#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int ctprfs_(char *uplo, char *trans, char *diag, integer *n, integer *nrhs, complex *ap, complex *b, integer *ldb, complex *x, integer *ldx, real *ferr, real *berr, complex *work, real *rwork, 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 ======= CTPRFS provides error bounds and backward error estimates for the solution to a system of linear equations with a triangular packed coefficient matrix. The solution matrix X must be computed by CTPTRS or some other means before entering this routine. CTPRFS does not do iterative refinement because doing so cannot improve the backward error. Arguments ========= UPLO (input) CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular. 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) DIAG (input) CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular. 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 matrices B and X. NRHS >= 0. AP (input) COMPLEX array, dimension (N*(N+1)/2) The upper or lower triangular matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. If DIAG = 'U', the diagonal elements of A are not referenced and are assumed to be 1. B (input) COMPLEX 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) COMPLEX array, dimension (LDX,NRHS) The solution matrix X. LDX (input) INTEGER The leading dimension of the array X. LDX >= max(1,N). 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) REAL array, dimension (N) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value ===================================================================== Test the input parameters. Parameter adjustments */ /* Table of constant values */ static integer c__1 = 1; /* System generated locals */ integer b_dim1, b_offset, x_dim1, x_offset, i__1, i__2, i__3, i__4, i__5; real r__1, r__2, r__3, r__4; complex q__1; /* Builtin functions */ double r_imag(complex *); /* Local variables */ static integer kase; static real safe1, safe2; static integer i__, j, k; static real s; extern logical lsame_(char *, char *); extern /* Subroutine */ int ccopy_(integer *, complex *, integer *, complex *, integer *), caxpy_(integer *, complex *, complex *, integer *, complex *, integer *), ctpmv_(char *, char *, char *, integer *, complex *, complex *, integer *); static logical upper; extern /* Subroutine */ int ctpsv_(char *, char *, char *, integer *, complex *, complex *, integer *); static integer kc; extern /* Subroutine */ int clacon_(integer *, complex *, complex *, real *, integer *); static real xk; extern doublereal slamch_(char *); static integer nz; static real safmin; extern /* Subroutine */ int xerbla_(char *, integer *); static logical notran; static char transn[1], transt[1]; static logical nounit; static real lstres, eps; #define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1 #define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)] #define x_subscr(a_1,a_2) (a_2)*x_dim1 + a_1 #define x_ref(a_1,a_2) x[x_subscr(a_1,a_2)] --ap; 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; --rwork; /* Function Body */ *info = 0; upper = lsame_(uplo, "U"); notran = lsame_(trans, "N"); nounit = lsame_(diag, "N"); if (! upper && ! lsame_(uplo, "L")) { *info = -1; } else if (! notran && ! lsame_(trans, "T") && ! lsame_(trans, "C")) { *info = -2; } else if (! nounit && ! lsame_(diag, "U")) { *info = -3; } else if (*n < 0) { *info = -4; } else if (*nrhs < 0) { *info = -5; } else if (*ldb < max(1,*n)) { *info = -8; } else if (*ldx < max(1,*n)) { *info = -10; } if (*info != 0) { i__1 = -(*info); xerbla_("CTPRFS", &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.f; berr[j] = 0.f; /* L10: */ } return 0; } if (notran) { *(unsigned char *)transn = 'N'; *(unsigned char *)transt = 'C'; } else { *(unsigned char *)transn = 'C'; *(unsigned char *)transt = 'N'; } /* NZ = maximum number of nonzero elements in each row of A, plus 1 */ nz = *n + 1; eps = slamch_("Epsilon"); safmin = slamch_("Safe minimum"); safe1 = nz * safmin; safe2 = safe1 / eps; /* Do for each right hand side */ i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { /* Compute residual R = B - op(A) * X, where op(A) = A, A**T, or A**H, depending on TRANS. */ ccopy_(n, &x_ref(1, j), &c__1, &work[1], &c__1); ctpmv_(uplo, trans, diag, n, &ap[1], &work[1], &c__1); q__1.r = -1.f, q__1.i = 0.f; caxpy_(n, &q__1, &b_ref(1, j), &c__1, &work[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__) { i__3 = b_subscr(i__, j); rwork[i__] = (r__1 = b[i__3].r, dabs(r__1)) + (r__2 = r_imag(& b_ref(i__, j)), dabs(r__2)); /* L20: */ } if (notran) { /* Compute abs(A)*abs(X) + abs(B). */ if (upper) { kc = 1; if (nounit) { i__2 = *n; for (k = 1; k <= i__2; ++k) { i__3 = x_subscr(k, j); xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(& x_ref(k, j)), dabs(r__2)); i__3 = k; for (i__ = 1; i__ <= i__3; ++i__) { i__4 = kc + i__ - 1; rwork[i__] += ((r__1 = ap[i__4].r, dabs(r__1)) + ( r__2 = r_imag(&ap[kc + i__ - 1]), dabs( r__2))) * xk; /* L30: */ } kc += k; /* L40: */ } } else { i__2 = *n; for (k = 1; k <= i__2; ++k) { i__3 = x_subscr(k, j); xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(& x_ref(k, j)), dabs(r__2)); i__3 = k - 1; for (i__ = 1; i__ <= i__3; ++i__) { i__4 = kc + i__ - 1; rwork[i__] += ((r__1 = ap[i__4].r, dabs(r__1)) + ( r__2 = r_imag(&ap[kc + i__ - 1]), dabs( r__2))) * xk; /* L50: */ } rwork[k] += xk; kc += k; /* L60: */ } } } else { kc = 1; if (nounit) { i__2 = *n; for (k = 1; k <= i__2; ++k) { i__3 = x_subscr(k, j); xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(& x_ref(k, j)), dabs(r__2)); i__3 = *n; for (i__ = k; i__ <= i__3; ++i__) { i__4 = kc + i__ - k; rwork[i__] += ((r__1 = ap[i__4].r, dabs(r__1)) + ( r__2 = r_imag(&ap[kc + i__ - k]), dabs( r__2))) * xk; /* L70: */ } kc = kc + *n - k + 1; /* L80: */ } } else { i__2 = *n; for (k = 1; k <= i__2; ++k) { i__3 = x_subscr(k, j); xk = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(& x_ref(k, j)), dabs(r__2)); i__3 = *n; for (i__ = k + 1; i__ <= i__3; ++i__) { i__4 = kc + i__ - k; rwork[i__] += ((r__1 = ap[i__4].r, dabs(r__1)) + ( r__2 = r_imag(&ap[kc + i__ - k]), dabs( r__2))) * xk; /* L90: */ } rwork[k] += xk; kc = kc + *n - k + 1; /* L100: */ } } } } else { /* Compute abs(A**H)*abs(X) + abs(B). */ if (upper) { kc = 1; if (nounit) { i__2 = *n; for (k = 1; k <= i__2; ++k) { s = 0.f; i__3 = k; for (i__ = 1; i__ <= i__3; ++i__) { i__4 = kc + i__ - 1; i__5 = x_subscr(i__, j); s += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 = r_imag(&ap[kc + i__ - 1]), dabs(r__2))) * ((r__3 = x[i__5].r, dabs(r__3)) + (r__4 = r_imag(&x_ref(i__, j)), dabs(r__4))); /* L110: */ } rwork[k] += s; kc += k; /* L120: */ } } else { i__2 = *n; for (k = 1; k <= i__2; ++k) { i__3 = x_subscr(k, j); s = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(& x_ref(k, j)), dabs(r__2)); i__3 = k - 1; for (i__ = 1; i__ <= i__3; ++i__) { i__4 = kc + i__ - 1; i__5 = x_subscr(i__, j); s += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 = r_imag(&ap[kc + i__ - 1]), dabs(r__2))) * ((r__3 = x[i__5].r, dabs(r__3)) + (r__4 = r_imag(&x_ref(i__, j)), dabs(r__4))); /* L130: */ } rwork[k] += s; kc += k; /* L140: */ } } } else { kc = 1; if (nounit) { i__2 = *n; for (k = 1; k <= i__2; ++k) { s = 0.f; i__3 = *n; for (i__ = k; i__ <= i__3; ++i__) { i__4 = kc + i__ - k; i__5 = x_subscr(i__, j); s += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 = r_imag(&ap[kc + i__ - k]), dabs(r__2))) * ((r__3 = x[i__5].r, dabs(r__3)) + (r__4 = r_imag(&x_ref(i__, j)), dabs(r__4))); /* L150: */ } rwork[k] += s; kc = kc + *n - k + 1; /* L160: */ } } else { i__2 = *n; for (k = 1; k <= i__2; ++k) { i__3 = x_subscr(k, j); s = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(& x_ref(k, j)), dabs(r__2)); i__3 = *n; for (i__ = k + 1; i__ <= i__3; ++i__) { i__4 = kc + i__ - k; i__5 = x_subscr(i__, j); s += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 = r_imag(&ap[kc + i__ - k]), dabs(r__2))) * ((r__3 = x[i__5].r, dabs(r__3)) + (r__4 = r_imag(&x_ref(i__, j)), dabs(r__4))); /* L170: */ } rwork[k] += s; kc = kc + *n - k + 1; /* L180: */ } } } } s = 0.f; i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { if (rwork[i__] > safe2) { /* Computing MAX */ i__3 = i__; r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 = r_imag(&work[i__]), dabs(r__2))) / rwork[i__]; s = dmax(r__3,r__4); } else { /* Computing MAX */ i__3 = i__; r__3 = s, r__4 = ((r__1 = work[i__3].r, dabs(r__1)) + (r__2 = r_imag(&work[i__]), dabs(r__2)) + safe1) / (rwork[i__] + safe1); s = dmax(r__3,r__4); } /* L190: */ } berr[j] = s; /* 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 CLACON 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 (rwork[i__] > safe2) { i__3 = i__; rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 = r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[ i__]; } else { i__3 = i__; rwork[i__] = (r__1 = work[i__3].r, dabs(r__1)) + (r__2 = r_imag(&work[i__]), dabs(r__2)) + nz * eps * rwork[ i__] + safe1; } /* L200: */ } kase = 0; L210: clacon_(n, &work[*n + 1], &work[1], &ferr[j], &kase); if (kase != 0) { if (kase == 1) { /* Multiply by diag(W)*inv(op(A)**H). */ ctpsv_(uplo, transt, diag, n, &ap[1], &work[1], &c__1); i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { i__3 = i__; i__4 = i__; i__5 = i__; q__1.r = rwork[i__4] * work[i__5].r, q__1.i = rwork[i__4] * work[i__5].i; work[i__3].r = q__1.r, work[i__3].i = q__1.i; /* L220: */ } } else { /* Multiply by inv(op(A))*diag(W). */ i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { i__3 = i__; i__4 = i__; i__5 = i__; q__1.r = rwork[i__4] * work[i__5].r, q__1.i = rwork[i__4] * work[i__5].i; work[i__3].r = q__1.r, work[i__3].i = q__1.i; /* L230: */ } ctpsv_(uplo, transn, diag, n, &ap[1], &work[1], &c__1); } goto L210; } /* Normalize error. */ lstres = 0.f; i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { /* Computing MAX */ i__3 = x_subscr(i__, j); r__3 = lstres, r__4 = (r__1 = x[i__3].r, dabs(r__1)) + (r__2 = r_imag(&x_ref(i__, j)), dabs(r__2)); lstres = dmax(r__3,r__4); /* L240: */ } if (lstres != 0.f) { ferr[j] /= lstres; } /* L250: */ } return 0; /* End of CTPRFS */ } /* ctprfs_ */ #undef x_ref #undef x_subscr #undef b_ref #undef b_subscr