dpbrfs.c

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/* dpbrfs.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"

/* Table of constant values */

static integer c__1 = 1;
static doublereal c_b12 = -1.;
static doublereal c_b14 = 1.;

/* Subroutine */ int dpbrfs_(char *uplo, integer *n, integer *kd, integer *
        nrhs, doublereal *ab, integer *ldab, doublereal *afb, integer *ldafb, 
        doublereal *b, integer *ldb, doublereal *x, integer *ldx, doublereal *
        ferr, doublereal *berr, doublereal *work, integer *iwork, integer *
        info)
{
    /* System generated locals */
    integer ab_dim1, ab_offset, afb_dim1, afb_offset, b_dim1, b_offset, 
            x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5;
    doublereal d__1, d__2, d__3;

    /* Local variables */
    integer i__, j, k, l;
    doublereal s, xk;
    integer nz;
    doublereal eps;
    integer kase;
    doublereal safe1, safe2;
    extern logical lsame_(char *, char *);
    integer isave[3];
    extern /* Subroutine */ int dsbmv_(char *, integer *, integer *, 
            doublereal *, doublereal *, integer *, doublereal *, integer *, 
            doublereal *, doublereal *, integer *), dcopy_(integer *, 
            doublereal *, integer *, doublereal *, integer *), daxpy_(integer 
            *, doublereal *, doublereal *, integer *, doublereal *, integer *)
            ;
    integer count;
    logical upper;
    extern /* Subroutine */ int dlacn2_(integer *, doublereal *, doublereal *, 
             integer *, doublereal *, integer *, integer *);
    extern doublereal dlamch_(char *);
    doublereal safmin;
    extern /* Subroutine */ int xerbla_(char *, integer *), dpbtrs_(
            char *, integer *, integer *, integer *, doublereal *, integer *, 
            doublereal *, integer *, integer *);
    doublereal lstres;


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

/*     Modified to call DLACN2 in place of DLACON, 5 Feb 03, SJH. */

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

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

/*  DPBRFS improves the computed solution to a system of linear */
/*  equations when the coefficient matrix is symmetric positive definite */
/*  and banded, and provides error bounds and backward error estimates */
/*  for the solution. */

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

/*  UPLO    (input) CHARACTER*1 */
/*          = 'U':  Upper triangle of A is stored; */
/*          = 'L':  Lower triangle of A is stored. */

/*  N       (input) INTEGER */
/*          The order of the matrix A.  N >= 0. */

/*  KD      (input) INTEGER */
/*          The number of superdiagonals of the matrix A if UPLO = 'U', */
/*          or the number of subdiagonals if UPLO = 'L'.  KD >= 0. */

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

/*  AB      (input) DOUBLE PRECISION array, dimension (LDAB,N) */
/*          The upper or lower triangle of the symmetric band matrix A, */
/*          stored in the first KD+1 rows of the array.  The j-th column */
/*          of A is stored in the j-th column of the array AB as follows: */
/*          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; */
/*          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd). */

/*  LDAB    (input) INTEGER */
/*          The leading dimension of the array AB.  LDAB >= KD+1. */

/*  AFB     (input) DOUBLE PRECISION array, dimension (LDAFB,N) */
/*          The triangular factor U or L from the Cholesky factorization */
/*          A = U**T*U or A = L*L**T of the band matrix A as computed by */
/*          DPBTRF, in the same storage format as A (see AB). */

/*  LDAFB   (input) INTEGER */
/*          The leading dimension of the array AFB.  LDAFB >= KD+1. */

/*  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS) */
/*          The right hand side matrix B. */

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

/*  X       (input/output) DOUBLE PRECISION array, dimension (LDX,NRHS) */
/*          On entry, the solution matrix X, as computed by DPBTRS. */
/*          On exit, the improved solution matrix X. */

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

/*  FERR    (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (3*N) */

/*  IWORK   (workspace) INTEGER array, dimension (N) */

/*  INFO    (output) INTEGER */
/*          = 0:  successful exit */
/*          < 0:  if INFO = -i, the i-th argument had an illegal value */

/*  Internal Parameters */
/*  =================== */

/*  ITMAX is the maximum number of steps of iterative refinement. */

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

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

/*     Test the input parameters. */

    /* Parameter adjustments */
    ab_dim1 = *ldab;
    ab_offset = 1 + ab_dim1;
    ab -= ab_offset;
    afb_dim1 = *ldafb;
    afb_offset = 1 + afb_dim1;
    afb -= afb_offset;
    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;
    --iwork;

    /* Function Body */
    *info = 0;
    upper = lsame_(uplo, "U");
    if (! upper && ! lsame_(uplo, "L")) {
        *info = -1;
    } else if (*n < 0) {
        *info = -2;
    } else if (*kd < 0) {
        *info = -3;
    } else if (*nrhs < 0) {
        *info = -4;
    } else if (*ldab < *kd + 1) {
        *info = -6;
    } else if (*ldafb < *kd + 1) {
        *info = -8;
    } else if (*ldb < max(1,*n)) {
        *info = -10;
    } else if (*ldx < max(1,*n)) {
        *info = -12;
    }
    if (*info != 0) {
        i__1 = -(*info);
        xerbla_("DPBRFS", &i__1);
        return 0;
    }

/*     Quick return if possible */

    if (*n == 0 || *nrhs == 0) {
        i__1 = *nrhs;
        for (j = 1; j <= i__1; ++j) {
            ferr[j] = 0.;
            berr[j] = 0.;
/* L10: */
        }
        return 0;
    }

/*     NZ = maximum number of nonzero elements in each row of A, plus 1 */

/* Computing MIN */
    i__1 = *n + 1, i__2 = (*kd << 1) + 2;
    nz = min(i__1,i__2);
    eps = dlamch_("Epsilon");
    safmin = dlamch_("Safe minimum");
    safe1 = nz * safmin;
    safe2 = safe1 / eps;

/*     Do for each right hand side */

    i__1 = *nrhs;
    for (j = 1; j <= i__1; ++j) {

        count = 1;
        lstres = 3.;
L20:

/*        Loop until stopping criterion is satisfied. */

/*        Compute residual R = B - A * X */

        dcopy_(n, &b[j * b_dim1 + 1], &c__1, &work[*n + 1], &c__1);
        dsbmv_(uplo, n, kd, &c_b12, &ab[ab_offset], ldab, &x[j * x_dim1 + 1], 
                &c__1, &c_b14, &work[*n + 1], &c__1);

/*        Compute componentwise relative backward error from formula */

/*        max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) ) */

/*        where abs(Z) is the componentwise absolute value of the matrix */
/*        or vector Z.  If the i-th component of the denominator is less */
/*        than SAFE2, then SAFE1 is added to the i-th components of the */
/*        numerator and denominator before dividing. */

        i__2 = *n;
        for (i__ = 1; i__ <= i__2; ++i__) {
            work[i__] = (d__1 = b[i__ + j * b_dim1], abs(d__1));
/* L30: */
        }

/*        Compute abs(A)*abs(X) + abs(B). */

        if (upper) {
            i__2 = *n;
            for (k = 1; k <= i__2; ++k) {
                s = 0.;
                xk = (d__1 = x[k + j * x_dim1], abs(d__1));
                l = *kd + 1 - k;
/* Computing MAX */
                i__3 = 1, i__4 = k - *kd;
                i__5 = k - 1;
                for (i__ = max(i__3,i__4); i__ <= i__5; ++i__) {
                    work[i__] += (d__1 = ab[l + i__ + k * ab_dim1], abs(d__1))
                             * xk;
                    s += (d__1 = ab[l + i__ + k * ab_dim1], abs(d__1)) * (
                            d__2 = x[i__ + j * x_dim1], abs(d__2));
/* L40: */
                }
                work[k] = work[k] + (d__1 = ab[*kd + 1 + k * ab_dim1], abs(
                        d__1)) * xk + s;
/* L50: */
            }
        } else {
            i__2 = *n;
            for (k = 1; k <= i__2; ++k) {
                s = 0.;
                xk = (d__1 = x[k + j * x_dim1], abs(d__1));
                work[k] += (d__1 = ab[k * ab_dim1 + 1], abs(d__1)) * xk;
                l = 1 - k;
/* Computing MIN */
                i__3 = *n, i__4 = k + *kd;
                i__5 = min(i__3,i__4);
                for (i__ = k + 1; i__ <= i__5; ++i__) {
                    work[i__] += (d__1 = ab[l + i__ + k * ab_dim1], abs(d__1))
                             * xk;
                    s += (d__1 = ab[l + i__ + k * ab_dim1], abs(d__1)) * (
                            d__2 = x[i__ + j * x_dim1], abs(d__2));
/* L60: */
                }
                work[k] += s;
/* L70: */
            }
        }
        s = 0.;
        i__2 = *n;
        for (i__ = 1; i__ <= i__2; ++i__) {
            if (work[i__] > safe2) {
/* Computing MAX */
                d__2 = s, d__3 = (d__1 = work[*n + i__], abs(d__1)) / work[
                        i__];
                s = max(d__2,d__3);
            } else {
/* Computing MAX */
                d__2 = s, d__3 = ((d__1 = work[*n + i__], abs(d__1)) + safe1) 
                        / (work[i__] + safe1);
                s = max(d__2,d__3);
            }
/* L80: */
        }
        berr[j] = s;

/*        Test stopping criterion. Continue iterating if */
/*           1) The residual BERR(J) is larger than machine epsilon, and */
/*           2) BERR(J) decreased by at least a factor of 2 during the */
/*              last iteration, and */
/*           3) At most ITMAX iterations tried. */

        if (berr[j] > eps && berr[j] * 2. <= lstres && count <= 5) {

/*           Update solution and try again. */

            dpbtrs_(uplo, n, kd, &c__1, &afb[afb_offset], ldafb, &work[*n + 1]
, n, info);
            daxpy_(n, &c_b14, &work[*n + 1], &c__1, &x[j * x_dim1 + 1], &c__1)
                    ;
            lstres = berr[j];
            ++count;
            goto L20;
        }

/*        Bound error from formula */

/*        norm(X - XTRUE) / norm(X) .le. FERR = */
/*        norm( abs(inv(A))* */
/*           ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X) */

/*        where */
/*          norm(Z) is the magnitude of the largest component of Z */
/*          inv(A) is the inverse of A */
/*          abs(Z) is the componentwise absolute value of the matrix or */
/*             vector Z */
/*          NZ is the maximum number of nonzeros in any row of A, plus 1 */
/*          EPS is machine epsilon */

/*        The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B)) */
/*        is incremented by SAFE1 if the i-th component of */
/*        abs(A)*abs(X) + abs(B) is less than SAFE2. */

/*        Use DLACN2 to estimate the infinity-norm of the matrix */
/*           inv(A) * diag(W), */
/*        where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) */

        i__2 = *n;
        for (i__ = 1; i__ <= i__2; ++i__) {
            if (work[i__] > safe2) {
                work[i__] = (d__1 = work[*n + i__], abs(d__1)) + nz * eps * 
                        work[i__];
            } else {
                work[i__] = (d__1 = work[*n + i__], abs(d__1)) + nz * eps * 
                        work[i__] + safe1;
            }
/* L90: */
        }

        kase = 0;
L100:
        dlacn2_(n, &work[(*n << 1) + 1], &work[*n + 1], &iwork[1], &ferr[j], &
                kase, isave);
        if (kase != 0) {
            if (kase == 1) {

/*              Multiply by diag(W)*inv(A'). */

                dpbtrs_(uplo, n, kd, &c__1, &afb[afb_offset], ldafb, &work[*n 
                        + 1], n, info);
                i__2 = *n;
                for (i__ = 1; i__ <= i__2; ++i__) {
                    work[*n + i__] *= work[i__];
/* L110: */
                }
            } else if (kase == 2) {

/*              Multiply by inv(A)*diag(W). */

                i__2 = *n;
                for (i__ = 1; i__ <= i__2; ++i__) {
                    work[*n + i__] *= work[i__];
/* L120: */
                }
                dpbtrs_(uplo, n, kd, &c__1, &afb[afb_offset], ldafb, &work[*n 
                        + 1], n, info);
            }
            goto L100;
        }

/*        Normalize error. */

        lstres = 0.;
        i__2 = *n;
        for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
            d__2 = lstres, d__3 = (d__1 = x[i__ + j * x_dim1], abs(d__1));
            lstres = max(d__2,d__3);
/* L130: */
        }
        if (lstres != 0.) {
            ferr[j] /= lstres;
        }

/* L140: */
    }

    return 0;

/*     End of DPBRFS */

} /* dpbrfs_ */