zlatrs.c

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#include "f2c.h"
#include "netlib.h"

/* Modified by Peter Vanroose, June 2001: manual optimisation and clean-up */

/* Table of constant values */
static integer c__1 = 1;
static doublereal c_b36 = .5;

/* Subroutine */ void zlatrs_(uplo, trans, diag, normin, n, a, lda, x, scale, cnorm, info)
const char *uplo, *trans, *diag, *normin;
const integer *n;
const doublecomplex *a;
const integer *lda;
doublecomplex *x;
doublereal *scale, *cnorm;
integer *info;
{
    /* System generated locals */
    integer i__1;
    doublereal d__1;
    doublecomplex z__1;

    /* Local variables */
    static integer jinc;
    static doublereal xbnd;
    static integer imax;
    static doublereal tmax;
    static doublecomplex tjjs;
    static doublereal xmax, grow;
    static integer i, j;
    static doublereal tscal;
    static doublecomplex uscal;
    static integer jlast;
    static doublecomplex csumj;
    static logical upper;
    static doublereal xj;
    static doublereal bignum;
    static logical notran;
    static integer jfirst;
    static doublereal smlnum;
    static logical nounit;
    static doublereal rec, tjj;

/*  -- LAPACK auxiliary routine (version 2.0) -- */
/*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., */
/*     Courant Institute, Argonne National Lab, and Rice University */
/*     June 30, 1992 */

/*  ===================================================================== */
/*                                                                        */
/*  Purpose                                                               */
/*  =======                                                               */
/*                                                                        */
/*  ZLATRS solves one of the triangular systems                           */
/*                                                                        */
/*     A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,                 */
/*                                                                        */
/*  with scaling to prevent overflow.  Here A is an upper or lower        */
/*  triangular matrix, A**T denotes the transpose of A, A**H denotes the  */
/*  conjugate transpose of A, x and b are n-element vectors, and s is a   */
/*  scaling factor, usually less than or equal to 1, chosen so that the   */
/*  components of x will be less than the overflow threshold.  If the     */
/*  unscaled problem will not cause overflow, the Level 2 BLAS routine    */
/*  ZTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j), */
/*  then s is set to 0 and a non-trivial solution to A*x = 0 is returned. */
/*                                                                        */
/*  Arguments                                                             */
/*  =========                                                             */
/*                                                                        */
/*  UPLO    (input) CHARACTER*1                                           */
/*          Specifies whether the matrix A is upper or lower triangular.  */
/*          = 'U':  Upper triangular                                      */
/*          = 'L':  Lower triangular                                      */
/*                                                                        */
/*  TRANS   (input) CHARACTER*1                                           */
/*          Specifies the operation applied to A.                         */
/*          = 'N':  Solve A * x = s*b     (No transpose)                  */
/*          = 'T':  Solve A**T * x = s*b  (Transpose)                     */
/*          = 'C':  Solve A**H * x = s*b  (Conjugate transpose)           */
/*                                                                        */
/*  DIAG    (input) CHARACTER*1                                           */
/*          Specifies whether or not the matrix A is unit triangular.     */
/*          = 'N':  Non-unit triangular                                   */
/*          = 'U':  Unit triangular                                       */
/*                                                                        */
/*  NORMIN  (input) CHARACTER*1                                           */
/*          Specifies whether CNORM has been set or not.                  */
/*          = 'Y':  CNORM contains the column norms on entry              */
/*          = 'N':  CNORM is not set on entry.  On exit, the norms will   */
/*                  be computed and stored in CNORM.                      */
/*                                                                        */
/*  N       (input) INTEGER                                               */
/*          The order of the matrix A.  N >= 0.                           */
/*                                                                        */
/*  A       (input) COMPLEX*16 array, dimension (LDA,N)                   */
/*          The triangular matrix A.  If UPLO = 'U', the leading n by n   */
/*          upper triangular part of the array A contains the upper       */
/*          triangular matrix, and the strictly lower triangular part of  */
/*          A is not referenced.  If UPLO = 'L', the leading n by n lower */
/*          triangular part of the array A contains the lower triangular  */
/*          matrix, and the strictly upper triangular part of A is not    */
/*          referenced.  If DIAG = 'U', the diagonal elements of A are    */
/*          also not referenced and are assumed to be 1.                  */
/*                                                                        */
/*  LDA     (input) INTEGER                                               */
/*          The leading dimension of the array A.  LDA >= max (1,N).      */
/*                                                                        */
/*  X       (input/output) COMPLEX*16 array, dimension (N)                */
/*          On entry, the right hand side b of the triangular system.     */
/*          On exit, X is overwritten by the solution vector x.           */
/*                                                                        */
/*  SCALE   (output) DOUBLE PRECISION                                     */
/*          The scaling factor s for the triangular system                */
/*             A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.         */
/*          If SCALE = 0, the matrix A is singular or badly scaled, and   */
/*          the vector x is an exact or approximate solution to A*x = 0.  */
/*                                                                        */
/*  CNORM   (input or output) DOUBLE PRECISION array, dimension (N)       */
/*                                                                        */
/*          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)      */
/*          contains the norm of the off-diagonal part of the j-th column */
/*          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal */
/*          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)     */
/*          must be greater than or equal to the 1-norm.                  */
/*                                                                        */
/*          If NORMIN = 'N', CNORM is an output argument and CNORM(j)     */
/*          returns the 1-norm of the offdiagonal part of the j-th column */
/*          of A.                                                         */
/*                                                                        */
/*  INFO    (output) INTEGER                                              */
/*          = 0:  successful exit                                         */
/*          < 0:  if INFO = -k, the k-th argument had an illegal value    */
/*                                                                        */
/*  Further Details                                                       */
/*  ======= =======                                                       */
/*                                                                        */
/*  A rough bound on x is computed; if that is less than overflow, ZTRSV  */
/*  is called, otherwise, specific code is used which checks for possible */
/*  overflow or divide-by-zero at every operation.                        */
/*                                                                        */
/*  A columnwise scheme is used for solving A*x = b.  The basic algorithm */
/*  if A is lower triangular is                                           */
/*                                                                        */
/*       x[1:n] := b[1:n]                                                 */
/*       for j = 1, ..., n                                                */
/*            x(j) := x(j) / A(j,j)                                       */
/*            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]                    */
/*       end                                                              */
/*                                                                        */
/*  Define bounds on the components of x after j iterations of the loop:  */
/*     M(j) = bound on x[1:j]                                             */
/*     G(j) = bound on x[j+1:n]                                           */
/*  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.              */
/*                                                                        */
/*  Then for iteration j+1 we have                                        */
/*     M(j+1) <= G(j) / | A(j+1,j+1) |                                    */
/*     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |                         */
/*            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )                 */
/*                                                                        */
/*  where CNORM(j+1) is greater than or equal to the infinity-norm of     */
/*  column j+1 of A, not counting the diagonal.  Hence                    */
/*                                                                        */
/*     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )                 */
/*                  1<=i<=j                                               */
/*  and                                                                   */
/*                                                                        */
/*     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )  */
/*                                   1<=i< j                              */
/*                                                                        */
/*  Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTRSV if the    */
/*  reciprocal of the largest M(j), j=1,..,n, is larger than              */
/*  max(underflow, 1/overflow).                                           */
/*                                                                        */
/*  The bound on x(j) is also used to determine when a step in the        */
/*  columnwise method can be performed without fear of overflow.  If      */
/*  the computed bound is greater than a large constant, x is scaled to   */
/*  prevent overflow, but if the bound overflows, x is set to 0, x(j) to  */
/*  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.    */
/*                                                                        */
/*  Similarly, a row-wise scheme is used to solve A**T *x = b  or         */
/*  A**H *x = b.  The basic algorithm for A upper triangular is           */
/*                                                                        */
/*       for j = 1, ..., n                                                */
/*            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)          */
/*       end                                                              */
/*                                                                        */
/*  We simultaneously compute two bounds                                  */
/*       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j       */
/*       M(j) = bound on x(i), 1<=i<=j                                    */
/*                                                                        */
/*  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we   */
/*  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.      */
/*  Then the bound on x(j) is                                             */
/*                                                                        */
/*       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |                   */
/*                                                                        */
/*            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )           */
/*                      1<=i<=j                                           */
/*                                                                        */
/*  and we can safely call ZTRSV if 1/M(n) and 1/G(n) are both greater    */
/*  than max(underflow, 1/overflow).                                      */
/*                                                                        */
/*  ===================================================================== */

    *info = 0;
    upper = lsame_(uplo, "U");
    notran = lsame_(trans, "N");
    nounit = lsame_(diag, "N");

/*     Test the input parameters. */

    if (! upper && ! lsame_(uplo, "L")) {
        *info = -1;
    } else if (! notran && ! lsame_(trans, "T") && ! lsame_(trans, "C")) {
        *info = -2;
    } else if (! nounit && ! lsame_(diag, "U")) {
        *info = -3;
    } else if (! lsame_(normin, "Y") && ! lsame_(normin, "N")) {
        *info = -4;
    } else if (*n < 0) {
        *info = -5;
    } else if (*lda < max(1,*n)) {
        *info = -7;
    }
    if (*info != 0) {
        i__1 = -(*info);
        xerbla_("ZLATRS", &i__1);
        return;
    }

/*     Quick return if possible */

    if (*n == 0) {
        return;
    }

/*     Determine machine dependent parameters to control overflow. */

    smlnum = dlamch_("Safe minimum");
    bignum = 1. / smlnum;
    dlabad_(&smlnum, &bignum);
    smlnum /= dlamch_("Precision");
    bignum = 1. / smlnum;
    *scale = 1.;

    if (lsame_(normin, "N")) {

/*        Compute the 1-norm of each column, not including the diagonal. */

        if (upper) {

/*           A is upper triangular. */

            for (j = 0; j < *n; ++j) {
                cnorm[j] = dzasum_(&j, &a[j * *lda], &c__1);
            }
        } else {

/*           A is lower triangular. */

            for (j = 0; j < *n - 1; ++j) {
                i__1 = *n - j - 1;
                cnorm[j] = dzasum_(&i__1, &a[j + 1 + j * *lda], &c__1);
            }
            cnorm[*n-1] = 0.;
        }
    }

/*     Scale the column norms by TSCAL if the maximum element in CNORM is */
/*     greater than BIGNUM/2. */

    imax = idamax_(n, &cnorm[1], &c__1) - 1;
    tmax = cnorm[imax];
    if (tmax <= bignum * .5) {
        tscal = 1.;
    } else {
        tscal = .5 / (smlnum * tmax);
        dscal_(n, &tscal, cnorm, &c__1);
    }

/*     Compute a bound on the computed solution vector to see if the */
/*     Level 2 BLAS routine ZTRSV can be used. */

    xmax = 0.;
    for (j = 0; j < *n; ++j) {
        xmax = max(xmax, abs(x[j].r / 2.) + abs(x[j].i / 2.));
    }
    xbnd = xmax;

    if (notran) {

/*        Compute the growth in A * x = b. */

        if (upper) {
            jfirst = *n - 1;
            jlast = 0;
            jinc = -1;
        } else {
            jfirst = 0;
            jlast = *n - 1;
            jinc = 1;
        }

        if (tscal != 1.) {