#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int dgtrfs_(char *trans, integer *n, integer *nrhs, doublereal *dl, doublereal *d__, doublereal *du, doublereal *dlf, doublereal *df, doublereal *duf, doublereal *du2, integer *ipiv, doublereal *b, integer *ldb, doublereal *x, integer *ldx, doublereal * ferr, doublereal *berr, doublereal *work, integer *iwork, integer * info) { /* -- LAPACK routine (version 3.0) -- Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., Courant Institute, Argonne National Lab, and Rice University September 30, 1994 Purpose ======= DGTRFS improves the computed solution to a system of linear equations when the coefficient matrix is tridiagonal, and provides error bounds and backward error estimates for the solution. Arguments ========= 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 = Transpose) 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. DL (input) DOUBLE PRECISION array, dimension (N-1) The (n-1) subdiagonal elements of A. D (input) DOUBLE PRECISION array, dimension (N) The diagonal elements of A. DU (input) DOUBLE PRECISION array, dimension (N-1) The (n-1) superdiagonal elements of A. DLF (input) DOUBLE PRECISION array, dimension (N-1) The (n-1) multipliers that define the matrix L from the LU factorization of A as computed by DGTTRF. DF (input) DOUBLE PRECISION array, dimension (N) The n diagonal elements of the upper triangular matrix U from the LU factorization of A. DUF (input) DOUBLE PRECISION array, dimension (N-1) The (n-1) elements of the first superdiagonal of U. DU2 (input) DOUBLE PRECISION array, dimension (N-2) The (n-2) elements of the second superdiagonal of U. IPIV (input) INTEGER array, dimension (N) The pivot indices; for 1 <= i <= n, row i of the matrix was interchanged with row IPIV(i). IPIV(i) will always be either i or i+1; IPIV(i) = i indicates a row interchange was not required. 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 DGTTRS. 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. ===================================================================== Test the input parameters. Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; static doublereal c_b18 = -1.; static doublereal c_b19 = 1.; /* System generated locals */ integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2; doublereal d__1, d__2, d__3, d__4; /* Local variables */ static integer kase; static doublereal safe1, safe2; static integer i__, j; static doublereal s; extern logical lsame_(char *, char *); extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *), daxpy_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *); static integer count; extern doublereal dlamch_(char *); extern /* Subroutine */ int dlacon_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *); static integer nz; extern /* Subroutine */ int dlagtm_(char *, integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, integer *); static doublereal safmin; extern /* Subroutine */ int xerbla_(char *, integer *); static logical notran; static char transn[1]; extern /* Subroutine */ int dgttrs_(char *, integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, integer *); static char transt[1]; static doublereal lstres, eps; #define b_ref(a_1,a_2) b[(a_2)*b_dim1 + a_1] #define x_ref(a_1,a_2) x[(a_2)*x_dim1 + a_1] --dl; --d__; --du; --dlf; --df; --duf; --du2; --ipiv; b_dim1 = *ldb; b_offset = 1 + b_dim1 * 1; b -= b_offset; x_dim1 = *ldx; x_offset = 1 + x_dim1 * 1; x -= x_offset; --ferr; --berr; --work; --iwork; /* Function Body */ *info = 0; notran = lsame_(trans, "N"); if (! notran && ! lsame_(trans, "T") && ! lsame_( trans, "C")) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*nrhs < 0) { *info = -3; } else if (*ldb < max(1,*n)) { *info = -13; } else if (*ldx < max(1,*n)) { *info = -15; } if (*info != 0) { i__1 = -(*info); xerbla_("DGTRFS", &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; } if (notran) { *(unsigned char *)transn = 'N'; *(unsigned char *)transt = 'T'; } else { *(unsigned char *)transn = 'T'; *(unsigned char *)transt = 'N'; } /* 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 - op(A) * X, where op(A) = A, A**T, or A**H, depending on TRANS. */ dcopy_(n, &b_ref(1, j), &c__1, &work[*n + 1], &c__1); dlagtm_(trans, n, &c__1, &c_b18, &dl[1], &d__[1], &du[1], &x_ref(1, j) , ldx, &c_b19, &work[*n + 1], n); /* Compute abs(op(A))*abs(x) + abs(b) for use in the backward error bound. */ if (notran) { if (*n == 1) { work[1] = (d__1 = b_ref(1, j), abs(d__1)) + (d__2 = d__[1] * x_ref(1, j), abs(d__2)); } else { work[1] = (d__1 = b_ref(1, j), abs(d__1)) + (d__2 = d__[1] * x_ref(1, j), abs(d__2)) + (d__3 = du[1] * x_ref(2, j), abs(d__3)); i__2 = *n - 1; for (i__ = 2; i__ <= i__2; ++i__) { work[i__] = (d__1 = b_ref(i__, j), abs(d__1)) + (d__2 = dl[i__ - 1] * x_ref(i__ - 1, j), abs(d__2)) + ( d__3 = d__[i__] * x_ref(i__, j), abs(d__3)) + ( d__4 = du[i__] * x_ref(i__ + 1, j), abs(d__4)); /* L30: */ } work[*n] = (d__1 = b_ref(*n, j), abs(d__1)) + (d__2 = dl[*n - 1] * x_ref(*n - 1, j), abs(d__2)) + (d__3 = d__[*n] * x_ref(*n, j), abs(d__3)); } } else { if (*n == 1) { work[1] = (d__1 = b_ref(1, j), abs(d__1)) + (d__2 = d__[1] * x_ref(1, j), abs(d__2)); } else { work[1] = (d__1 = b_ref(1, j), abs(d__1)) + (d__2 = d__[1] * x_ref(1, j), abs(d__2)) + (d__3 = dl[1] * x_ref(2, j), abs(d__3)); i__2 = *n - 1; for (i__ = 2; i__ <= i__2; ++i__) { work[i__] = (d__1 = b_ref(i__, j), abs(d__1)) + (d__2 = du[i__ - 1] * x_ref(i__ - 1, j), abs(d__2)) + ( d__3 = d__[i__] * x_ref(i__, j), abs(d__3)) + ( d__4 = dl[i__] * x_ref(i__ + 1, j), abs(d__4)); /* L40: */ } work[*n] = (d__1 = b_ref(*n, j), abs(d__1)) + (d__2 = du[*n - 1] * x_ref(*n - 1, j), abs(d__2)) + (d__3 = d__[*n] * x_ref(*n, j), abs(d__3)); } } /* Compute componentwise relative backward error from formula max(i) ( abs(R(i)) / ( abs(op(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); } /* L50: */ } 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. */ dgttrs_(trans, n, &c__1, &dlf[1], &df[1], &duf[1], &du2[1], &ipiv[ 1], &work[*n + 1], n, info); daxpy_(n, &c_b19, &work[*n + 1], &c__1, &x_ref(1, j), &c__1); lstres = berr[j]; ++count; goto L20; } /* Bound error from formula norm(X - XTRUE) / norm(X) .le. FERR = norm( abs(inv(op(A)))* ( abs(R) + NZ*EPS*( abs(op(A))*abs(X)+abs(B) ))) / norm(X) where norm(Z) is the magnitude of the largest component of Z inv(op(A)) is the inverse of op(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(op(A))*abs(X)+abs(B)) is incremented by SAFE1 if the i-th component of abs(op(A))*abs(X) + abs(B) is less than SAFE2. Use DLACON to estimate the infinity-norm of the matrix inv(op(A)) * diag(W), where W = abs(R) + NZ*EPS*( abs(op(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; } /* L60: */ } kase = 0; L70: dlacon_(n, &work[(*n << 1) + 1], &work[*n + 1], &iwork[1], &ferr[j], & kase); if (kase != 0) { if (kase == 1) { /* Multiply by diag(W)*inv(op(A)**T). */ dgttrs_(transt, n, &c__1, &dlf[1], &df[1], &duf[1], &du2[1], & ipiv[1], &work[*n + 1], n, info); i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { work[*n + i__] = work[i__] * work[*n + i__]; /* L80: */ } } else { /* Multiply by inv(op(A))*diag(W). */ i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { work[*n + i__] = work[i__] * work[*n + i__]; /* L90: */ } dgttrs_(transn, n, &c__1, &dlf[1], &df[1], &duf[1], &du2[1], & ipiv[1], &work[*n + 1], n, info); } goto L70; } /* Normalize error. */ lstres = 0.; i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { /* Computing MAX */ d__2 = lstres, d__3 = (d__1 = x_ref(i__, j), abs(d__1)); lstres = max(d__2,d__3); /* L100: */ } if (lstres != 0.) { ferr[j] /= lstres; } /* L110: */ } return 0; /* End of DGTRFS */ } /* dgtrfs_ */ #undef x_ref #undef b_ref