/* Subroutine */ int dposvx_(char *fact, char *uplo, integer *n, integer * nrhs, doublereal *a, integer *lda, doublereal *af, integer *ldaf, char *equed, doublereal *s, doublereal *b, integer *ldb, doublereal * x, integer *ldx, doublereal *rcond, doublereal *ferr, doublereal * berr, doublereal *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, i__1, i__2; doublereal d__1, d__2; /* Local variables */ integer i__, j; doublereal amax, smin, smax; doublereal scond, anorm; logical equil, rcequ; logical nofact; doublereal bignum; integer infequ; doublereal smlnum; /* -- LAPACK driver routine (version 3.2) -- */ /* November 2006 */ /* Purpose */ /* ======= */ /* DPOSVX 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. */ /* Error bounds on the solution and a condition estimate are also */ /* provided. */ /* 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. If the reciprocal of the condition number is less than machine */ /* precision, INFO = N+1 is returned as a warning, but 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. Iterative refinement is applied to improve the computed solution */ /* matrix and calculate error bounds and backward error estimates */ /* for it. */ /* 6. If equilibration was used, the matrix X is premultiplied by */ /* diag(S) so that it solves the original system before */ /* equilibration. */ /* Arguments */ /* ========= */ /* 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 = 'Y', the matrix A has been equilibrated */ /* with scaling factors given by S. A and AF will not */ /* be 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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': Equilibration was done, 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) DOUBLE PRECISION array, dimension (N) */ /* The scale factors for A; not accessed if EQUED = 'N'. 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. */ /* B (input/output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS) */ /* If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to */ /* the original system of equations. Note that if EQUED = 'Y', */ /* A and B are modified on exit, 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) DOUBLE PRECISION */ /* The estimate of the reciprocal condition number of the matrix */ /* A after equilibration (if done). If RCOND is less than the */ /* machine precision (in particular, if RCOND = 0), the matrix */ /* is singular to working precision. This condition is */ /* indicated by a return code of INFO > 0. */ /* FERR (output) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (3*N) */ /* IWORK (workspace) INTEGER array, dimension (N) */ /* INFO (output) INTEGER */ /* = 0: successful exit */ /* < 0: if INFO = -i, the i-th argument had an illegal value */ /* > 0: if INFO = i, and i is */ /* <= N: the leading minor of order i of A is */ /* not positive definite, so the factorization */ /* could not be completed, and the solution has not */ /* been computed. RCOND = 0 is returned. */ /* = N+1: U is nonsingular, but RCOND is less than machine */ /* precision, meaning that the matrix is singular */ /* to working precision. Nevertheless, the */ /* solution and error bounds are computed because */ /* there are a number of situations where the */ /* computed solution can be more accurate than the */ /* value of RCOND would suggest. */ /* ===================================================================== */ /* Parameter adjustments */ 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; --ferr; --berr; --work; --iwork; /* Function Body */ *info = 0; nofact = lsame_(fact, "N"); equil = lsame_(fact, "E"); if (nofact || equil) { *(unsigned char *)equed = 'N'; rcequ = FALSE_; } else { rcequ = lsame_(equed, "Y"); smlnum = dlamch_("Safe minimum"); bignum = 1. / smlnum; } /* Test the input parameters. */ 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.; i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MIN */ d__1 = smin, d__2 = s[j]; smin = min(d__1,d__2); /* Computing MAX */ d__1 = smax, d__2 = s[j]; smax = max(d__1,d__2); } if (smin <= 0.) { *info = -10; } else if (*n > 0) { scond = max(smin,smlnum) / min(smax,bignum); } else { scond = 1.; } } 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_("DPOSVX", &i__1); return 0; } if (equil) { /* Compute row and column scalings to equilibrate the matrix A. */ dpoequ_(n, &a[a_offset], lda, &s[1], &scond, &amax, &infequ); if (infequ == 0) { /* Equilibrate the matrix. */ dlaqsy_(uplo, n, &a[a_offset], lda, &s[1], &scond, &amax, equed); rcequ = lsame_(equed, "Y"); } } /* Scale the right hand side. */ if (rcequ) { i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { b[i__ + j * b_dim1] = s[i__] * b[i__ + j * b_dim1]; } } } if (nofact || equil) { /* Compute the Cholesky factorization A = U'*U or A = L*L'. */ dlacpy_(uplo, n, n, &a[a_offset], lda, &af[af_offset], ldaf); dpotrf_(uplo, n, &af[af_offset], ldaf, info); /* Return if INFO is non-zero. */ if (*info > 0) { *rcond = 0.; return 0; } } /* Compute the norm of the matrix A. */ anorm = dlansy_("1", uplo, n, &a[a_offset], lda, &work[1]); /* Compute the reciprocal of the condition number of A. */ dpocon_(uplo, n, &af[af_offset], ldaf, &anorm, rcond, &work[1], &iwork[1], info); /* Compute the solution matrix X. */ dlacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); dpotrs_(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. */ dporfs_(uplo, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, &b[ b_offset], ldb, &x[x_offset], ldx, &ferr[1], &berr[1], &work[1], & iwork[1], info); /* Transform the solution matrix X to a solution of the original */ /* system. */ if (rcequ) { i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { i__2 = *n; for (i__ = 1; i__ <= i__2; ++i__) { x[i__ + j * x_dim1] = s[i__] * x[i__ + j * x_dim1]; } } i__1 = *nrhs; for (j = 1; j <= i__1; ++j) { ferr[j] /= scond; } } /* Set INFO = N+1 if the matrix is singular to working precision. */ if (*rcond < dlamch_("Epsilon")) { *info = *n + 1; } return 0; /* End of DPOSVX */ } /* dposvx_ */
/* Subroutine */ int ddrvpo_(logical *dotype, integer *nn, integer *nval, integer *nrhs, doublereal *thresh, logical *tsterr, integer *nmax, doublereal *a, doublereal *afac, doublereal *asav, doublereal *b, doublereal *bsav, doublereal *x, doublereal *xact, doublereal *s, doublereal *work, doublereal *rwork, integer *iwork, integer *nout) { /* Initialized data */ static integer iseedy[4] = { 1988,1989,1990,1991 }; static char uplos[1*2] = "U" "L"; static char facts[1*3] = "F" "N" "E"; static char equeds[1*2] = "N" "Y"; /* Format strings */ static char fmt_9999[] = "(1x,a,\002, UPLO='\002,a1,\002', N =\002,i5" ",\002, type \002,i1,\002, test(\002,i1,\002)=\002,g12.5)"; static char fmt_9997[] = "(1x,a,\002, FACT='\002,a1,\002', UPLO='\002," "a1,\002', N=\002,i5,\002, EQUED='\002,a1,\002', type \002,i1," "\002, test(\002,i1,\002) =\002,g12.5)"; static char fmt_9998[] = "(1x,a,\002, FACT='\002,a1,\002', UPLO='\002," "a1,\002', N=\002,i5,\002, type \002,i1,\002, test(\002,i1,\002)" "=\002,g12.5)"; /* System generated locals */ address a__1[2]; integer i__1, i__2, i__3, i__4, i__5[2]; char ch__1[2]; /* Builtin functions */ /* Subroutine */ int s_copy(char *, char *, ftnlen, ftnlen); integer s_wsfe(cilist *), do_fio(integer *, char *, ftnlen), e_wsfe(void); /* Subroutine */ int s_cat(char *, char **, integer *, integer *, ftnlen); /* Local variables */ integer i__, k, n, k1, nb, in, kl, ku, nt, lda; char fact[1]; integer ioff, mode; doublereal amax; char path[3]; integer imat, info; char dist[1], uplo[1], type__[1]; integer nrun, ifact; extern /* Subroutine */ int dget04_(integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *); integer nfail, iseed[4], nfact; extern doublereal dget06_(doublereal *, doublereal *); extern logical lsame_(char *, char *); char equed[1]; integer nbmin; doublereal rcond, roldc, scond; integer nimat; extern /* Subroutine */ int dpot01_(char *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *), dpot02_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *), dpot05_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *); doublereal anorm; logical equil; integer iuplo, izero, nerrs; extern /* Subroutine */ int dposv_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, integer *); logical zerot; char xtype[1]; extern /* Subroutine */ int dlatb4_(char *, integer *, integer *, integer *, char *, integer *, integer *, doublereal *, integer *, doublereal *, char *), aladhd_(integer *, char *), alaerh_(char *, char *, integer *, integer *, char *, integer *, integer *, integer *, integer *, integer *, integer *, integer *, integer *, integer *); logical prefac; doublereal rcondc; logical nofact; integer iequed; extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), dlarhs_(char *, char *, char *, char *, integer *, integer *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, integer *), dlaset_(char *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *), alasvm_(char *, integer *, integer *, integer *, integer *); doublereal cndnum; extern /* Subroutine */ int dlatms_(integer *, integer *, char *, integer *, char *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *, char *, doublereal *, integer *, doublereal *, integer *); doublereal ainvnm; extern doublereal dlansy_(char *, char *, integer *, doublereal *, integer *, doublereal *); extern /* Subroutine */ int dlaqsy_(char *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, char *), dpoequ_(integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, integer *), dpotrf_( char *, integer *, doublereal *, integer *, integer *), dpotri_(char *, integer *, doublereal *, integer *, integer *), xlaenv_(integer *, integer *), derrvx_(char *, integer *); doublereal result[6]; extern /* Subroutine */ int dposvx_(char *, char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, char *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *); /* Fortran I/O blocks */ static cilist io___48 = { 0, 0, 0, fmt_9999, 0 }; static cilist io___51 = { 0, 0, 0, fmt_9997, 0 }; static cilist io___52 = { 0, 0, 0, fmt_9998, 0 }; /* -- LAPACK test routine (version 3.1) -- */ /* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */ /* November 2006 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* Purpose */ /* ======= */ /* DDRVPO tests the driver routines DPOSV and -SVX. */ /* Arguments */ /* ========= */ /* DOTYPE (input) LOGICAL array, dimension (NTYPES) */ /* The matrix types to be used for testing. Matrices of type j */ /* (for 1 <= j <= NTYPES) are used for testing if DOTYPE(j) = */ /* .TRUE.; if DOTYPE(j) = .FALSE., then type j is not used. */ /* NN (input) INTEGER */ /* The number of values of N contained in the vector NVAL. */ /* NVAL (input) INTEGER array, dimension (NN) */ /* The values of the matrix dimension N. */ /* NRHS (input) INTEGER */ /* The number of right hand side vectors to be generated for */ /* each linear system. */ /* THRESH (input) DOUBLE PRECISION */ /* The threshold value for the test ratios. A result is */ /* included in the output file if RESULT >= THRESH. To have */ /* every test ratio printed, use THRESH = 0. */ /* TSTERR (input) LOGICAL */ /* Flag that indicates whether error exits are to be tested. */ /* NMAX (input) INTEGER */ /* The maximum value permitted for N, used in dimensioning the */ /* work arrays. */ /* A (workspace) DOUBLE PRECISION array, dimension (NMAX*NMAX) */ /* AFAC (workspace) DOUBLE PRECISION array, dimension (NMAX*NMAX) */ /* ASAV (workspace) DOUBLE PRECISION array, dimension (NMAX*NMAX) */ /* B (workspace) DOUBLE PRECISION array, dimension (NMAX*NRHS) */ /* BSAV (workspace) DOUBLE PRECISION array, dimension (NMAX*NRHS) */ /* X (workspace) DOUBLE PRECISION array, dimension (NMAX*NRHS) */ /* XACT (workspace) DOUBLE PRECISION array, dimension (NMAX*NRHS) */ /* S (workspace) DOUBLE PRECISION array, dimension (NMAX) */ /* WORK (workspace) DOUBLE PRECISION array, dimension */ /* (NMAX*max(3,NRHS)) */ /* RWORK (workspace) DOUBLE PRECISION array, dimension (NMAX+2*NRHS) */ /* IWORK (workspace) INTEGER array, dimension (NMAX) */ /* NOUT (input) INTEGER */ /* The unit number for output. */ /* ===================================================================== */ /* .. Parameters .. */ /* .. */ /* .. Local Scalars .. */ /* .. */ /* .. Local Arrays .. */ /* .. */ /* .. External Functions .. */ /* .. */ /* .. External Subroutines .. */ /* .. */ /* .. Intrinsic Functions .. */ /* .. */ /* .. Scalars in Common .. */ /* .. */ /* .. Common blocks .. */ /* .. */ /* .. Data statements .. */ /* Parameter adjustments */ --iwork; --rwork; --work; --s; --xact; --x; --bsav; --b; --asav; --afac; --a; --nval; --dotype; /* Function Body */ /* .. */ /* .. Executable Statements .. */ /* Initialize constants and the random number seed. */ s_copy(path, "Double precision", (ftnlen)1, (ftnlen)16); s_copy(path + 1, "PO", (ftnlen)2, (ftnlen)2); nrun = 0; nfail = 0; nerrs = 0; for (i__ = 1; i__ <= 4; ++i__) { iseed[i__ - 1] = iseedy[i__ - 1]; /* L10: */ } /* Test the error exits */ if (*tsterr) { derrvx_(path, nout); } infoc_1.infot = 0; /* Set the block size and minimum block size for testing. */ nb = 1; nbmin = 2; xlaenv_(&c__1, &nb); xlaenv_(&c__2, &nbmin); /* Do for each value of N in NVAL */ i__1 = *nn; for (in = 1; in <= i__1; ++in) { n = nval[in]; lda = max(n,1); *(unsigned char *)xtype = 'N'; nimat = 9; if (n <= 0) { nimat = 1; } i__2 = nimat; for (imat = 1; imat <= i__2; ++imat) { /* Do the tests only if DOTYPE( IMAT ) is true. */ if (! dotype[imat]) { goto L120; } /* Skip types 3, 4, or 5 if the matrix size is too small. */ zerot = imat >= 3 && imat <= 5; if (zerot && n < imat - 2) { goto L120; } /* Do first for UPLO = 'U', then for UPLO = 'L' */ for (iuplo = 1; iuplo <= 2; ++iuplo) { *(unsigned char *)uplo = *(unsigned char *)&uplos[iuplo - 1]; /* Set up parameters with DLATB4 and generate a test matrix */ /* with DLATMS. */ dlatb4_(path, &imat, &n, &n, type__, &kl, &ku, &anorm, &mode, &cndnum, dist); s_copy(srnamc_1.srnamt, "DLATMS", (ftnlen)32, (ftnlen)6); dlatms_(&n, &n, dist, iseed, type__, &rwork[1], &mode, & cndnum, &anorm, &kl, &ku, uplo, &a[1], &lda, &work[1], &info); /* Check error code from DLATMS. */ if (info != 0) { alaerh_(path, "DLATMS", &info, &c__0, uplo, &n, &n, &c_n1, &c_n1, &c_n1, &imat, &nfail, &nerrs, nout); goto L110; } /* For types 3-5, zero one row and column of the matrix to */ /* test that INFO is returned correctly. */ if (zerot) { if (imat == 3) { izero = 1; } else if (imat == 4) { izero = n; } else { izero = n / 2 + 1; } ioff = (izero - 1) * lda; /* Set row and column IZERO of A to 0. */ if (iuplo == 1) { i__3 = izero - 1; for (i__ = 1; i__ <= i__3; ++i__) { a[ioff + i__] = 0.; /* L20: */ } ioff += izero; i__3 = n; for (i__ = izero; i__ <= i__3; ++i__) { a[ioff] = 0.; ioff += lda; /* L30: */ } } else { ioff = izero; i__3 = izero - 1; for (i__ = 1; i__ <= i__3; ++i__) { a[ioff] = 0.; ioff += lda; /* L40: */ } ioff -= izero; i__3 = n; for (i__ = izero; i__ <= i__3; ++i__) { a[ioff + i__] = 0.; /* L50: */ } } } else { izero = 0; } /* Save a copy of the matrix A in ASAV. */ dlacpy_(uplo, &n, &n, &a[1], &lda, &asav[1], &lda); for (iequed = 1; iequed <= 2; ++iequed) { *(unsigned char *)equed = *(unsigned char *)&equeds[ iequed - 1]; if (iequed == 1) { nfact = 3; } else { nfact = 1; } i__3 = nfact; for (ifact = 1; ifact <= i__3; ++ifact) { *(unsigned char *)fact = *(unsigned char *)&facts[ ifact - 1]; prefac = lsame_(fact, "F"); nofact = lsame_(fact, "N"); equil = lsame_(fact, "E"); if (zerot) { if (prefac) { goto L90; } rcondc = 0.; } else if (! lsame_(fact, "N")) { /* Compute the condition number for comparison with */ /* the value returned by DPOSVX (FACT = 'N' reuses */ /* the condition number from the previous iteration */ /* with FACT = 'F'). */ dlacpy_(uplo, &n, &n, &asav[1], &lda, &afac[1], & lda); if (equil || iequed > 1) { /* Compute row and column scale factors to */ /* equilibrate the matrix A. */ dpoequ_(&n, &afac[1], &lda, &s[1], &scond, & amax, &info); if (info == 0 && n > 0) { if (iequed > 1) { scond = 0.; } /* Equilibrate the matrix. */ dlaqsy_(uplo, &n, &afac[1], &lda, &s[1], & scond, &amax, equed); } } /* Save the condition number of the */ /* non-equilibrated system for use in DGET04. */ if (equil) { roldc = rcondc; } /* Compute the 1-norm of A. */ anorm = dlansy_("1", uplo, &n, &afac[1], &lda, & rwork[1]); /* Factor the matrix A. */ dpotrf_(uplo, &n, &afac[1], &lda, &info); /* Form the inverse of A. */ dlacpy_(uplo, &n, &n, &afac[1], &lda, &a[1], &lda); dpotri_(uplo, &n, &a[1], &lda, &info); /* Compute the 1-norm condition number of A. */ ainvnm = dlansy_("1", uplo, &n, &a[1], &lda, & rwork[1]); if (anorm <= 0. || ainvnm <= 0.) { rcondc = 1.; } else { rcondc = 1. / anorm / ainvnm; } } /* Restore the matrix A. */ dlacpy_(uplo, &n, &n, &asav[1], &lda, &a[1], &lda); /* Form an exact solution and set the right hand side. */ s_copy(srnamc_1.srnamt, "DLARHS", (ftnlen)32, (ftnlen) 6); dlarhs_(path, xtype, uplo, " ", &n, &n, &kl, &ku, nrhs, &a[1], &lda, &xact[1], &lda, &b[1], & lda, iseed, &info); *(unsigned char *)xtype = 'C'; dlacpy_("Full", &n, nrhs, &b[1], &lda, &bsav[1], &lda); if (nofact) { /* --- Test DPOSV --- */ /* Compute the L*L' or U'*U factorization of the */ /* matrix and solve the system. */ dlacpy_(uplo, &n, &n, &a[1], &lda, &afac[1], &lda); dlacpy_("Full", &n, nrhs, &b[1], &lda, &x[1], & lda); s_copy(srnamc_1.srnamt, "DPOSV ", (ftnlen)32, ( ftnlen)6); dposv_(uplo, &n, nrhs, &afac[1], &lda, &x[1], & lda, &info); /* Check error code from DPOSV . */ if (info != izero) { alaerh_(path, "DPOSV ", &info, &izero, uplo, & n, &n, &c_n1, &c_n1, nrhs, &imat, & nfail, &nerrs, nout); goto L70; } else if (info != 0) { goto L70; } /* Reconstruct matrix from factors and compute */ /* residual. */ dpot01_(uplo, &n, &a[1], &lda, &afac[1], &lda, & rwork[1], result); /* Compute residual of the computed solution. */ dlacpy_("Full", &n, nrhs, &b[1], &lda, &work[1], & lda); dpot02_(uplo, &n, nrhs, &a[1], &lda, &x[1], &lda, &work[1], &lda, &rwork[1], &result[1]); /* Check solution from generated exact solution. */ dget04_(&n, nrhs, &x[1], &lda, &xact[1], &lda, & rcondc, &result[2]); nt = 3; /* Print information about the tests that did not */ /* pass the threshold. */ i__4 = nt; for (k = 1; k <= i__4; ++k) { if (result[k - 1] >= *thresh) { if (nfail == 0 && nerrs == 0) { aladhd_(nout, path); } io___48.ciunit = *nout; s_wsfe(&io___48); do_fio(&c__1, "DPOSV ", (ftnlen)6); do_fio(&c__1, uplo, (ftnlen)1); do_fio(&c__1, (char *)&n, (ftnlen)sizeof( integer)); do_fio(&c__1, (char *)&imat, (ftnlen) sizeof(integer)); do_fio(&c__1, (char *)&k, (ftnlen)sizeof( integer)); do_fio(&c__1, (char *)&result[k - 1], ( ftnlen)sizeof(doublereal)); e_wsfe(); ++nfail; } /* L60: */ } nrun += nt; L70: ; } /* --- Test DPOSVX --- */ if (! prefac) { dlaset_(uplo, &n, &n, &c_b50, &c_b50, &afac[1], & lda); } dlaset_("Full", &n, nrhs, &c_b50, &c_b50, &x[1], &lda); if (iequed > 1 && n > 0) { /* Equilibrate the matrix if FACT='F' and */ /* EQUED='Y'. */ dlaqsy_(uplo, &n, &a[1], &lda, &s[1], &scond, & amax, equed); } /* Solve the system and compute the condition number */ /* and error bounds using DPOSVX. */ s_copy(srnamc_1.srnamt, "DPOSVX", (ftnlen)32, (ftnlen) 6); dposvx_(fact, uplo, &n, nrhs, &a[1], &lda, &afac[1], & lda, equed, &s[1], &b[1], &lda, &x[1], &lda, & rcond, &rwork[1], &rwork[*nrhs + 1], &work[1], &iwork[1], &info); /* Check the error code from DPOSVX. */ if (info != izero) { /* Writing concatenation */ i__5[0] = 1, a__1[0] = fact; i__5[1] = 1, a__1[1] = uplo; s_cat(ch__1, a__1, i__5, &c__2, (ftnlen)2); alaerh_(path, "DPOSVX", &info, &izero, ch__1, &n, &n, &c_n1, &c_n1, nrhs, &imat, &nfail, & nerrs, nout); goto L90; } if (info == 0) { if (! prefac) { /* Reconstruct matrix from factors and compute */ /* residual. */ dpot01_(uplo, &n, &a[1], &lda, &afac[1], &lda, &rwork[(*nrhs << 1) + 1], result); k1 = 1; } else { k1 = 2; } /* Compute residual of the computed solution. */ dlacpy_("Full", &n, nrhs, &bsav[1], &lda, &work[1] , &lda); dpot02_(uplo, &n, nrhs, &asav[1], &lda, &x[1], & lda, &work[1], &lda, &rwork[(*nrhs << 1) + 1], &result[1]); /* Check solution from generated exact solution. */ if (nofact || prefac && lsame_(equed, "N")) { dget04_(&n, nrhs, &x[1], &lda, &xact[1], &lda, &rcondc, &result[2]); } else { dget04_(&n, nrhs, &x[1], &lda, &xact[1], &lda, &roldc, &result[2]); } /* Check the error bounds from iterative */ /* refinement. */ dpot05_(uplo, &n, nrhs, &asav[1], &lda, &b[1], & lda, &x[1], &lda, &xact[1], &lda, &rwork[ 1], &rwork[*nrhs + 1], &result[3]); } else { k1 = 6; } /* Compare RCOND from DPOSVX with the computed value */ /* in RCONDC. */ result[5] = dget06_(&rcond, &rcondc); /* Print information about the tests that did not pass */ /* the threshold. */ for (k = k1; k <= 6; ++k) { if (result[k - 1] >= *thresh) { if (nfail == 0 && nerrs == 0) { aladhd_(nout, path); } if (prefac) { io___51.ciunit = *nout; s_wsfe(&io___51); do_fio(&c__1, "DPOSVX", (ftnlen)6); do_fio(&c__1, fact, (ftnlen)1); do_fio(&c__1, uplo, (ftnlen)1); do_fio(&c__1, (char *)&n, (ftnlen)sizeof( integer)); do_fio(&c__1, equed, (ftnlen)1); do_fio(&c__1, (char *)&imat, (ftnlen) sizeof(integer)); do_fio(&c__1, (char *)&k, (ftnlen)sizeof( integer)); do_fio(&c__1, (char *)&result[k - 1], ( ftnlen)sizeof(doublereal)); e_wsfe(); } else { io___52.ciunit = *nout; s_wsfe(&io___52); do_fio(&c__1, "DPOSVX", (ftnlen)6); do_fio(&c__1, fact, (ftnlen)1); do_fio(&c__1, uplo, (ftnlen)1); do_fio(&c__1, (char *)&n, (ftnlen)sizeof( integer)); do_fio(&c__1, (char *)&imat, (ftnlen) sizeof(integer)); do_fio(&c__1, (char *)&k, (ftnlen)sizeof( integer)); do_fio(&c__1, (char *)&result[k - 1], ( ftnlen)sizeof(doublereal)); e_wsfe(); } ++nfail; } /* L80: */ } nrun = nrun + 7 - k1; L90: ; } /* L100: */ } L110: ; } L120: ; } /* L130: */ } /* Print a summary of the results. */ alasvm_(path, nout, &nfail, &nrun, &nerrs); return 0; /* End of DDRVPO */ } /* ddrvpo_ */
/* Subroutine */ int dsysvxx_(char *fact, char *uplo, integer *n, integer * nrhs, doublereal *a, integer *lda, doublereal *af, integer *ldaf, integer *ipiv, char *equed, doublereal *s, doublereal *b, integer * ldb, doublereal *x, integer *ldx, doublereal *rcond, doublereal * rpvgrw, doublereal *berr, integer *n_err_bnds__, doublereal * err_bnds_norm__, doublereal *err_bnds_comp__, integer *nparams, doublereal *params, doublereal *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; doublereal d__1, d__2; /* Local variables */ integer j; doublereal amax, smin, smax; doublereal scond; logical equil, rcequ; logical nofact; doublereal bignum; integer infequ; doublereal smlnum; /* -- 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. -- */ /* Purpose */ /* ======= */ /* DSYSVXX uses the diagonal pivoting factorization to compute the */ /* solution to a double precision system of linear equations A * X = B, where A */ /* is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices. */ /* If requested, both normwise and maximum componentwise error bounds */ /* are returned. DSYSVXX 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. */ /* DSYSVXX 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 */ /* DSYSVXX 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 DSYSVXX would itself produce. */ /* Description */ /* =========== */ /* The following steps are performed: */ /* 1. If FACT = 'E', double precision 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 LU decomposition is used to factor */ /* the matrix A (after equilibration if FACT = 'E') as */ /* A = U * D * U**T, if UPLO = 'U', or */ /* A = L * D * L**T, if UPLO = 'L', */ /* where U (or L) is a product of permutation and unit upper (lower) */ /* triangular matrices, and D is symmetric and block diagonal with */ /* 1-by-1 and 2-by-2 diagonal blocks. */ /* 3. If some D(i,i)=0, so that D is exactly singular, 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(R) 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 and IPIV contain the factored form of A. */ /* If EQUED is not 'N', the matrix A has been */ /* equilibrated with scaling factors given by S. */ /* A, AF, and IPIV 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) DOUBLE PRECISION 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. */ /* 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) DOUBLE PRECISION array, dimension (LDAF,N) */ /* If FACT = 'F', then AF is an input argument and on entry */ /* 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 DSYTRF. */ /* If FACT = 'N', then AF is an output argument and on exit */ /* returns 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. */ /* LDAF (input) INTEGER */ /* The leading dimension of the array AF. LDAF >= max(1,N). */ /* IPIV (input or output) INTEGER array, dimension (N) */ /* If FACT = 'F', then IPIV is an input argument and on entry */ /* contains details of the interchanges and the block */ /* structure of D, as determined by DSYTRF. If IPIV(k) > 0, */ /* then rows and columns k and IPIV(k) were interchanged and */ /* D(k,k) is a 1-by-1 diagonal block. If UPLO = 'U' and */ /* IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and */ /* -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 */ /* diagonal block. If UPLO = 'L' and IPIV(k) = IPIV(k+1) < 0, */ /* then rows and columns k+1 and -IPIV(k) were interchanged */ /* and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. */ /* If FACT = 'N', then IPIV is an output argument and on exit */ /* contains details of the interchanges and the block */ /* structure of D, as determined by DSYTRF. */ /* 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) DOUBLE PRECISION array, dimension (N) */ /* The 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) 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. */ /* RPVGRW (output) DOUBLE PRECISION */ /* 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<INFO<=N, then this contains the reciprocal pivot growth factor */ /* for the leading INFO columns of A. */ /* 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 extra-precise refinement algorithm. */ /* (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) DOUBLE PRECISION array, dimension (4*N) */ /* IWORK (workspace) INTEGER array, dimension (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. */ /* ================================================================== */ /* 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; --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 = dlamch_("Safe minimum"); bignum = 1. / 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 DSYRFSX. */ *rpvgrw = 0.; /* Test the input parameters. PARAMS is not tested until DSYRFSX. */ 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.; i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MIN */ d__1 = smin, d__2 = s[j]; smin = min(d__1,d__2); /* Computing MAX */ d__1 = smax, d__2 = s[j]; smax = max(d__1,d__2); } if (smin <= 0.) { *info = -10; } else if (*n > 0) { scond = max(smin,smlnum) / min(smax,bignum); } else { scond = 1.; } } 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_("DSYSVXX", &i__1); return 0; } if (equil) { /* Compute row and column scalings to equilibrate the matrix A. */ dsyequb_(uplo, n, &a[a_offset], lda, &s[1], &scond, &amax, &work[1], & infequ); if (infequ == 0) { /* Equilibrate the matrix. */ dlaqsy_(uplo, n, &a[a_offset], lda, &s[1], &scond, &amax, equed); rcequ = lsame_(equed, "Y"); } } /* Scale the right-hand side. */ if (rcequ) { dlascl2_(n, nrhs, &s[1], &b[b_offset], ldb); } if (nofact || equil) { /* Compute the LU factorization of A. */ dlacpy_(uplo, n, n, &a[a_offset], lda, &af[af_offset], ldaf); i__1 = max(1,*n) * 5; dsytrf_(uplo, n, &af[af_offset], ldaf, &ipiv[1], &work[1], &i__1, 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. */ if (*n > 0) { *rpvgrw = dla_syrpvgrw__(uplo, n, info, &a[a_offset], lda, & af[af_offset], ldaf, &ipiv[1], &work[1], (ftnlen)1); } return 0; } } /* Compute the reciprocal pivot growth factor RPVGRW. */ if (*n > 0) { *rpvgrw = dla_syrpvgrw__(uplo, n, info, &a[a_offset], lda, &af[ af_offset], ldaf, &ipiv[1], &work[1], (ftnlen)1); } /* Compute the solution matrix X. */ dlacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); dsytrs_(uplo, n, nrhs, &af[af_offset], ldaf, &ipiv[1], &x[x_offset], ldx, info); /* Use iterative refinement to improve the computed solution and */ /* compute error bounds and backward error estimates for it. */ dsyrfsx_(uplo, equed, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, & ipiv[1], &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) { dlascl2_(n, nrhs, &s[1], &x[x_offset], ldx); } return 0; /* End of DSYSVXX */ } /* dsysvxx_ */
/* Subroutine */ int dsysvxx_(char *fact, char *uplo, integer *n, integer * nrhs, doublereal *a, integer *lda, doublereal *af, integer *ldaf, integer *ipiv, char *equed, doublereal *s, doublereal *b, integer * ldb, doublereal *x, integer *ldx, doublereal *rcond, doublereal * rpvgrw, doublereal *berr, integer *n_err_bnds__, doublereal * err_bnds_norm__, doublereal *err_bnds_comp__, integer *nparams, doublereal *params, doublereal *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; doublereal d__1, d__2; /* Local variables */ integer j; doublereal amax, smin, smax; extern doublereal dla_syrpvgrw_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, doublereal *); extern logical lsame_(char *, char *); doublereal scond; logical equil, rcequ; extern doublereal dlamch_(char *); logical nofact; extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), xerbla_(char *, integer *); doublereal bignum; integer infequ; extern /* Subroutine */ int dlaqsy_(char *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, char *); doublereal smlnum; extern /* Subroutine */ int dsytrf_(char *, integer *, doublereal *, integer *, integer *, doublereal *, integer *, integer *), dlascl2_(integer *, integer *, doublereal *, doublereal *, integer *), dsytrs_(char *, integer *, integer *, doublereal *, integer *, integer *, doublereal *, integer *, integer *), dsyequb_(char *, integer *, doublereal *, integer *, doublereal * , doublereal *, doublereal *, doublereal *, integer *), dsyrfsx_(char *, char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, integer *, doublereal *, doublereal *, integer *, integer *); /* -- LAPACK driver routine (version 3.4.2) -- */ /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */ /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */ /* September 2012 */ /* .. Scalar Arguments .. */ /* .. */ /* .. Array Arguments .. */ /* .. */ /* ================================================================== */ /* .. 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; --ipiv; --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 = dlamch_("Safe minimum"); bignum = 1. / 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 DSYRFSX. */ *rpvgrw = 0.; /* Test the input parameters. PARAMS is not tested until DSYRFSX. */ 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.; i__1 = *n; for (j = 1; j <= i__1; ++j) { /* Computing MIN */ d__1 = smin; d__2 = s[j]; // , expr subst smin = min(d__1,d__2); /* Computing MAX */ d__1 = smax; d__2 = s[j]; // , expr subst smax = max(d__1,d__2); /* L10: */ } if (smin <= 0.) { *info = -10; } else if (*n > 0) { scond = max(smin,smlnum) / min(smax,bignum); } else { scond = 1.; } } 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_("DSYSVXX", &i__1); return 0; } if (equil) { /* Compute row and column scalings to equilibrate the matrix A. */ dsyequb_(uplo, n, &a[a_offset], lda, &s[1], &scond, &amax, &work[1], & infequ); if (infequ == 0) { /* Equilibrate the matrix. */ dlaqsy_(uplo, n, &a[a_offset], lda, &s[1], &scond, &amax, equed); rcequ = lsame_(equed, "Y"); } } /* Scale the right-hand side. */ if (rcequ) { dlascl2_(n, nrhs, &s[1], &b[b_offset], ldb); } if (nofact || equil) { /* Compute the LDL^T or UDU^T factorization of A. */ dlacpy_(uplo, n, n, &a[a_offset], lda, &af[af_offset], ldaf); i__1 = max(1,*n) * 5; dsytrf_(uplo, n, &af[af_offset], ldaf, &ipiv[1], &work[1], &i__1, 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. */ if (*n > 0) { *rpvgrw = dla_syrpvgrw_(uplo, n, info, &a[a_offset], lda, & af[af_offset], ldaf, &ipiv[1], &work[1]); } return 0; } } /* Compute the reciprocal pivot growth factor RPVGRW. */ if (*n > 0) { *rpvgrw = dla_syrpvgrw_(uplo, n, info, &a[a_offset], lda, &af[ af_offset], ldaf, &ipiv[1], &work[1]); } /* Compute the solution matrix X. */ dlacpy_("Full", n, nrhs, &b[b_offset], ldb, &x[x_offset], ldx); dsytrs_(uplo, n, nrhs, &af[af_offset], ldaf, &ipiv[1], &x[x_offset], ldx, info); /* Use iterative refinement to improve the computed solution and */ /* compute error bounds and backward error estimates for it. */ dsyrfsx_(uplo, equed, n, nrhs, &a[a_offset], lda, &af[af_offset], ldaf, & ipiv[1], &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) { dlascl2_(n, nrhs, &s[1], &x[x_offset], ldx); } return 0; /* End of DSYSVXX */ }