cgesvx.c

/* cgesvx.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 cgesvx_(char *fact, char *trans, integer *n, integer *
	nrhs, complex *a, integer *lda, complex *af, integer *ldaf, integer *
	ipiv, char *equed, real *r__, real *c__, complex *b, integer *ldb, 
	complex *x, integer *ldx, real *rcond, real *ferr, real *berr, 
	complex *work, real *rwork, integer *info)
{
    /* System generated locals */
    integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1, 
	    x_offset, i__1, i__2, i__3, i__4, i__5;
    real r__1, r__2;
    complex q__1;

    /* Local variables */
    integer i__, j;
    real amax;
    char norm[1];
    extern logical lsame_(char *, char *);
    real rcmin, rcmax, anorm;
    logical equil;
    extern doublereal clange_(char *, integer *, integer *, complex *, 
	    integer *, real *);
    extern /* Subroutine */ int claqge_(integer *, integer *, complex *, 
	    integer *, real *, real *, real *, real *, real *, char *)
	    , cgecon_(char *, integer *, complex *, integer *, real *, real *, 
	     complex *, real *, integer *);
    real colcnd;
    extern doublereal slamch_(char *);
    extern /* Subroutine */ int cgeequ_(integer *, integer *, complex *, 
	    integer *, real *, real *, real *, real *, real *, integer *);
    logical nofact;
    extern /* Subroutine */ int cgerfs_(char *, integer *, integer *, complex 
	    *, integer *, complex *, integer *, integer *, complex *, integer 
	    *, complex *, integer *, real *, real *, complex *, real *, 
	    integer *), cgetrf_(integer *, integer *, complex *, 
	    integer *, integer *, integer *), clacpy_(char *, integer *, 
	    integer *, complex *, integer *, complex *, integer *), 
	    xerbla_(char *, integer *);
    real bignum;
    extern doublereal clantr_(char *, char *, char *, integer *, integer *, 
	    complex *, integer *, real *);
    integer infequ;
    logical colequ;
    extern /* Subroutine */ int cgetrs_(char *, integer *, integer *, complex 
	    *, integer *, integer *, complex *, integer *, integer *);
    real rowcnd;
    logical notran;
    real smlnum;
    logical rowequ;
    real rpvgrw;


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

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

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

/*  CGESVX uses the LU factorization to compute the solution to a complex */
/*  system of linear equations */
/*     A * X = B, */
/*  where A is an N-by-N matrix and X and B are N-by-NRHS matrices. */

/*  Error bounds on the solution and a condition estimate are also */
/*  provided. */

/*  Description */
/*  =========== */

/*  The following steps are performed: */

/*  1. If FACT = 'E', real scaling factors are computed to equilibrate */
/*     the system: */
/*        TRANS = 'N':  diag(R)*A*diag(C)     *inv(diag(C))*X = diag(R)*B */
/*        TRANS = 'T': (diag(R)*A*diag(C))**T *inv(diag(R))*X = diag(C)*B */
/*        TRANS = 'C': (diag(R)*A*diag(C))**H *inv(diag(R))*X = diag(C)*B */
/*     Whether or not the system will be equilibrated depends on the */
/*     scaling of the matrix A, but if equilibration is used, A is */
/*     overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if TRANS='N') */
/*     or diag(C)*B (if TRANS = 'T' or 'C'). */

/*  2. If FACT = 'N' or 'E', the LU decomposition is used to factor the */
/*     matrix A (after equilibration if FACT = 'E') as */
/*        A = P * L * U, */
/*     where P is a permutation matrix, L is a unit lower triangular */
/*     matrix, and U is upper triangular. */

/*  3. If some U(i,i)=0, so that U is exactly singular, then the routine */
/*     returns with INFO = i. Otherwise, the factored form of A is used */
/*     to estimate the condition number of the matrix A.  If the */
/*     reciprocal of the condition number is less than machine precision, */
/*     INFO = N+1 is returned as a warning, but the routine still goes on */
/*     to solve for X and compute error bounds as described below. */

/*  4. The system of equations is solved for X using the factored form */
/*     of A. */

/*  5. Iterative refinement is applied to improve the computed solution */
/*     matrix and calculate error bounds and backward error estimates */
/*     for it. */

/*  6. If equilibration was used, the matrix X is premultiplied by */
/*     diag(C) (if TRANS = 'N') or diag(R) (if TRANS = 'T' or 'C') so */
/*     that it solves the original system before equilibration. */

/*  Arguments */
/*  ========= */

/*  FACT    (input) CHARACTER*1 */
/*          Specifies whether or not the factored form of the matrix A is */
/*          supplied on entry, and if not, whether the matrix A should be */
/*          equilibrated before it is factored. */
/*          = 'F':  On entry, AF and IPIV contain the factored form of A. */
/*                  If EQUED is not 'N', the matrix A has been */
/*                  equilibrated with scaling factors given by R and C. */
/*                  A, AF, and IPIV are not modified. */
/*          = 'N':  The matrix A will be copied to AF and factored. */
/*          = 'E':  The matrix A will be equilibrated if necessary, then */
/*                  copied to AF and factored. */

/*  TRANS   (input) CHARACTER*1 */
/*          Specifies the form of the system of equations: */
/*          = 'N':  A * X = B     (No transpose) */
/*          = 'T':  A**T * X = B  (Transpose) */
/*          = 'C':  A**H * X = B  (Conjugate transpose) */

/*  N       (input) INTEGER */
/*          The number of linear equations, i.e., the order of the */
/*          matrix A.  N >= 0. */

/*  NRHS    (input) INTEGER */
/*          The number of right hand sides, i.e., the number of columns */
/*          of the matrices B and X.  NRHS >= 0. */

/*  A       (input/output) COMPLEX array, dimension (LDA,N) */
/*          On entry, the N-by-N matrix A.  If FACT = 'F' and EQUED is */
/*          not 'N', then A must have been equilibrated by the scaling */
/*          factors in R and/or C.  A is not modified if FACT = 'F' or */
/*          'N', or if FACT = 'E' and EQUED = 'N' on exit. */

/*          On exit, if EQUED .ne. 'N', A is scaled as follows: */
/*          EQUED = 'R':  A := diag(R) * A */
/*          EQUED = 'C':  A := A * diag(C) */
/*          EQUED = 'B':  A := diag(R) * A * diag(C). */

/*  LDA     (input) INTEGER */
/*          The leading dimension of the array A.  LDA >= max(1,N). */

/*  AF      (input or output) COMPLEX array, dimension (LDAF,N) */
/*          If FACT = 'F', then AF is an input argument and on entry */
/*          contains the factors L and U from the factorization */
/*          A = P*L*U as computed by CGETRF.  If EQUED .ne. 'N', then */
/*          AF is the factored form of the equilibrated matrix A. */

/*          If FACT = 'N', then AF is an output argument and on exit */
/*          returns the factors L and U from the factorization A = P*L*U */
/*          of the original matrix A. */

/*          If FACT = 'E', then AF is an output argument and on exit */
/*          returns the factors L and U from the factorization A = P*L*U */
/*          of the equilibrated matrix A (see the description of A for */
/*          the form of the equilibrated matrix). */

/*  LDAF    (input) INTEGER */
/*          The leading dimension of the array AF.  LDAF >= max(1,N). */

/*  IPIV    (input or output) INTEGER array, dimension (N) */
/*          If FACT = 'F', then IPIV is an input argument and on entry */
/*          contains the pivot indices from the factorization A = P*L*U */
/*          as computed by CGETRF; row i of the matrix was interchanged */
/*          with row IPIV(i). */

/*          If FACT = 'N', then IPIV is an output argument and on exit */
/*          contains the pivot indices from the factorization A = P*L*U */
/*          of the original matrix A. */

/*          If FACT = 'E', then IPIV is an output argument and on exit */
/*          contains the pivot indices from the factorization A = P*L*U */
/*          of the equilibrated matrix A. */

/*  EQUED   (input or output) CHARACTER*1 */
/*          Specifies the form of equilibration that was done. */
/*          = 'N':  No equilibration (always true if FACT = 'N'). */
/*          = 'R':  Row equilibration, i.e., A has been premultiplied by */
/*                  diag(R). */
/*          = 'C':  Column equilibration, i.e., A has been postmultiplied */
/*                  by diag(C). */
/*          = 'B':  Both row and column equilibration, i.e., A has been */
/*                  replaced by diag(R) * A * diag(C). */
/*          EQUED is an input argument if FACT = 'F'; otherwise, it is an */
/*          output argument. */

/*  R       (input or output) REAL array, dimension (N) */
/*          The row scale factors for A.  If EQUED = 'R' or 'B', A is */
/*          multiplied on the left by diag(R); if EQUED = 'N' or 'C', R */
/*          is not accessed.  R is an input argument if FACT = 'F'; */
/*          otherwise, R is an output argument.  If FACT = 'F' and */
/*          EQUED = 'R' or 'B', each element of R must be positive. */

/*  C       (input or output) REAL array, dimension (N) */
/*          The column scale factors for A.  If EQUED = 'C' or 'B', A is */
/*          multiplied on the right by diag(C); if EQUED = 'N' or 'R', C */
/*          is not accessed.  C is an input argument if FACT = 'F'; */
/*          otherwise, C is an output argument.  If FACT = 'F' and */
/*          EQUED = 'C' or 'B', each element of C must be positive. */

/*  B       (input/output) COMPLEX array, dimension (LDB,NRHS) */
/*          On entry, the N-by-NRHS right hand side matrix B. */
/*          On exit, */
/*          if EQUED = 'N', B is not modified; */
/*          if TRANS = 'N' and EQUED = 'R' or 'B', B is overwritten by */
/*          diag(R)*B; */
/*          if TRANS = 'T' or 'C' and EQUED = 'C' or 'B', B is */
/*          overwritten by diag(C)*B. */

/*  LDB     (input) INTEGER */
/*          The leading dimension of the array B.  LDB >= max(1,N). */

/*  X       (output) COMPLEX array, dimension (LDX,NRHS) */
/*          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X */
/*          to the original system of equations.  Note that A and B are */
/*          modified on exit if EQUED .ne. 'N', and the solution to the */
/*          equilibrated system is inv(diag(C))*X if TRANS = 'N' and */
/*          EQUED = 'C' or 'B', or inv(diag(R))*X if TRANS = 'T' or 'C' */
/*          and EQUED = 'R' or 'B'. */

/*  LDX     (input) INTEGER */
/*          The leading dimension of the array X.  LDX >= max(1,N). */

/*  RCOND   (output) REAL */
/*          The estimate of the reciprocal condition number of the matrix */
/*          A after equilibration (if done).  If RCOND is less than the */
/*          machine precision (in particular, if RCOND = 0), the matrix */
/*          is singular to working precision.  This condition is */
/*          indicated by a return code of INFO > 0. */

/*  FERR    (output) REAL array, dimension (NRHS) */
/*          The estimated forward error bound for each solution vector */
/*          X(j) (the j-th column of the solution matrix X). */
/*          If XTRUE is the true solution corresponding to X(j), FERR(j) */
/*          is an estimated upper bound for the magnitude of the largest */
/*          element in (X(j) - XTRUE) divided by the magnitude of the */
/*          largest element in X(j).  The estimate is as reliable as */
/*          the estimate for RCOND, and is almost always a slight */
/*          overestimate of the true error. */

/*  BERR    (output) REAL array, dimension (NRHS) */
/*          The componentwise relative backward error of each solution */
/*          vector X(j) (i.e., the smallest relative change in */
/*          any element of A or B that makes X(j) an exact solution). */

/*  WORK    (workspace) COMPLEX array, dimension (2*N) */

/*  RWORK   (workspace/output) REAL array, dimension (2*N) */
/*          On exit, RWORK(1) contains the reciprocal pivot growth */
/*          factor norm(A)/norm(U). The "max absolute element" norm is */
/*          used. If RWORK(1) is much less than 1, then the stability */
/*          of the LU factorization of the (equilibrated) matrix A */
/*          could be poor. This also means that the solution X, condition */
/*          estimator RCOND, and forward error bound FERR could be */
/*          unreliable. If factorization fails with 0<INFO<=N, then */
/*          RWORK(1) contains the reciprocal pivot growth factor for the */
/*          leading INFO columns of A. */

/*  INFO    (output) INTEGER */
/*          = 0:  successful exit */
/*          < 0:  if INFO = -i, the i-th argument had an illegal value */
/*          > 0:  if INFO = i, and i is */
/*                <= N:  U(i,i) is exactly zero.  The factorization has */
/*                       been completed, but the factor U is exactly */
/*                       singular, so the solution and error bounds */
/*                       could not be computed. RCOND = 0 is returned. */
/*                = N+1: U is nonsingular, but RCOND is less than machine */
/*                       precision, meaning that the matrix is singular */
/*                       to working precision.  Nevertheless, the */
/*                       solution and error bounds are computed because */
/*                       there are a number of situations where the */
/*                       computed solution can be more accurate than the */
/*                       value of RCOND would suggest. */

/*  ===================================================================== */

/*     .. Parameters .. */
/*     .. */
/*     .. Local Scalars .. */
/*     .. */
/*     .. External Functions .. */
/*     .. */
/*     .. External Subroutines .. */
/*     .. */
/*     .. Intrinsic Functions .. */
/*     .. */
/*     .. Executable Statements .. */

    /* Parameter adjustments */
    a_dim1 = *lda;
    a_offset = 1 + a_dim1;
    a -= a_offset;
    af_dim1 = *ldaf;
    af_offset = 1 + af_dim1;
    af -= af_offset;
    --ipiv;
    --r__;
    --c__;
    b_dim1 = *ldb;
    b_offset = 1 + b_dim1;
    b -= b_offset;
    x_dim1 = *ldx;
    x_offset = 1 + x_dim1;
    x -= x_offset;
    --ferr;
    --berr;
    --work;
    --rwork;

    /* Function Body */
    *info = 0;
    nofact = lsame_(fact, "N");
    equil = lsame_(fact, "E");
    notran = lsame_(trans, "N");
    if (nofact || equil) {
	*(unsigned char *)equed = 'N';
	rowequ = FALSE_;
	colequ = FALSE_;
    } else {
	rowequ = lsame_(equed, "R") || lsame_(equed, 
		"B");
	colequ = lsame_(equed, "C") || lsame_(equed, 
		"B");
	smlnum = slamch_("Safe minimum");
	bignum = 1.f / smlnum;
    }

/*     Test the input parameters. */

    if (! nofact && ! equil && ! lsame_(fact, "F")) {
	*info = -1;
    } else if (! notran && ! lsame_(trans, "T") && ! 
	    lsame_(trans, "C")) {
	*info = -2;
    } else if (*n < 0) {
	*info = -3;
    } else if (*nrhs < 0) {
	*info = -4;
    } else if (*lda < max(1,*n)) {
	*info = -6;
    } else if (*ldaf < max(1,*n)) {
	*info = -8;
    } else if (lsame_(fact, "F") && ! (rowequ || colequ 
	    || lsame_(equed, "N"))) {
	*info = -10;
    } else {
	if (rowequ) {
	    rcmin = bignum;
	    rcmax = 0.f;
	    i__1 = *n;
	    for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
		r__1 = rcmin, r__2 = r__[j];
		rcmin = dmin(r__1,r__2);
/* Computing MAX */
		r__1 = rcmax, r__2 = r__[j];
		rcmax = dmax(r__1,r__2);
/* L10: */
	    }
	    if (rcmin <= 0.f) {
		*info = -11;
	    } else if (*n > 0) {
		rowcnd = dmax(rcmin,smlnum) / dmin(rcmax,bignum);
	    } else {
		rowcnd = 1.f;
	    }
	}
	if (colequ && *info == 0) {
	    rcmin = bignum;
	    rcmax = 0.f;
	    i__1 = *n;
	    for (j = 1; j <= i__1; ++j) {
/* Computing MIN */
		r__1 = rcmin, r__2 = c__[j];
		rcmin = dmin(r__1,r__2);
/* Computing MAX */
		r__1 = rcmax, r__2 = c__[j];
		rcmax = dmax(r__1,r__2);
/* L20: */
	    }
	    if (rcmin <= 0.f) {
		*info = -12;
	    } else if (*n > 0) {
		colcnd = dmax(rcmin,smlnum) / dmin(rcmax,bignum);
	    } else {
		colcnd = 1.f;
	    }
	}
	if (*info == 0) {
	    if (*ldb < max(1,*n)) {
		*info = -14;
	    } else if (*ldx < max(1,*n)) {
		*info = -16;
	    }
	}
    }

    if (*info != 0) {
	i__1 = -(*info);
	xerbla_("CGESVX", &i__1);
	return 0;
    }

    if (equil) {

/*        Compute row and column scalings to equilibrate the matrix A. */

	cgeequ_(n, n, &a[a_offset], lda, &r__[1], &c__[1], &rowcnd, &colcnd, &
		amax, &infequ);
	if (infequ == 0) {

/*           Equilibrate the matrix. */

	    claqge_(n, n, &a[a_offset], lda, &r__[1], &c__[1], &rowcnd, &
		    colcnd, &amax, equed);
	    rowequ = lsame_(equed, "R") || lsame_(equed, 
		     "B");
	    colequ = lsame_(equed, "C") || lsame_(equed, 
		     "B");
	}
    }

/*     Scale the right hand side. */

    if (notran) {
	if (rowequ) {
	    i__1 = *nrhs;
	    for (j = 1; j <= i__1; ++j) {
		i__2 = *n;
		for (i__ = 1; i__ <= i__2; ++i__) {
		    i__3 = i__ + j * b_dim1;
		    i__4 = i__;
		    i__5 = i__ + j * b_dim1;
		    q__1.r = r__[i__4] * b[i__5].r, q__1.i = r__[i__4] * b[
			    i__5].i;
		    b[i__3].r = q__1.r, b[i__3].i = q__1.i;
/* L30: */
		}
/* L40: */
	    }
	}
    } else if (colequ) {
	i__1 = *nrhs;
	for (j = 1; j <= i__1; ++j) {
	    i__2 = *n;
	    for (i__ = 1; i__ <= i__2; ++i__) {
		i__3 = i__ + j * b_dim1;
		i__4 = i__;
		i__5 = i__ + j * b_dim1;
		q__1.r = c__[i__4] * b[i__5].r, q__1.i = c__[i__4] * b[i__5]
			.i;
		b[i__3].r = q__1.r, b[i__3].i = q__1.i;
/* L50: */
	    }
/* L60: */
	}
    }

    if (nofact || equil) {

/*        Compute the LU factorization of A. */

	clacpy_("Full", n, n, &a[a_offset], lda, &af[af_offset], ldaf);
	cgetrf_(n, n, &af[af_offset], ldaf, &ipiv[1], info);

/*        Return if INFO is non-zero. */

	if (*info > 0) {

/*           Compute the reciprocal pivot growth factor of the */
/*           leading rank-deficient INFO columns of A. */

	    rpvgrw = clantr_("M", "U", "N", info, info, &af[af_offset], ldaf, 
		    &rwork[1]);
	    if (rpvgrw == 0.f) {
		rpvgrw = 1.f;
	    } else {
		rpvgrw = clange_("M", n, info, &a[a_offset], lda, &rwork[1]) / rpvgrw;
	    }
	    rwork[1] = rpvgrw;
	    *rcond = 0.f;
	    return 0;
	}
    }

/*     Compute the norm of the matrix A and the */
/*     reciprocal pivot growth factor RPVGRW. */

    if (notran) {
	*(unsigned char *)norm = '1';
    } else {
	*(unsigned char *)norm = 'I';
    }
    anorm = clange_(norm, n, n, &a[a_offset], lda, &rwork[1]);
    rpvgrw = clantr_("M", "U", "N", n, n, &af[af_offset], ldaf, &rwork[1]);
    if (rpvgrw == 0.f) {
	rpvgrw = 1.f;
    } else {
	rpvgrw = clange_("M", n, n, &a[a_offset], lda, &rwork[1]) /
		 rpvgrw;
    }

/*     Compute the reciprocal of the condition number of A. */

    cgecon_(norm, n, &af[af_offset], ldaf, &anorm, rcond, &work[1], &rwork[1], 
	     info);

/*     Compute the solution matrix X. */

    clacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx);
    cgetrs_(trans, n, nrhs, &af[af_offset], ldaf, &ipiv[1], &x[x_offset], ldx, 
	     info);

/*     Use iterative refinement to improve the computed solution and */
/*     compute error bounds and backward error estimates for it. */

    cgerfs_(trans, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, &ipiv[1], 
	     &b[b_offset], ldb, &x[x_offset], ldx, &ferr[1], &berr[1], &work[
	    1], &rwork[1], info);

/*     Transform the solution matrix X to a solution of the original */
/*     system. */

    if (notran) {
	if (colequ) {
	    i__1 = *nrhs;
	    for (j = 1; j <= i__1; ++j) {
		i__2 = *n;
		for (i__ = 1; i__ <= i__2; ++i__) {
		    i__3 = i__ + j * x_dim1;
		    i__4 = i__;
		    i__5 = i__ + j * x_dim1;
		    q__1.r = c__[i__4] * x[i__5].r, q__1.i = c__[i__4] * x[
			    i__5].i;
		    x[i__3].r = q__1.r, x[i__3].i = q__1.i;
/* L70: */
		}
/* L80: */
	    }
	    i__1 = *nrhs;
	    for (j = 1; j <= i__1; ++j) {
		ferr[j] /= colcnd;
/* L90: */
	    }
	}
    } else if (rowequ) {
	i__1 = *nrhs;
	for (j = 1; j <= i__1; ++j) {
	    i__2 = *n;
	    for (i__ = 1; i__ <= i__2; ++i__) {
		i__3 = i__ + j * x_dim1;
		i__4 = i__;
		i__5 = i__ + j * x_dim1;
		q__1.r = r__[i__4] * x[i__5].r, q__1.i = r__[i__4] * x[i__5]
			.i;
		x[i__3].r = q__1.r, x[i__3].i = q__1.i;
/* L100: */
	    }
/* L110: */
	}
	i__1 = *nrhs;
	for (j = 1; j <= i__1; ++j) {
	    ferr[j] /= rowcnd;
/* L120: */
	}
    }

/*     Set INFO = N+1 if the matrix is singular to working precision. */

    if (*rcond < slamch_("Epsilon")) {
	*info = *n + 1;
    }

    rwork[1] = rpvgrw;
    return 0;

/*     End of CGESVX */

} /* cgesvx_ */