/* sposvxx.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" /* Subroutine */ int sposvxx_(char *fact, char *uplo, integer *n, integer * nrhs, real *a, integer *lda, real *af, integer *ldaf, char *equed, real *s, real *b, integer *ldb, real *x, integer *ldx, real *rcond, real *rpvgrw, real *berr, integer *n_err_bnds__, real * err_bnds_norm__, real *err_bnds_comp__, integer *nparams, real * params, real *work, integer *iwork, 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; real r__1, r__2; /* Local variables */ integer j; real amax, smin, smax; extern doublereal sla_porpvgrw__(char *, integer *, real *, integer *, real *, integer *, real *, ftnlen); extern logical lsame_(char *, char *); real scond; logical equil, rcequ; extern doublereal slamch_(char *); logical nofact; extern /* Subroutine */ int xerbla_(char *, integer *); real bignum; integer infequ; extern /* Subroutine */ int slacpy_(char *, integer *, integer *, real *, integer *, real *, integer *); real smlnum; extern /* Subroutine */ int slaqsy_(char *, integer *, real *, integer *, real *, real *, real *, char *), spotrf_(char *, integer *, real *, integer *, integer *), spotrs_(char *, integer *, integer *, real *, integer *, real *, integer *, integer *), slascl2_(integer *, integer *, real *, real *, integer *), spoequb_(integer *, real *, integer *, real *, real * , real *, integer *), sporfsx_(char *, char *, integer *, integer *, real *, integer *, real *, integer *, real *, real *, integer * , real *, integer *, real *, real *, integer *, real *, real *, integer *, real *, real *, integer *, integer *); /* -- LAPACK driver routine (version 3.2) -- */ /* -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- */ /* -- Jason Riedy of Univ. of California Berkeley. -- */ /* -- November 2008 -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley and NAG Ltd. -- */ /* .. */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T */ /* to compute the solution to a real system of linear equations */ /* A * X = B, where A is an N-by-N symmetric positive definite matrix */ /* and X and B are N-by-NRHS matrices. */ /* If requested, both normwise and maximum componentwise error bounds */ /* are returned. SPOSVXX will return a solution with a tiny */ /* guaranteed error (O(eps) where eps is the working machine */ /* precision) unless the matrix is very ill-conditioned, in which */ /* case a warning is returned. Relevant condition numbers also are */ /* calculated and returned. */ /* SPOSVXX accepts user-provided factorizations and equilibration */ /* factors; see the definitions of the FACT and EQUED options. */ /* Solving with refinement and using a factorization from a previous */ /* SPOSVXX call will also produce a solution with either O(eps) */ /* errors or warnings, but we cannot make that claim for general */ /* user-provided factorizations and equilibration factors if they */ /* differ from what SPOSVXX would itself produce. */ /* Description */ /* =========== */ /* The following steps are performed: */ /* 1. If FACT = 'E', real scaling factors are computed to equilibrate */ /* the system: */ /* diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B */ /* Whether or not the system will be equilibrated depends on the */ /* scaling of the matrix A, but if equilibration is used, A is */ /* overwritten by diag(S)*A*diag(S) and B by diag(S)*B. */ /* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to */ /* factor the matrix A (after equilibration if FACT = 'E') as */ /* A = U**T* U, if UPLO = 'U', or */ /* A = L * L**T, if UPLO = 'L', */ /* where U is an upper triangular matrix and L is a lower triangular */ /* matrix. */ /* 3. If the leading i-by-i principal minor is not positive definite, */ /* then the routine returns with INFO = i. Otherwise, the factored */ /* form of A is used to estimate the condition number of the matrix */ /* A (see argument RCOND). If the reciprocal of the condition number */ /* is less than machine precision, the routine still goes on to solve */ /* for X and compute error bounds as described below. */ /* 4. The system of equations is solved for X using the factored form */ /* of A. */ /* 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero), */ /* the routine will use iterative refinement to try to get a small */ /* error and error bounds. Refinement calculates the residual to at */ /* least twice the working precision. */ /* 6. If equilibration was used, the matrix X is premultiplied by */ /* diag(S) so that it solves the original system before */ /* equilibration. */ /* 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. */ /* FACT (input) CHARACTER*1 */ /* Specifies whether or not the factored form of the matrix A is */ /* supplied on entry, and if not, whether the matrix A should be */ /* equilibrated before it is factored. */ /* = 'F': On entry, AF contains the factored form of A. */ /* If EQUED is not 'N', the matrix A has been */ /* equilibrated with scaling factors given by S. */ /* A and AF are not modified. */ /* = 'N': The matrix A will be copied to AF and factored. */ /* = 'E': The matrix A will be equilibrated if necessary, then */ /* copied to AF and factored. */ /* 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 matrices B and X. NRHS >= 0. */ /* A (input/output) REAL array, dimension (LDA,N) */ /* On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = */ /* 'Y', then A must contain the equilibrated matrix */ /* diag(S)*A*diag(S). 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. A is */ /* not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = */ /* 'N' on exit. */ /* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by */ /* diag(S)*A*diag(S). */ /* LDA (input) INTEGER */ /* The leading dimension of the array A. LDA >= max(1,N). */ /* AF (input or output) REAL array, dimension (LDAF,N) */ /* If FACT = 'F', then AF is an input argument and on entry */ /* contains the triangular factor U or L from the Cholesky */ /* factorization A = U**T*U or A = L*L**T, in the same storage */ /* format as A. If EQUED .ne. 'N', then AF is the factored */ /* form of the equilibrated matrix diag(S)*A*diag(S). */ /* If FACT = 'N', then AF is an output argument and on exit */ /* returns the triangular factor U or L from the Cholesky */ /* factorization A = U**T*U or A = L*L**T of the original */ /* matrix A. */ /* If FACT = 'E', then AF is an output argument and on exit */ /* returns the triangular factor U or L from the Cholesky */ /* factorization A = U**T*U or A = L*L**T of the equilibrated */ /* matrix A (see the description of A for the form of the */ /* equilibrated matrix). */ /* LDAF (input) INTEGER */ /* The leading dimension of the array AF. LDAF >= max(1,N). */ /* EQUED (input or output) CHARACTER*1 */ /* Specifies the form of equilibration that was done. */ /* = 'N': No equilibration (always true if FACT = 'N'). */ /* = 'Y': Both row and column equilibration, i.e., A has been */ /* replaced by diag(S) * A * diag(S). */ /* EQUED is an input argument if FACT = 'F'; otherwise, it is an */ /* output argument. */ /* S (input or output) REAL 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/output) REAL array, dimension (LDB,NRHS) */ /* On entry, the N-by-NRHS right hand side matrix B. */ /* On exit, */ /* if EQUED = 'N', B is not modified; */ /* if EQUED = 'Y', B is overwritten by diag(S)*B; */ /* LDB (input) INTEGER */ /* The leading dimension of the array B. LDB >= max(1,N). */ /* X (output) REAL array, dimension (LDX,NRHS) */ /* If INFO = 0, the N-by-NRHS solution matrix X to the original */ /* system of equations. Note that A and B are modified on exit if */ /* EQUED .ne. 'N', and the solution to the equilibrated system is */ /* inv(diag(S))*X. */ /* LDX (input) INTEGER */ /* The leading dimension of the array X. LDX >= max(1,N). */ /* RCOND (output) 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. */ /* RPVGRW (output) REAL */ /* Reciprocal pivot growth. On exit, this contains the reciprocal */ /* pivot growth factor norm(A)/norm(U). The "max absolute element" */ /* norm is used. If this is much less than 1, then the stability of */ /* the LU factorization of the (equilibrated) matrix A could be poor. */ /* This also means that the solution X, estimated condition numbers, */ /* and error bounds could be unreliable. If factorization fails with */ /* 0 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 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; --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; --iwork; /* Function Body */ *info = 0; nofact = lsame_(fact, "N"); equil = lsame_(fact, "E"); smlnum = slamch_("Safe minimum"); bignum = 1.f / smlnum; if (nofact || equil) { *(unsigned char *)equed = 'N'; rcequ = FALSE_; } else { rcequ = lsame_(equed, "Y"); } /* Default is failure. If an input parameter is wrong or */ /* factorization fails, make everything look horrible. Only the */ /* pivot growth is set here, the rest is initialized in SPORFSX. */ *rpvgrw = 0.f; /* Test the input parameters. PARAMS is not tested until SPORFSX. */ if (! nofact && ! equil && ! lsame_(fact, "F")) { *info = -1; } else if (! lsame_(uplo, "U") && ! lsame_(uplo, "L")) { *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 (lsame_(fact, "F") && ! (rcequ || lsame_( equed, "N"))) { *info = -9; } else { if (rcequ) { smin = bignum; smax = 0.f; i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MIN */ r__1 = smin, r__2 = s[j]; smin = dmin(r__1,r__2); /* Computing MAX */ r__1 = smax, r__2 = s[j]; smax = dmax(r__1,r__2); /* L10: */ } if (smin <= 0.f) { *info = -10; } else if (*n > 0) { scond = dmax(smin,smlnum) / dmin(smax,bignum); } else { scond = 1.f; } } if (*info == 0) { if (*ldb < max(1,*n)) { *info = -12; } else if (*ldx < max(1,*n)) { *info = -14; } } } if (*info != 0) { i__1 = -(*info); xerbla_("SPOSVXX", &i__1); return 0; } if (equil) { /* Compute row and column scalings to equilibrate the matrix A. */ spoequb_(n, &a[a_offset], lda, &s[1], &scond, &amax, &infequ); if (infequ == 0) { /* Equilibrate the matrix. */ slaqsy_(uplo, n, &a[a_offset], lda, &s[1], &scond, &amax, equed); rcequ = lsame_(equed, "Y"); } } /* Scale the right-hand side. */ if (rcequ) { slascl2_(n, nrhs, &s[1], &b[b_offset], ldb); } if (nofact || equil) { /* Compute the LU factorization of A. */ slacpy_(uplo, n, n, &a[a_offset], lda, &af[af_offset], ldaf); spotrf_(uplo, n, &af[af_offset], ldaf, info); /* Return if INFO is non-zero. */ if (*info != 0) { /* Pivot in column INFO is exactly 0 */ /* Compute the reciprocal pivot growth factor of the */ /* leading rank-deficient INFO columns of A. */ *rpvgrw = sla_porpvgrw__(uplo, info, &a[a_offset], lda, &af[ af_offset], ldaf, &work[1], (ftnlen)1); return 0; } } /* Compute the reciprocal growth factor RPVGRW. */ *rpvgrw = sla_porpvgrw__(uplo, n, &a[a_offset], lda, &af[af_offset], ldaf, &work[1], (ftnlen)1); /* Compute the solution matrix X. */ slacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); spotrs_(uplo, n, nrhs, &af[af_offset], ldaf, &x[x_offset], ldx, info); /* Use iterative refinement to improve the computed solution and */ /* compute error bounds and backward error estimates for it. */ sporfsx_(uplo, equed, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, & s[1], &b[b_offset], ldb, &x[x_offset], ldx, rcond, &berr[1], n_err_bnds__, &err_bnds_norm__[err_bnds_norm_offset], & err_bnds_comp__[err_bnds_comp_offset], nparams, ¶ms[1], &work[ 1], &iwork[1], info); /* Scale solutions. */ if (rcequ) { slascl2_(n, nrhs, &s[1], &x[x_offset], ldx); } return 0; /* End of SPOSVXX */ } /* sposvxx_ */