#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int dgbrfs_(char *trans, integer *n, integer *kl, integer * ku, integer *nrhs, doublereal *ab, integer *ldab, doublereal *afb, integer *ldafb, 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 ======= DGBRFS improves the computed solution to a system of linear equations when the coefficient matrix is banded, 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. KL (input) INTEGER The number of subdiagonals within the band of A. KL >= 0. KU (input) INTEGER The number of superdiagonals within the band of A. KU >= 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 original band matrix A, stored in rows 1 to KL+KU+1. The j-th column of A is stored in the j-th column of the array AB as follows: AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). LDAB (input) INTEGER The leading dimension of the array AB. LDAB >= KL+KU+1. AFB (input) DOUBLE PRECISION array, dimension (LDAFB,N) Details of the LU factorization of the band matrix A, as computed by DGBTRF. U is stored as an upper triangular band matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and the multipliers used during the factorization are stored in rows KL+KU+2 to 2*KL+KU+1. LDAFB (input) INTEGER The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. IPIV (input) INTEGER array, dimension (N) The pivot indices from DGBTRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i). 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 DGBTRS. 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_b15 = -1.; static doublereal c_b17 = 1.; /* 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, i__6, i__7; doublereal d__1, d__2, d__3; /* Local variables */ static integer kase; static doublereal safe1, safe2; static integer i__, j, k; static doublereal s; extern /* Subroutine */ int dgbmv_(char *, integer *, integer *, integer * , integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *); extern logical lsame_(char *, char *); extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, doublereal *, integer *), daxpy_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *); static integer count, kk; extern doublereal dlamch_(char *); extern /* Subroutine */ int dlacon_(integer *, doublereal *, doublereal *, integer *, doublereal *, integer *); static doublereal xk; static integer nz; static doublereal safmin; extern /* Subroutine */ int xerbla_(char *, integer *), dgbtrs_( char *, integer *, integer *, integer *, integer *, doublereal *, integer *, integer *, doublereal *, integer *, integer *); static logical notran; 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] #define ab_ref(a_1,a_2) ab[(a_2)*ab_dim1 + a_1] ab_dim1 = *ldab; ab_offset = 1 + ab_dim1 * 1; ab -= ab_offset; afb_dim1 = *ldafb; afb_offset = 1 + afb_dim1 * 1; afb -= afb_offset; --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 (*kl < 0) { *info = -3; } else if (*ku < 0) { *info = -4; } else if (*nrhs < 0) { *info = -5; } else if (*ldab < *kl + *ku + 1) { *info = -7; } else if (*ldafb < (*kl << 1) + *ku + 1) { *info = -9; } else if (*ldb < max(1,*n)) { *info = -12; } else if (*ldx < max(1,*n)) { *info = -14; } if (*info != 0) { i__1 = -(*info); xerbla_("DGBRFS", &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 *)transt = 'T'; } else { *(unsigned char *)transt = 'N'; } /* NZ = maximum number of nonzero elements in each row of A, plus 1 Computing MIN */ i__1 = *kl + *ku + 2, i__2 = *n + 1; 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 - 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); dgbmv_(trans, n, n, kl, ku, &c_b15, &ab[ab_offset], ldab, &x_ref(1, j) , &c__1, &c_b17, &work[*n + 1], &c__1); /* 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. */ i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { work[i__] = (d__1 = b_ref(i__, j), abs(d__1)); /* L30: */ } /* Compute abs(op(A))*abs(X) + abs(B). */ if (notran) { i__2 = *n; for (k = 1; k <= i__2; ++k) { kk = *ku + 1 - k; xk = (d__1 = x_ref(k, j), abs(d__1)); /* Computing MAX */ i__3 = 1, i__4 = k - *ku; /* Computing MIN */ i__6 = *n, i__7 = k + *kl; i__5 = min(i__6,i__7); for (i__ = max(i__3,i__4); i__ <= i__5; ++i__) { work[i__] += (d__1 = ab_ref(kk + i__, k), abs(d__1)) * xk; /* L40: */ } /* L50: */ } } else { i__2 = *n; for (k = 1; k <= i__2; ++k) { s = 0.; kk = *ku + 1 - k; /* Computing MAX */ i__5 = 1, i__3 = k - *ku; /* Computing MIN */ i__6 = *n, i__7 = k + *kl; i__4 = min(i__6,i__7); for (i__ = max(i__5,i__3); i__ <= i__4; ++i__) { s += (d__1 = ab_ref(kk + i__, k), abs(d__1)) * (d__2 = x_ref(i__, j), 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. */ dgbtrs_(trans, n, kl, ku, &c__1, &afb[afb_offset], ldafb, &ipiv[1] , &work[*n + 1], n, info); daxpy_(n, &c_b17, &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; } /* L90: */ } kase = 0; L100: 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). */ dgbtrs_(transt, n, kl, ku, &c__1, &afb[afb_offset], ldafb, & ipiv[1], &work[*n + 1], n, info); i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { work[*n + i__] *= work[i__]; /* L110: */ } } else { /* Multiply by inv(op(A))*diag(W). */ i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { work[*n + i__] *= work[i__]; /* L120: */ } dgbtrs_(trans, n, kl, ku, &c__1, &afb[afb_offset], ldafb, & ipiv[1], &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_ref(i__, j), abs(d__1)); lstres = max(d__2,d__3); /* L130: */ } if (lstres != 0.) { ferr[j] /= lstres; } /* L140: */ } return 0; /* End of DGBRFS */ } /* dgbrfs_ */ #undef ab_ref #undef x_ref #undef b_ref