#include "blaswrap.h" #include "f2c.h" /* Subroutine */ int csprfs_(char *uplo, integer *n, integer *nrhs, complex * ap, complex *afp, integer *ipiv, 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 ======= CSPRFS improves the computed solution to a system of linear equations when the coefficient matrix is symmetric indefinite and packed, 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. 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 triangle of the symmetric 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)*(2*n-j)/2) = A(i,j) for j<=i<=n. AFP (input) COMPLEX array, dimension (N*(N+1)/2) The factored form of the matrix A. AFP contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by CSPTRF, stored as a packed triangular matrix. IPIV (input) INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by CSPTRF. 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/output) COMPLEX array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by CSPTRS. On exit, the improved 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 Internal Parameters =================== ITMAX is the maximum number of steps of iterative refinement. ===================================================================== Test the input parameters. Parameter adjustments */ /* Table of constant values */ static complex c_b1 = {1.f,0.f}; 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 *); static integer count; extern /* Subroutine */ int cspmv_(char *, integer *, complex *, complex * , complex *, integer *, complex *, complex *, integer *); static logical upper; static integer ik, kk; 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 real lstres; extern /* Subroutine */ int csptrs_(char *, integer *, integer *, complex *, integer *, complex *, integer *, integer *); static real 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; --afp; --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; --rwork; /* Function Body */ *info = 0; upper = lsame_(uplo, "U"); if (! upper && ! lsame_(uplo, "L")) { *info = -1; } else if (*n < 0) { *info = -2; } else if (*nrhs < 0) { *info = -3; } else if (*ldb < max(1,*n)) { *info = -8; } else if (*ldx < max(1,*n)) { *info = -10; } if (*info != 0) { i__1 = -(*info); xerbla_("CSPRFS", &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; } /* 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) { count = 1; lstres = 3.f; L20: /* Loop until stopping criterion is satisfied. Compute residual R = B - A * X */ ccopy_(n, &b_ref(1, j), &c__1, &work[1], &c__1); q__1.r = -1.f, q__1.i = 0.f; cspmv_(uplo, n, &q__1, &ap[1], &x_ref(1, j), &c__1, &c_b1, &work[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__) { 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)); /* L30: */ } /* Compute abs(A)*abs(X) + abs(B). */ kk = 1; if (upper) { i__2 = *n; for (k = 1; k <= i__2; ++k) { s = 0.f; 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)); ik = kk; i__3 = k - 1; for (i__ = 1; i__ <= i__3; ++i__) { i__4 = ik; rwork[i__] += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 = r_imag(&ap[ik]), dabs(r__2))) * xk; i__4 = ik; i__5 = x_subscr(i__, j); s += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 = r_imag(& ap[ik]), dabs(r__2))) * ((r__3 = x[i__5].r, dabs( r__3)) + (r__4 = r_imag(&x_ref(i__, j)), dabs( r__4))); ++ik; /* L40: */ } i__3 = kk + k - 1; rwork[k] = rwork[k] + ((r__1 = ap[i__3].r, dabs(r__1)) + ( r__2 = r_imag(&ap[kk + k - 1]), dabs(r__2))) * xk + s; kk += k; /* L50: */ } } else { i__2 = *n; for (k = 1; k <= i__2; ++k) { s = 0.f; 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 = kk; rwork[k] += ((r__1 = ap[i__3].r, dabs(r__1)) + (r__2 = r_imag( &ap[kk]), dabs(r__2))) * xk; ik = kk + 1; i__3 = *n; for (i__ = k + 1; i__ <= i__3; ++i__) { i__4 = ik; rwork[i__] += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 = r_imag(&ap[ik]), dabs(r__2))) * xk; i__4 = ik; i__5 = x_subscr(i__, j); s += ((r__1 = ap[i__4].r, dabs(r__1)) + (r__2 = r_imag(& ap[ik]), dabs(r__2))) * ((r__3 = x[i__5].r, dabs( r__3)) + (r__4 = r_imag(&x_ref(i__, j)), dabs( r__4))); ++ik; /* L60: */ } rwork[k] += s; kk += *n - k + 1; /* L70: */ } } 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); } /* 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.f <= lstres && count <= 5) { /* Update solution and try again. */ csptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[1], n, info); caxpy_(n, &c_b1, &work[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(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 CLACON 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 (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; } /* L90: */ } kase = 0; L100: clacon_(n, &work[*n + 1], &work[1], &ferr[j], &kase); if (kase != 0) { if (kase == 1) { /* Multiply by diag(W)*inv(A'). */ csptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[1], n, info); 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; /* L110: */ } } else if (kase == 2) { /* Multiply by inv(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; /* L120: */ } csptrs_(uplo, n, &c__1, &afp[1], &ipiv[1], &work[1], n, info); } goto L100; } /* 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); /* L130: */ } if (lstres != 0.f) { ferr[j] /= lstres; } /* L140: */ } return 0; /* End of CSPRFS */ } /* csprfs_ */ #undef x_ref #undef x_subscr #undef b_ref #undef b_subscr