/* sla_syrfsx_extended.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 real c_b9 = -1.f; static real c_b11 = 1.f; /* Subroutine */ int sla_syrfsx_extended__(integer *prec_type__, char *uplo, integer *n, integer *nrhs, real *a, integer *lda, real *af, integer * ldaf, integer *ipiv, logical *colequ, real *c__, real *b, integer * ldb, real *y, integer *ldy, real *berr_out__, integer *n_norms__, real *err_bnds_norm__, real *err_bnds_comp__, real *res, real *ayb, real *dy, real *y_tail__, real *rcond, integer *ithresh, real * rthresh, real *dz_ub__, logical *ignore_cwise__, integer *info, ftnlen uplo_len) { /* System generated locals */ integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, y_dim1, y_offset, err_bnds_norm_dim1, err_bnds_norm_offset, err_bnds_comp_dim1, err_bnds_comp_offset, i__1, i__2, i__3; real r__1, r__2; /* Local variables */ real dxratmax, dzratmax; integer i__, j; logical incr_prec__; extern /* Subroutine */ int sla_syamv__(integer *, integer *, real *, real *, integer *, real *, integer *, real *, real *, integer *); real prev_dz_z__, yk, final_dx_x__, final_dz_z__; extern /* Subroutine */ int sla_wwaddw__(integer *, real *, real *, real * ); real prevnormdx; integer cnt; real dyk, eps, incr_thresh__, dx_x__, dz_z__, ymin; extern /* Subroutine */ int sla_lin_berr__(integer *, integer *, integer * , real *, real *, real *); integer y_prec_state__, uplo2; extern /* Subroutine */ int blas_ssymv_x__(integer *, integer *, real *, real *, integer *, real *, integer *, real *, real *, integer *, integer *); extern logical lsame_(char *, char *); real dxrat, dzrat; extern /* Subroutine */ int blas_ssymv2_x__(integer *, integer *, real *, real *, integer *, real *, real *, integer *, real *, real *, integer *, integer *), scopy_(integer *, real *, integer *, real * , integer *); real normx, normy; extern /* Subroutine */ int saxpy_(integer *, real *, real *, integer *, real *, integer *), ssymv_(char *, integer *, real *, real *, integer *, real *, integer *, real *, real *, integer *); extern doublereal slamch_(char *); real normdx; extern /* Subroutine */ int ssytrs_(char *, integer *, integer *, real *, integer *, integer *, real *, integer *, integer *); real hugeval; extern integer ilauplo_(char *); integer x_state__, z_state__; /* -- LAPACK routine (version 3.2.1) -- */ /* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- */ /* -- Jason Riedy of Univ. of California Berkeley. -- */ /* -- April 2009 -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley and NAG Ltd. -- */ /* .. */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* SLA_SYRFSX_EXTENDED improves the computed solution to a system of */ /* linear equations by performing extra-precise iterative refinement */ /* and provides error bounds and backward error estimates for the solution. */ /* This subroutine is called by SSYRFSX to perform iterative refinement. */ /* In addition to normwise error bound, the code provides maximum */ /* componentwise error bound if possible. See comments for ERR_BNDS_NORM */ /* and ERR_BNDS_COMP for details of the error bounds. Note that this */ /* subroutine is only resonsible for setting the second fields of */ /* ERR_BNDS_NORM and ERR_BNDS_COMP. */ /* Arguments */ /* ========= */ /* PREC_TYPE (input) INTEGER */ /* Specifies the intermediate precision to be used in refinement. */ /* The value is defined by ILAPREC(P) where P is a CHARACTER and */ /* P = 'S': Single */ /* = 'D': Double */ /* = 'I': Indigenous */ /* = 'X', 'E': Extra */ /* UPLO (input) CHARACTER*1 */ /* = 'U': Upper triangle of A is stored; */ /* = 'L': Lower triangle of A is stored. */ /* N (input) INTEGER */ /* The number of linear equations, i.e., 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. */ /* A (input) REAL array, dimension (LDA,N) */ /* On entry, the N-by-N matrix A. */ /* LDA (input) INTEGER */ /* The leading dimension of the array A. LDA >= max(1,N). */ /* AF (input) REAL array, dimension (LDAF,N) */ /* The block diagonal matrix D and the multipliers used to */ /* obtain the factor U or L as computed by SSYTRF. */ /* LDAF (input) INTEGER */ /* The leading dimension of the array AF. LDAF >= max(1,N). */ /* IPIV (input) INTEGER array, dimension (N) */ /* Details of the interchanges and the block structure of D */ /* as determined by SSYTRF. */ /* COLEQU (input) LOGICAL */ /* If .TRUE. then column equilibration was done to A before calling */ /* this routine. This is needed to compute the solution and error */ /* bounds correctly. */ /* C (input) REAL array, dimension (N) */ /* The column scale factors for A. If COLEQU = .FALSE., C */ /* is not accessed. If C is input, each element of C should be a power */ /* of the radix to ensure a reliable solution and error estimates. */ /* Scaling by powers of the radix does not cause rounding errors unless */ /* the result underflows or overflows. Rounding errors during scaling */ /* lead to refining with a matrix that is not equivalent to the */ /* input matrix, producing error estimates that may not be */ /* reliable. */ /* B (input) REAL array, dimension (LDB,NRHS) */ /* The right-hand-side matrix B. */ /* LDB (input) INTEGER */ /* The leading dimension of the array B. LDB >= max(1,N). */ /* Y (input/output) REAL array, dimension (LDY,NRHS) */ /* On entry, the solution matrix X, as computed by SSYTRS. */ /* On exit, the improved solution matrix Y. */ /* LDY (input) INTEGER */ /* The leading dimension of the array Y. LDY >= max(1,N). */ /* BERR_OUT (output) REAL array, dimension (NRHS) */ /* On exit, BERR_OUT(j) contains the componentwise relative backward */ /* error for right-hand-side j from the formula */ /* max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */ /* where abs(Z) is the componentwise absolute value of the matrix */ /* or vector Z. This is computed by SLA_LIN_BERR. */ /* N_NORMS (input) INTEGER */ /* Determines which error bounds to return (see ERR_BNDS_NORM */ /* and ERR_BNDS_COMP). */ /* If N_NORMS >= 1 return normwise error bounds. */ /* If N_NORMS >= 2 return componentwise error bounds. */ /* ERR_BNDS_NORM (input/output) REAL array, dimension (NRHS, N_ERR_BNDS) */ /* For each right-hand side, this array contains information about */ /* various error bounds and condition numbers corresponding to the */ /* normwise relative error, which is defined as follows: */ /* Normwise relative error in the ith solution vector: */ /* max_j (abs(XTRUE(j,i) - X(j,i))) */ /* ------------------------------ */ /* max_j abs(X(j,i)) */ /* The array is indexed by the type of error information as described */ /* below. There currently are up to three pieces of information */ /* returned. */ /* The first index in ERR_BNDS_NORM(i,:) corresponds to the ith */ /* right-hand side. */ /* The second index in ERR_BNDS_NORM(:,err) contains the following */ /* three fields: */ /* err = 1 "Trust/don't trust" boolean. Trust the answer if the */ /* reciprocal condition number is less than the threshold */ /* sqrt(n) * slamch('Epsilon'). */ /* err = 2 "Guaranteed" error bound: The estimated forward error, */ /* almost certainly within a factor of 10 of the true error */ /* so long as the next entry is greater than the threshold */ /* sqrt(n) * slamch('Epsilon'). This error bound should only */ /* be trusted if the previous boolean is true. */ /* err = 3 Reciprocal condition number: Estimated normwise */ /* reciprocal condition number. Compared with the threshold */ /* sqrt(n) * slamch('Epsilon') to determine if the error */ /* estimate is "guaranteed". These reciprocal condition */ /* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */ /* appropriately scaled matrix Z. */ /* Let Z = S*A, where S scales each row by a power of the */ /* radix so all absolute row sums of Z are approximately 1. */ /* This subroutine is only responsible for setting the second field */ /* above. */ /* See Lapack Working Note 165 for further details and extra */ /* cautions. */ /* ERR_BNDS_COMP (input/output) REAL array, dimension (NRHS, N_ERR_BNDS) */ /* For each right-hand side, this array contains information about */ /* various error bounds and condition numbers corresponding to the */ /* componentwise relative error, which is defined as follows: */ /* Componentwise relative error in the ith solution vector: */ /* abs(XTRUE(j,i) - X(j,i)) */ /* max_j ---------------------- */ /* abs(X(j,i)) */ /* The array is indexed by the right-hand side i (on which the */ /* componentwise relative error depends), and the type of error */ /* information as described below. There currently are up to three */ /* pieces of information returned for each right-hand side. If */ /* componentwise accuracy is not requested (PARAMS(3) = 0.0), then */ /* ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most */ /* the first (:,N_ERR_BNDS) entries are returned. */ /* The first index in ERR_BNDS_COMP(i,:) corresponds to the ith */ /* right-hand side. */ /* The second index in ERR_BNDS_COMP(:,err) contains the following */ /* three fields: */ /* err = 1 "Trust/don't trust" boolean. Trust the answer if the */ /* reciprocal condition number is less than the threshold */ /* sqrt(n) * slamch('Epsilon'). */ /* err = 2 "Guaranteed" error bound: The estimated forward error, */ /* almost certainly within a factor of 10 of the true error */ /* so long as the next entry is greater than the threshold */ /* sqrt(n) * slamch('Epsilon'). This error bound should only */ /* be trusted if the previous boolean is true. */ /* err = 3 Reciprocal condition number: Estimated componentwise */ /* reciprocal condition number. Compared with the threshold */ /* sqrt(n) * slamch('Epsilon') to determine if the error */ /* estimate is "guaranteed". These reciprocal condition */ /* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some */ /* appropriately scaled matrix Z. */ /* Let Z = S*(A*diag(x)), where x is the solution for the */ /* current right-hand side and S scales each row of */ /* A*diag(x) by a power of the radix so all absolute row */ /* sums of Z are approximately 1. */ /* This subroutine is only responsible for setting the second field */ /* above. */ /* See Lapack Working Note 165 for further details and extra */ /* cautions. */ /* RES (input) REAL array, dimension (N) */ /* Workspace to hold the intermediate residual. */ /* AYB (input) REAL array, dimension (N) */ /* Workspace. This can be the same workspace passed for Y_TAIL. */ /* DY (input) REAL array, dimension (N) */ /* Workspace to hold the intermediate solution. */ /* Y_TAIL (input) REAL array, dimension (N) */ /* Workspace to hold the trailing bits of the intermediate solution. */ /* RCOND (input) REAL */ /* Reciprocal scaled condition number. This is an estimate of the */ /* reciprocal Skeel condition number of the matrix A after */ /* equilibration (if done). If this is less than the machine */ /* precision (in particular, if it is zero), the matrix is singular */ /* to working precision. Note that the error may still be small even */ /* if this number is very small and the matrix appears ill- */ /* conditioned. */ /* ITHRESH (input) INTEGER */ /* The maximum number of residual computations allowed for */ /* refinement. The default is 10. For 'aggressive' set to 100 to */ /* permit convergence using approximate factorizations or */ /* factorizations other than LU. If the factorization uses a */ /* technique other than Gaussian elimination, the guarantees in */ /* ERR_BNDS_NORM and ERR_BNDS_COMP may no longer be trustworthy. */ /* RTHRESH (input) REAL */ /* Determines when to stop refinement if the error estimate stops */ /* decreasing. Refinement will stop when the next solution no longer */ /* satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is */ /* the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The */ /* default value is 0.5. For 'aggressive' set to 0.9 to permit */ /* convergence on extremely ill-conditioned matrices. See LAWN 165 */ /* for more details. */ /* DZ_UB (input) REAL */ /* Determines when to start considering componentwise convergence. */ /* Componentwise convergence is only considered after each component */ /* of the solution Y is stable, which we definte as the relative */ /* change in each component being less than DZ_UB. The default value */ /* is 0.25, requiring the first bit to be stable. See LAWN 165 for */ /* more details. */ /* IGNORE_CWISE (input) LOGICAL */ /* If .TRUE. then ignore componentwise convergence. Default value */ /* is .FALSE.. */ /* INFO (output) INTEGER */ /* = 0: Successful exit. */ /* < 0: if INFO = -i, the ith argument to SSYTRS had an illegal */ /* value */ /* ===================================================================== */ /* .. Local Scalars .. */ /* .. */ /* .. Parameters .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Parameter adjustments */ err_bnds_comp_dim1 = *nrhs; err_bnds_comp_offset = 1 + err_bnds_comp_dim1; err_bnds_comp__ -= err_bnds_comp_offset; err_bnds_norm_dim1 = *nrhs; err_bnds_norm_offset = 1 + err_bnds_norm_dim1; err_bnds_norm__ -= err_bnds_norm_offset; a_dim1 = *lda; a_offset = 1 + a_dim1; a -= a_offset; af_dim1 = *ldaf; af_offset = 1 + af_dim1; af -= af_offset; --ipiv; --c__; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; y_dim1 = *ldy; y_offset = 1 + y_dim1; y -= y_offset; --berr_out__; --res; --ayb; --dy; --y_tail__; /* Function Body */ if (*info != 0) { return 0; } eps = slamch_("Epsilon"); hugeval = slamch_("Overflow"); /* Force HUGEVAL to Inf */ hugeval *= hugeval; /* Using HUGEVAL may lead to spurious underflows. */ incr_thresh__ = (real) (*n) * eps; if (lsame_(uplo, "L")) { uplo2 = ilauplo_("L"); } else { uplo2 = ilauplo_("U"); } i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { y_prec_state__ = 1; if (y_prec_state__ == 2) { i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { y_tail__[i__] = 0.f; } } dxrat = 0.f; dxratmax = 0.f; dzrat = 0.f; dzratmax = 0.f; final_dx_x__ = hugeval; final_dz_z__ = hugeval; prevnormdx = hugeval; prev_dz_z__ = hugeval; dz_z__ = hugeval; dx_x__ = hugeval; x_state__ = 1; z_state__ = 0; incr_prec__ = FALSE_; i__2 = *ithresh; for (cnt = 1; cnt <= i__2; ++cnt) { /* Compute residual RES = B_s - op(A_s) * Y, */ /* op(A) = A, A**T, or A**H depending on TRANS (and type). */ scopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1); if (y_prec_state__ == 0) { ssymv_(uplo, n, &c_b9, &a[a_offset], lda, &y[j * y_dim1 + 1], &c__1, &c_b11, &res[1], &c__1); } else if (y_prec_state__ == 1) { blas_ssymv_x__(&uplo2, n, &c_b9, &a[a_offset], lda, &y[j * y_dim1 + 1], &c__1, &c_b11, &res[1], &c__1, prec_type__); } else { blas_ssymv2_x__(&uplo2, n, &c_b9, &a[a_offset], lda, &y[j * y_dim1 + 1], &y_tail__[1], &c__1, &c_b11, &res[1], & c__1, prec_type__); } /* XXX: RES is no longer needed. */ scopy_(n, &res[1], &c__1, &dy[1], &c__1); ssytrs_(uplo, n, nrhs, &af[af_offset], ldaf, &ipiv[1], &dy[1], n, info); /* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT. */ normx = 0.f; normy = 0.f; normdx = 0.f; dz_z__ = 0.f; ymin = hugeval; i__3 = *n; for (i__ = 1; i__ <= i__3; ++i__) { yk = (r__1 = y[i__ + j * y_dim1], dabs(r__1)); dyk = (r__1 = dy[i__], dabs(r__1)); if (yk != 0.f) { /* Computing MAX */ r__1 = dz_z__, r__2 = dyk / yk; dz_z__ = dmax(r__1,r__2); } else if (dyk != 0.f) { dz_z__ = hugeval; } ymin = dmin(ymin,yk); normy = dmax(normy,yk); if (*colequ) { /* Computing MAX */ r__1 = normx, r__2 = yk * c__[i__]; normx = dmax(r__1,r__2); /* Computing MAX */ r__1 = normdx, r__2 = dyk * c__[i__]; normdx = dmax(r__1,r__2); } else { normx = normy; normdx = dmax(normdx,dyk); } } if (normx != 0.f) { dx_x__ = normdx / normx; } else if (normdx == 0.f) { dx_x__ = 0.f; } else { dx_x__ = hugeval; } dxrat = normdx / prevnormdx; dzrat = dz_z__ / prev_dz_z__; /* Check termination criteria. */ if (ymin * *rcond < incr_thresh__ * normy && y_prec_state__ < 2) { incr_prec__ = TRUE_; } if (x_state__ == 3 && dxrat <= *rthresh) { x_state__ = 1; } if (x_state__ == 1) { if (dx_x__ <= eps) { x_state__ = 2; } else if (dxrat > *rthresh) { if (y_prec_state__ != 2) { incr_prec__ = TRUE_; } else { x_state__ = 3; } } else { if (dxrat > dxratmax) { dxratmax = dxrat; } } if (x_state__ > 1) { final_dx_x__ = dx_x__; } } if (z_state__ == 0 && dz_z__ <= *dz_ub__) { z_state__ = 1; } if (z_state__ == 3 && dzrat <= *rthresh) { z_state__ = 1; } if (z_state__ == 1) { if (dz_z__ <= eps) { z_state__ = 2; } else if (dz_z__ > *dz_ub__) { z_state__ = 0; dzratmax = 0.f; final_dz_z__ = hugeval; } else if (dzrat > *rthresh) { if (y_prec_state__ != 2) { incr_prec__ = TRUE_; } else { z_state__ = 3; } } else { if (dzrat > dzratmax) { dzratmax = dzrat; } } if (z_state__ > 1) { final_dz_z__ = dz_z__; } } if (x_state__ != 1 && (*ignore_cwise__ || z_state__ != 1)) { goto L666; } if (incr_prec__) { incr_prec__ = FALSE_; ++y_prec_state__; i__3 = *n; for (i__ = 1; i__ <= i__3; ++i__) { y_tail__[i__] = 0.f; } } prevnormdx = normdx; prev_dz_z__ = dz_z__; /* Update soluton. */ if (y_prec_state__ < 2) { saxpy_(n, &c_b11, &dy[1], &c__1, &y[j * y_dim1 + 1], &c__1); } else { sla_wwaddw__(n, &y[j * y_dim1 + 1], &y_tail__[1], &dy[1]); } } /* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT. */ L666: /* Set final_* when cnt hits ithresh. */ if (x_state__ == 1) { final_dx_x__ = dx_x__; } if (z_state__ == 1) { final_dz_z__ = dz_z__; } /* Compute error bounds. */ if (*n_norms__ >= 1) { err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = final_dx_x__ / ( 1 - dxratmax); } if (*n_norms__ >= 2) { err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = final_dz_z__ / ( 1 - dzratmax); } /* Compute componentwise relative backward error from formula */ /* max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) */ /* where abs(Z) is the componentwise absolute value of the matrix */ /* or vector Z. */ /* Compute residual RES = B_s - op(A_s) * Y, */ /* op(A) = A, A**T, or A**H depending on TRANS (and type). */ scopy_(n, &b[j * b_dim1 + 1], &c__1, &res[1], &c__1); ssymv_(uplo, n, &c_b9, &a[a_offset], lda, &y[j * y_dim1 + 1], &c__1, & c_b11, &res[1], &c__1); i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { ayb[i__] = (r__1 = b[i__ + j * b_dim1], dabs(r__1)); } /* Compute abs(op(A_s))*abs(Y) + abs(B_s). */ sla_syamv__(&uplo2, n, &c_b11, &a[a_offset], lda, &y[j * y_dim1 + 1], &c__1, &c_b11, &ayb[1], &c__1); sla_lin_berr__(n, n, &c__1, &res[1], &ayb[1], &berr_out__[j]); /* End of loop for each RHS. */ } return 0; } /* sla_syrfsx_extended__ */