dptrfs.c

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/* dptrfs.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_b11 = 1.;

/* Subroutine */ int dptrfs_(integer *n, integer *nrhs, doublereal *d__, 
        doublereal *e, doublereal *df, doublereal *ef, doublereal *b, integer 
        *ldb, doublereal *x, integer *ldx, doublereal *ferr, doublereal *berr, 
         doublereal *work, integer *info)
{
    /* System generated locals */
    integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2;
    doublereal d__1, d__2, d__3;

    /* Local variables */
    integer i__, j;
    doublereal s, bi, cx, dx, ex;
    integer ix, nz;
    doublereal eps, safe1, safe2;
    extern /* Subroutine */ int daxpy_(integer *, doublereal *, doublereal *, 
            integer *, doublereal *, integer *);
    integer count;
    extern doublereal dlamch_(char *);
    extern integer idamax_(integer *, doublereal *, integer *);
    doublereal safmin;
    extern /* Subroutine */ int xerbla_(char *, integer *);
    doublereal lstres;
    extern /* Subroutine */ int dpttrs_(integer *, integer *, doublereal *, 
            doublereal *, doublereal *, integer *, integer *);


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

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

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

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

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

/*  N       (input) INTEGER */
/*          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 matrix B.  NRHS >= 0. */

/*  D       (input) DOUBLE PRECISION array, dimension (N) */
/*          The n diagonal elements of the tridiagonal matrix A. */

/*  E       (input) DOUBLE PRECISION array, dimension (N-1) */
/*          The (n-1) subdiagonal elements of the tridiagonal matrix A. */

/*  DF      (input) DOUBLE PRECISION array, dimension (N) */
/*          The n diagonal elements of the diagonal matrix D from the */
/*          factorization computed by DPTTRF. */

/*  EF      (input) DOUBLE PRECISION array, dimension (N-1) */
/*          The (n-1) subdiagonal elements of the unit bidiagonal factor */
/*          L from the factorization computed by DPTTRF. */

/*  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 DPTTRS. */
/*          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 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). */

/*  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 (2*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 .. */
/*     .. */
/*     .. External Subroutines .. */
/*     .. */
/*     .. Intrinsic Functions .. */
/*     .. */
/*     .. External Functions .. */
/*     .. */
/*     .. Executable Statements .. */

/*     Test the input parameters. */

    /* Parameter adjustments */
    --d__;
    --e;
    --df;
    --ef;
    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;

    /* Function Body */
    *info = 0;
    if (*n < 0) {
        *info = -1;
    } else if (*nrhs < 0) {
        *info = -2;
    } else if (*ldb < max(1,*n)) {
        *info = -8;
    } else if (*ldx < max(1,*n)) {
        *info = -10;
    }
    if (*info != 0) {
        i__1 = -(*info);
        xerbla_("DPTRFS", &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 */

    nz = 4;
    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.  Also compute */
/*        abs(A)*abs(x) + abs(b) for use in the backward error bound. */

        if (*n == 1) {
            bi = b[j * b_dim1 + 1];
            dx = d__[1] * x[j * x_dim1 + 1];
            work[*n + 1] = bi - dx;
            work[1] = abs(bi) + abs(dx);
        } else {
            bi = b[j * b_dim1 + 1];
            dx = d__[1] * x[j * x_dim1 + 1];
            ex = e[1] * x[j * x_dim1 + 2];
            work[*n + 1] = bi - dx - ex;
            work[1] = abs(bi) + abs(dx) + abs(ex);
            i__2 = *n - 1;
            for (i__ = 2; i__ <= i__2; ++i__) {
                bi = b[i__ + j * b_dim1];
                cx = e[i__ - 1] * x[i__ - 1 + j * x_dim1];
                dx = d__[i__] * x[i__ + j * x_dim1];
                ex = e[i__] * x[i__ + 1 + j * x_dim1];
                work[*n + i__] = bi - cx - dx - ex;
                work[i__] = abs(bi) + abs(cx) + abs(dx) + abs(ex);
/* L30: */
            }
            bi = b[*n + j * b_dim1];
            cx = e[*n - 1] * x[*n - 1 + j * x_dim1];
            dx = d__[*n] * x[*n + j * x_dim1];
            work[*n + *n] = bi - cx - dx;
            work[*n] = abs(bi) + abs(cx) + abs(dx);
        }

/*        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. */

        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);
            }
/* L40: */
        }
        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. */

            dpttrs_(n, &c__1, &df[1], &ef[1], &work[*n + 1], n, info);
            daxpy_(n, &c_b11, &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. */

        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;
            }
/* L50: */
        }
        ix = idamax_(n, &work[1], &c__1);
        ferr[j] = work[ix];

/*        Estimate the norm of inv(A). */

/*        Solve M(A) * x = e, where M(A) = (m(i,j)) is given by */

/*           m(i,j) =  abs(A(i,j)), i = j, */
/*           m(i,j) = -abs(A(i,j)), i .ne. j, */

/*        and e = [ 1, 1, ..., 1 ]'.  Note M(A) = M(L)*D*M(L)'. */

/*        Solve M(L) * x = e. */

        work[1] = 1.;
        i__2 = *n;
        for (i__ = 2; i__ <= i__2; ++i__) {
            work[i__] = work[i__ - 1] * (d__1 = ef[i__ - 1], abs(d__1)) + 1.;
/* L60: */
        }

/*        Solve D * M(L)' * x = b. */

        work[*n] /= df[*n];
        for (i__ = *n - 1; i__ >= 1; --i__) {
            work[i__] = work[i__] / df[i__] + work[i__ + 1] * (d__1 = ef[i__],
                     abs(d__1));
/* L70: */
        }

/*        Compute norm(inv(A)) = max(x(i)), 1<=i<=n. */

        ix = idamax_(n, &work[1], &c__1);
        ferr[j] *= (d__1 = work[ix], abs(d__1));

/*        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);
/* L80: */
        }
        if (lstres != 0.) {
            ferr[j] /= lstres;
        }

/* L90: */
    }

    return 0;

/*     End of DPTRFS */

} /* dptrfs_ */