/* zporfsx.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 logical c_true = TRUE_; static logical c_false = FALSE_; /* Subroutine */ int zporfsx_(char *uplo, char *equed, integer *n, integer * nrhs, doublecomplex *a, integer *lda, doublecomplex *af, integer * ldaf, doublereal *s, doublecomplex *b, integer *ldb, doublecomplex *x, integer *ldx, doublereal *rcond, doublereal *berr, integer * n_err_bnds__, doublereal *err_bnds_norm__, doublereal * err_bnds_comp__, integer *nparams, doublereal *params, doublecomplex * work, doublereal *rwork, integer *info) { /* System generated locals */ integer a_dim1, a_offset, af_dim1, af_offset, b_dim1, b_offset, x_dim1, x_offset, err_bnds_norm_dim1, err_bnds_norm_offset, err_bnds_comp_dim1, err_bnds_comp_offset, i__1; doublereal d__1, d__2; /* Builtin functions */ double sqrt(doublereal); /* Local variables */ doublereal illrcond_thresh__, unstable_thresh__, err_lbnd__; integer ref_type__; integer j; doublereal rcond_tmp__; integer prec_type__; doublereal cwise_wrong__; extern /* Subroutine */ int zla_porfsx_extended__(integer *, char *, integer *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, logical *, doublereal *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublecomplex *, doublereal *, doublecomplex *, doublecomplex *, doublereal *, integer *, doublereal *, doublereal *, logical *, integer *, ftnlen); char norm[1]; logical ignore_cwise__; extern logical lsame_(char *, char *); doublereal anorm; logical rcequ; extern doublereal zla_porcond_c__(char *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublereal *, logical *, integer *, doublecomplex *, doublereal *, ftnlen), zla_porcond_x__(char *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, integer *, doublecomplex *, doublereal *, ftnlen), dlamch_(char *); extern /* Subroutine */ int xerbla_(char *, integer *); extern doublereal zlanhe_(char *, char *, integer *, doublecomplex *, integer *, doublereal *); extern /* Subroutine */ int zpocon_(char *, integer *, doublecomplex *, integer *, doublereal *, doublereal *, doublecomplex *, doublereal *, integer *); extern integer ilaprec_(char *); integer ithresh, n_norms__; doublereal rthresh; /* -- 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 */ /* ======= */ /* ZPORFSX improves the computed solution to a system of linear */ /* equations when the coefficient matrix is symmetric positive */ /* definite, and provides error bounds and backward error estimates */ /* for the solution. 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. */ /* The original system of linear equations may have been equilibrated */ /* before calling this routine, as described by arguments EQUED and S */ /* below. In this case, the solution and error bounds returned are */ /* for the original unequilibrated system. */ /* Arguments */ /* ========= */ /* Some optional parameters are bundled in the PARAMS array. These */ /* settings determine how refinement is performed, but often the */ /* defaults are acceptable. If the defaults are acceptable, users */ /* can pass NPARAMS = 0 which prevents the source code from accessing */ /* the PARAMS argument. */ /* UPLO (input) CHARACTER*1 */ /* = 'U': Upper triangle of A is stored; */ /* = 'L': Lower triangle of A is stored. */ /* EQUED (input) CHARACTER*1 */ /* Specifies the form of equilibration that was done to A */ /* before calling this routine. This is needed to compute */ /* the solution and error bounds correctly. */ /* = 'N': No equilibration */ /* = 'Y': Both row and column equilibration, i.e., A has been */ /* replaced by diag(S) * A * diag(S). */ /* The right hand side B has been changed accordingly. */ /* 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. */ /* A (input) COMPLEX*16 array, dimension (LDA,N) */ /* The symmetric matrix A. If UPLO = 'U', the leading N-by-N */ /* upper triangular part of A contains the upper triangular part */ /* of the matrix A, and the strictly lower triangular part of A */ /* is not referenced. If UPLO = 'L', the leading N-by-N lower */ /* triangular part of A contains the lower triangular part of */ /* the matrix A, and the strictly upper triangular part of A is */ /* not referenced. */ /* LDA (input) INTEGER */ /* The leading dimension of the array A. LDA >= max(1,N). */ /* AF (input) COMPLEX*16 array, dimension (LDAF,N) */ /* The triangular factor U or L from the Cholesky factorization */ /* A = U**T*U or A = L*L**T, as computed by DPOTRF. */ /* LDAF (input) INTEGER */ /* The leading dimension of the array AF. LDAF >= max(1,N). */ /* S (input or output) DOUBLE PRECISION array, dimension (N) */ /* The row scale factors for A. If EQUED = 'Y', A is multiplied on */ /* the left and right by diag(S). S is an input argument if FACT = */ /* 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED */ /* = 'Y', each element of S must be positive. If S is output, each */ /* element of S is a power of the radix. If S is input, each element */ /* of S 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) COMPLEX*16 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*16 array, dimension (LDX,NRHS) */ /* On entry, the solution matrix X, as computed by DGETRS. */ /* On exit, the improved solution matrix X. */ /* LDX (input) INTEGER */ /* The leading dimension of the array X. LDX >= max(1,N). */ /* RCOND (output) DOUBLE PRECISION */ /* 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. */ /* BERR (output) DOUBLE PRECISION array, dimension (NRHS) */ /* Componentwise relative backward error. This is 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). */ /* N_ERR_BNDS (input) INTEGER */ /* Number of error bounds to return for each right hand side */ /* and each type (normwise or componentwise). See ERR_BNDS_NORM and */ /* ERR_BNDS_COMP below. */ /* ERR_BNDS_NORM (output) DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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. */ /* See Lapack Working Note 165 for further details and extra */ /* cautions. */ /* ERR_BNDS_COMP (output) DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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. */ /* See Lapack Working Note 165 for further details and extra */ /* cautions. */ /* NPARAMS (input) INTEGER */ /* Specifies the number of parameters set in PARAMS. If .LE. 0, the */ /* PARAMS array is never referenced and default values are used. */ /* PARAMS (input / output) DOUBLE PRECISION array, dimension NPARAMS */ /* Specifies algorithm parameters. If an entry is .LT. 0.0, then */ /* that entry will be filled with default value used for that */ /* parameter. Only positions up to NPARAMS are accessed; defaults */ /* are used for higher-numbered parameters. */ /* PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative */ /* refinement or not. */ /* Default: 1.0D+0 */ /* = 0.0 : No refinement is performed, and no error bounds are */ /* computed. */ /* = 1.0 : Use the double-precision refinement algorithm, */ /* possibly with doubled-single computations if the */ /* compilation environment does not support DOUBLE */ /* PRECISION. */ /* (other values are reserved for future use) */ /* PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual */ /* computations allowed for refinement. */ /* Default: 10 */ /* 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. */ /* PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code */ /* will attempt to find a solution with small componentwise */ /* relative error in the double-precision algorithm. Positive */ /* is true, 0.0 is false. */ /* Default: 1.0 (attempt componentwise convergence) */ /* WORK (workspace) COMPLEX*16 array, dimension (2*N) */ /* RWORK (workspace) DOUBLE PRECISION array, dimension (2*N) */ /* INFO (output) INTEGER */ /* = 0: Successful exit. The solution to every right-hand side is */ /* guaranteed. */ /* < 0: If INFO = -i, the i-th argument had an illegal value */ /* > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization */ /* has been completed, but the factor U is exactly singular, so */ /* the solution and error bounds could not be computed. RCOND = 0 */ /* is returned. */ /* = N+J: The solution corresponding to the Jth right-hand side is */ /* not guaranteed. The solutions corresponding to other right- */ /* hand sides K with K > J may not be guaranteed as well, but */ /* only the first such right-hand side is reported. If a small */ /* componentwise error is not requested (PARAMS(3) = 0.0) then */ /* the Jth right-hand side is the first with a normwise error */ /* bound that is not guaranteed (the smallest J such */ /* that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0) */ /* the Jth right-hand side is the first with either a normwise or */ /* componentwise error bound that is not guaranteed (the smallest */ /* J such that either ERR_BNDS_NORM(J,1) = 0.0 or */ /* ERR_BNDS_COMP(J,1) = 0.0). See the definition of */ /* ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information */ /* about all of the right-hand sides check ERR_BNDS_NORM or */ /* ERR_BNDS_COMP. */ /* ================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. Executable Statements .. */ /* Check the input parameters. */ /* 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; --s; b_dim1 = *ldb; b_offset = 1 + b_dim1; b -= b_offset; x_dim1 = *ldx; x_offset = 1 + x_dim1; x -= x_offset; --berr; --params; --work; --rwork; /* Function Body */ *info = 0; ref_type__ = 1; if (*nparams >= 1) { if (params[1] < 0.) { params[1] = 1.; } else { ref_type__ = (integer) params[1]; } } /* Set default parameters. */ illrcond_thresh__ = (doublereal) (*n) * dlamch_("Epsilon"); ithresh = 10; rthresh = .5; unstable_thresh__ = .25; ignore_cwise__ = FALSE_; if (*nparams >= 2) { if (params[2] < 0.) { params[2] = (doublereal) ithresh; } else { ithresh = (integer) params[2]; } } if (*nparams >= 3) { if (params[3] < 0.) { if (ignore_cwise__) { params[3] = 0.; } else { params[3] = 1.; } } else { ignore_cwise__ = params[3] == 0.; } } if (ref_type__ == 0 || *n_err_bnds__ == 0) { n_norms__ = 0; } else if (ignore_cwise__) { n_norms__ = 1; } else { n_norms__ = 2; } rcequ = lsame_(equed, "Y"); /* Test input parameters. */ if (! lsame_(uplo, "U") && ! lsame_(uplo, "L")) { *info = -1; } else if (! rcequ && ! lsame_(equed, "N")) { *info = -2; } else if (*n < 0) { *info = -3; } else if (*nrhs < 0) { *info = -4; } else if (*lda < max(1,*n)) { *info = -6; } else if (*ldaf < max(1,*n)) { *info = -8; } else if (*ldb < max(1,*n)) { *info = -11; } else if (*ldx < max(1,*n)) { *info = -13; } if (*info != 0) { i__1 = -(*info); xerbla_("ZPORFSX", &i__1); return 0; } /* Quick return if possible. */ if (*n == 0 || *nrhs == 0) { *rcond = 1.; i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { berr[j] = 0.; if (*n_err_bnds__ >= 1) { err_bnds_norm__[j + err_bnds_norm_dim1] = 1.; err_bnds_comp__[j + err_bnds_comp_dim1] = 1.; } else if (*n_err_bnds__ >= 2) { err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 0.; err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 0.; } else if (*n_err_bnds__ >= 3) { err_bnds_norm__[j + err_bnds_norm_dim1 * 3] = 1.; err_bnds_comp__[j + err_bnds_comp_dim1 * 3] = 1.; } } return 0; } /* Default to failure. */ *rcond = 0.; i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { berr[j] = 1.; if (*n_err_bnds__ >= 1) { err_bnds_norm__[j + err_bnds_norm_dim1] = 1.; err_bnds_comp__[j + err_bnds_comp_dim1] = 1.; } else if (*n_err_bnds__ >= 2) { err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 1.; err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 1.; } else if (*n_err_bnds__ >= 3) { err_bnds_norm__[j + err_bnds_norm_dim1 * 3] = 0.; err_bnds_comp__[j + err_bnds_comp_dim1 * 3] = 0.; } } /* Compute the norm of A and the reciprocal of the condition */ /* number of A. */ *(unsigned char *)norm = 'I'; anorm = zlanhe_(norm, uplo, n, &a[a_offset], lda, &rwork[1]); zpocon_(uplo, n, &af[af_offset], ldaf, &anorm, rcond, &work[1], &rwork[1], info); /* Perform refinement on each right-hand side */ if (ref_type__ != 0) { prec_type__ = ilaprec_("E"); zla_porfsx_extended__(&prec_type__, uplo, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, &rcequ, &s[1], &b[b_offset], ldb, &x[ x_offset], ldx, &berr[1], &n_norms__, &err_bnds_norm__[ err_bnds_norm_offset], &err_bnds_comp__[err_bnds_comp_offset], &work[1], &rwork[1], &work[*n + 1], (doublecomplex *)(&rwork[1]), rcond, &ithresh, & rthresh, &unstable_thresh__, &ignore_cwise__, info, (ftnlen)1) ; } /* Computing MAX */ d__1 = 10., d__2 = sqrt((doublereal) (*n)); err_lbnd__ = max(d__1,d__2) * dlamch_("Epsilon"); if (*n_err_bnds__ >= 1 && n_norms__ >= 1) { /* Compute scaled normwise condition number cond(A*C). */ if (rcequ) { rcond_tmp__ = zla_porcond_c__(uplo, n, &a[a_offset], lda, &af[ af_offset], ldaf, &s[1], &c_true, info, &work[1], &rwork[ 1], (ftnlen)1); } else { rcond_tmp__ = zla_porcond_c__(uplo, n, &a[a_offset], lda, &af[ af_offset], ldaf, &s[1], &c_false, info, &work[1], &rwork[ 1], (ftnlen)1); } i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { /* Cap the error at 1.0. */ if (*n_err_bnds__ >= 2 && err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] > 1.) { err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 1.; } /* Threshold the error (see LAWN). */ if (rcond_tmp__ < illrcond_thresh__) { err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = 1.; err_bnds_norm__[j + err_bnds_norm_dim1] = 0.; if (*info <= *n) { *info = *n + j; } } else if (err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] < err_lbnd__) { err_bnds_norm__[j + (err_bnds_norm_dim1 << 1)] = err_lbnd__; err_bnds_norm__[j + err_bnds_norm_dim1] = 1.; } /* Save the condition number. */ if (*n_err_bnds__ >= 3) { err_bnds_norm__[j + err_bnds_norm_dim1 * 3] = rcond_tmp__; } } } if (*n_err_bnds__ >= 1 && n_norms__ >= 2) { /* Compute componentwise condition number cond(A*diag(Y(:,J))) for */ /* each right-hand side using the current solution as an estimate of */ /* the true solution. If the componentwise error estimate is too */ /* large, then the solution is a lousy estimate of truth and the */ /* estimated RCOND may be too optimistic. To avoid misleading users, */ /* the inverse condition number is set to 0.0 when the estimated */ /* cwise error is at least CWISE_WRONG. */ cwise_wrong__ = sqrt(dlamch_("Epsilon")); i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { if (err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] < cwise_wrong__) { rcond_tmp__ = zla_porcond_x__(uplo, n, &a[a_offset], lda, &af[ af_offset], ldaf, &x[j * x_dim1 + 1], info, &work[1], &rwork[1], (ftnlen)1); } else { rcond_tmp__ = 0.; } /* Cap the error at 1.0. */ if (*n_err_bnds__ >= 2 && err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] > 1.) { err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 1.; } /* Threshold the error (see LAWN). */ if (rcond_tmp__ < illrcond_thresh__) { err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = 1.; err_bnds_comp__[j + err_bnds_comp_dim1] = 0.; if (params[3] == 1. && *info < *n + j) { *info = *n + j; } } else if (err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] < err_lbnd__) { err_bnds_comp__[j + (err_bnds_comp_dim1 << 1)] = err_lbnd__; err_bnds_comp__[j + err_bnds_comp_dim1] = 1.; } /* Save the condition number. */ if (*n_err_bnds__ >= 3) { err_bnds_comp__[j + err_bnds_comp_dim1 * 3] = rcond_tmp__; } } } return 0; /* End of ZPORFSX */ } /* zporfsx_ */