, doublereal *b, integer *ldb, 
        doublereal *wr, doublereal *wi, doublereal *x, integer *ldx, 
        doublereal *scale, doublereal *xnorm, integer *info)
{
    /* Initialized data */

    static logical zswap[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;
    doublereal d__1, d__2, d__3, d__4, d__5, d__6;
    static doublereal equiv_0[4], equiv_1[4];

    /* Local variables */
    integer j;
#define ci (equiv_0)
#define cr (equiv_1)
    doublereal bi1, bi2, br1, br2, xi1, xi2, xr1, xr2, ci21, ci22, cr21, cr22,
             li21, csi, ui11, lr21, ui12, ui22;
#define civ (equiv_0)
    doublereal csr, ur11, ur12, ur22;
#define crv (equiv_1)
    doublereal bbnd, cmax, ui11r, ui12s, temp, ur11r, ur12s, u22abs;
    integer icmax;
    doublereal bnorm, cnorm, smini;
    extern doublereal dlamch_(char *);
    extern /* Subroutine */ int dladiv_(doublereal *, doublereal *, 
            doublereal *, doublereal *, doublereal *, doublereal *);
    doublereal bignum, smlnum;


/*  -- LAPACK auxiliary routine (version 3.2) -- */
/*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/*     November 2006 */

/*     .. Scalar Arguments .. */
/*     .. */
/*     .. Array Arguments .. */
/*     .. */

/*  Purpose */
/*  ======= */

/*  DLALN2 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 DLALN2, 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) DOUBLE PRECISION */
/*          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) DOUBLE PRECISION */
/*          The coefficient c, which A is multiplied by. */

/*  A       (input) DOUBLE PRECISION 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) DOUBLE PRECISION */
/*          The 1,1 element in the diagonal matrix D. */

/*  D2      (input) DOUBLE PRECISION */
/*          The 2,2 element in the diagonal matrix D.  Not used if NW=1. */

/*  B       (input) DOUBLE PRECISION 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) DOUBLE PRECISION */
/*          The real part of the scalar "w". */

/*  WI      (input) DOUBLE PRECISION */
/*          The imaginary part of the scalar "w".  Not used if NW=1. */

/*  X       (output) DOUBLE PRECISION array, dimension (LDX,NW) */
/*          The NA x NW matrix X (unknowns), as computed by DLALN2. */
/*          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) DOUBLE PRECISION */
/*          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) DOUBLE PRECISION */
/*          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. * dlamch_("Safe minimum");
    bignum = 1. / smlnum;
    smini = max(*smin,smlnum);

/*     Don't check for input errors */

    *info = 0;

/*     Standard Initializations */

    *scale = 1.;

    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 = abs(csr);

/*           If | C | < SMINI, use C = SMINI */

            if (cnorm < smini) {
                csr = smini;
                cnorm = smini;
                *info = 1;
            }

/*           Check scaling for  X = B / C */

            bnorm = (d__1 = b[b_dim1 + 1], abs(d__1));
            if (cnorm < 1. && bnorm > 1.) {
                if (bnorm > bignum * cnorm) {
                    *scale = 1. / bnorm;
                }
            }

/*           Compute X */

            x[x_dim1 + 1] = b[b_dim1 + 1] * *scale / csr;
            *xnorm = (d__1 = x[x_dim1 + 1], abs(d__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 = abs(csr) + abs(csi);

/*           If | C | < SMINI, use C = SMINI */

            if (cnorm < smini) {
                csr = smini;
                csi = 0.;
                cnorm = smini;
                *info = 1;
            }

/*           Check scaling for  X = B / C */

            bnorm = (d__1 = b[b_dim1 + 1], abs(d__1)) + (d__2 = b[(b_dim1 << 
                    1) + 1], abs(d__2));
            if (cnorm < 1. && bnorm > 1.) {
                if (bnorm > bignum * cnorm) {
                    *scale = 1. / bnorm;
                }
            }

/*           Compute X */

            d__1 = *scale * b[b_dim1 + 1];
            d__2 = *scale * b[(b_dim1 << 1) + 1];
            dladiv_(&d__1, &d__2, &csr, &csi, &x[x_dim1 + 1], &x[(x_dim1 << 1)
                     + 1]);
            *xnorm = (d__1 = x[x_dim1 + 1], abs(d__1)) + (d__2 = x[(x_dim1 << 
                    1) + 1], abs(d__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.;
            icmax = 0;

            for (j = 1; j <= 4; ++j) {
                if ((d__1 = crv[j - 1], abs(d__1)) > cmax) {
                    cmax = (d__1 = crv[j - 1], abs(d__1));
                    icmax = j;
                }
/* L10: */
            }

/*           If norm(C) < SMINI, use SMINI*identity. */

            if (cmax < smini) {
/* Computing MAX */
                d__3 = (d__1 = b[b_dim1 + 1], abs(d__1)), d__4 = (d__2 = b[
                        b_dim1 + 2], abs(d__2));
                bnorm = max(d__3,d__4);
                if (smini < 1. && bnorm > 1.) {
                    if (bnorm > bignum * smini) {
                        *scale = 1. / 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. / ur11;
            lr21 = ur11r * cr21;
            ur22 = cr22 - ur12 * lr21;

/*           If smaller pivot < SMINI, use SMINI */

            if (abs(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 */
            d__2 = (d__1 = br1 * (ur22 * ur11r), abs(d__1)), d__3 = abs(br2);
            bbnd = max(d__2,d__3);
            if (bbnd > 1. && abs(ur22) < 1.) {
                if (bbnd >= bignum * abs(ur22)) {
                    *scale = 1. / bbnd;
                }
            }

            xr2 = br2 * *scale / ur22;
            xr1 = *scale * br1 * ur11r - xr2 * (ur11r * ur12);
            if (zswap[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 */
            d__1 = abs(xr1), d__2 = abs(xr2);
            *xnorm = max(d__1,d__2);

/*           Further scaling if  norm(A) norm(X) > overflow */

            if (*xnorm > 1. && cmax > 1.) {
                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.;
            ci[2] = 0.;
            ci[3] = -(*wi) * *d2;
            cmax = 0.;
            icmax = 0;

            for (j = 1; j <= 4; ++j) {
                if ((d__1 = crv[j - 1], abs(d__1)) + (d__2 = civ[j - 1], abs(
                        d__2)) > cmax) {
                    cmax = (d__1 = crv[j - 1], abs(d__1)) + (d__2 = civ[j - 1]
                            , abs(d__2));
                    icmax = j;
                }
/* L20: */
            }

/*           If norm(C) < SMINI, use SMINI*identity. */

            if (cmax < smini) {
/* Computing MAX */
                d__5 = (d__1 = b[b_dim1 + 1], abs(d__1)) + (d__2 = b[(b_dim1 
                        << 1) + 1], abs(d__2)), d__6 = (d__3 = b[b_dim1 + 2], 
                        abs(d__3)) + (d__4 = b[(b_dim1 << 1) + 2], abs(d__4));
                bnorm = max(d__5,d__6);
                if (smini < 1. && bnorm > 1.) {
                    if (bnorm > bignum * smini) {
                        *scale = 1. / 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 (abs(ur11) > abs(ui11)) {
                    temp = ui11 / ur11;
/* Computing 2nd power */
                    d__1 = temp;
                    ur11r = 1. / (ur11 * (d__1 * d__1 + 1.));
                    ui11r = -temp * ur11r;
                } else {
                    temp = ur11 / ui11;
/* Computing 2nd power */
                    d__1 = temp;
                    ui11r = -1. / (ui11 * (d__1 * d__1 + 1.));
                    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. / ur11;
                ui11r = 0.;
                lr21 = cr21 * ur11r;
                li21 = ci21 * ur11r;
                ur12s = ur12 * ur11r;
                ui12s = ui12 * ur11r;
                ur22 = cr22 - ur12 * lr21 + ui12 * li21;
                ui22 = -ur12 * li21 - ui12 * lr21;
            }
            u22abs = abs(ur22) + abs(ui22);

/*           If smaller pivot < SMINI, use SMINI */

            if (u22abs < smini) {
                ur22 = smini;
                ui22 = 0.;
                *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 */
            d__1 = (abs(br1) + abs(bi1)) * (u22abs * (abs(ur11r) + abs(ui11r))
                    ), d__2 = abs(br2) + abs(bi2);
            bbnd = max(d__1,d__2);
            if (bbnd > 1. && u22abs < 1.) {
                if (bbnd >= bignum * u22abs) {
                    *scale = 1. / bbnd;
                    br1 = *scale * br1;
                    bi1 = *scale * bi1;
                    br2 = *scale * br2;
                    bi2 = *scale * bi2;
                }
            }

            dladiv_(&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 (zswap[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 */
            d__1 = abs(xr1) + abs(xi1), d__2 = abs(xr2) + abs(xi2);
            *xnorm = max(d__1,d__2);

/*           Further scaling if  norm(A) norm(X) > overflow */

            if (*xnorm > 1. && cmax > 1.) {
                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 DLALN2 */

} /* dlaln2_ */

#undef crv
#undef civ
#undef cr
#undef ci