USAGE: rcond, info = NumRu::Lapack.dpocon( uplo, a, anorm, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DPOCON( UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO ) * Purpose * ======= * * DPOCON estimates the reciprocal of the condition number (in the * 1-norm) of a real symmetric positive definite matrix using the * Cholesky factorization A = U**T*U or A = L*L**T computed by DPOTRF. * * An estimate is obtained for norm(inv(A)), and the reciprocal of the * condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))). * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input) DOUBLE PRECISION array, dimension (LDA,N) * The triangular factor U or L from the Cholesky factorization * A = U**T*U or A = L*L**T, as computed by DPOTRF. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * ANORM (input) DOUBLE PRECISION * The 1-norm (or infinity-norm) of the symmetric matrix A. * * RCOND (output) DOUBLE PRECISION * The reciprocal of the condition number of the matrix A, * computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an * estimate of the 1-norm of inv(A) computed in this routine. * * 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 * * ===================================================================== *go to the page top

USAGE: s, scond, amax, info = NumRu::Lapack.dpoequ( a, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DPOEQU( N, A, LDA, S, SCOND, AMAX, INFO ) * Purpose * ======= * * DPOEQU computes row and column scalings intended to equilibrate a * symmetric positive definite matrix A and reduce its condition number * (with respect to the two-norm). S contains the scale factors, * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This * choice of S puts the condition number of B within a factor N of the * smallest possible condition number over all possible diagonal * scalings. * * Arguments * ========= * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input) DOUBLE PRECISION array, dimension (LDA,N) * The N-by-N symmetric positive definite matrix whose scaling * factors are to be computed. Only the diagonal elements of A * are referenced. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * S (output) DOUBLE PRECISION array, dimension (N) * If INFO = 0, S contains the scale factors for A. * * SCOND (output) DOUBLE PRECISION * If INFO = 0, S contains the ratio of the smallest S(i) to * the largest S(i). If SCOND >= 0.1 and AMAX is neither too * large nor too small, it is not worth scaling by S. * * AMAX (output) DOUBLE PRECISION * Absolute value of largest matrix element. If AMAX is very * close to overflow or very close to underflow, the matrix * should be scaled. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, the i-th diagonal element is nonpositive. * * ===================================================================== *go to the page top

USAGE: s, scond, amax, info = NumRu::Lapack.dpoequb( a, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO ) * Purpose * ======= * * DPOEQU computes row and column scalings intended to equilibrate a * symmetric positive definite matrix A and reduce its condition number * (with respect to the two-norm). S contains the scale factors, * S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with * elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This * choice of S puts the condition number of B within a factor N of the * smallest possible condition number over all possible diagonal * scalings. * * Arguments * ========= * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input) DOUBLE PRECISION array, dimension (LDA,N) * The N-by-N symmetric positive definite matrix whose scaling * factors are to be computed. Only the diagonal elements of A * are referenced. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * S (output) DOUBLE PRECISION array, dimension (N) * If INFO = 0, S contains the scale factors for A. * * SCOND (output) DOUBLE PRECISION * If INFO = 0, S contains the ratio of the smallest S(i) to * the largest S(i). If SCOND >= 0.1 and AMAX is neither too * large nor too small, it is not worth scaling by S. * * AMAX (output) DOUBLE PRECISION * Absolute value of largest matrix element. If AMAX is very * close to overflow or very close to underflow, the matrix * should be scaled. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, the i-th diagonal element is nonpositive. * * ===================================================================== *go to the page top

USAGE: ferr, berr, info, x = NumRu::Lapack.dporfs( uplo, a, af, b, x, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO ) * Purpose * ======= * * DPORFS 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. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrices B and X. NRHS >= 0. * * A (input) 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. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * AF (input) DOUBLE PRECISION 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). * * B (input) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDX,NRHS) * On entry, the solution matrix X, as computed by DPOTRS. * On exit, the improved solution matrix X. * * LDX (input) INTEGER * The leading dimension of the array X. LDX >= max(1,N). * * 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 * * Internal Parameters * =================== * * ITMAX is the maximum number of steps of iterative refinement. * * ===================================================================== *go to the page top

USAGE: rcond, berr, err_bnds_norm, err_bnds_comp, info, s, x, params = NumRu::Lapack.dporfsx( uplo, equed, a, af, s, b, x, params, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO ) * Purpose * ======= * * DPORFSX 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) 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. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * AF (input) DOUBLE PRECISION 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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) 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. * * ================================================================== *go to the page top

USAGE: info, a, b = NumRu::Lapack.dposv( uplo, a, b, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DPOSV( UPLO, N, NRHS, A, LDA, B, LDB, INFO ) * Purpose * ======= * * DPOSV computes 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. * * The Cholesky decomposition is used to factor A 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. The factored form of A is then used to solve the system of * equations A * X = B. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The number of linear equations, i.e., the order of the * matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrix B. NRHS >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, 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 INFO = 0, the factor U or L from the Cholesky * factorization A = U**T*U or A = L*L**T. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) * On entry, the N-by-NRHS right hand side matrix B. * On exit, if INFO = 0, the N-by-NRHS solution matrix X. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, 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. * * ===================================================================== * * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL DPOTRF, DPOTRS, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX * ..go to the page top

USAGE: x, rcond, ferr, berr, info, a, af, equed, s, b = NumRu::Lapack.dposvx( fact, uplo, a, af, equed, s, b, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO ) * 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. * * ===================================================================== *go to the page top

USAGE: x, rcond, rpvgrw, berr, err_bnds_norm, err_bnds_comp, info, a, af, equed, s, b, params = NumRu::Lapack.dposvxx( fact, uplo, a, af, equed, s, b, params, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO ) * Purpose * ======= * * DPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T * to compute the solution to a double precision 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. DPOSVXX 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. * * DPOSVXX 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 * DPOSVXX 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 DPOSVXX 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 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) 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': 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 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) 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 * 0go to the page top0 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. * * ================================================================== *

USAGE: info, a = NumRu::Lapack.dpotf2( uplo, a, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DPOTF2( UPLO, N, A, LDA, INFO ) * Purpose * ======= * * DPOTF2 computes the Cholesky factorization of a real symmetric * positive definite matrix A. * * The factorization has the form * A = U' * U , if UPLO = 'U', or * A = L * L', if UPLO = 'L', * where U is an upper triangular matrix and L is lower triangular. * * This is the unblocked version of the algorithm, calling Level 2 BLAS. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies whether the upper or lower triangular part of the * symmetric matrix A is stored. * = 'U': Upper triangular * = 'L': Lower triangular * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, 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 INFO = 0, the factor U or L from the Cholesky * factorization A = U'*U or A = L*L'. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -k, the k-th argument had an illegal value * > 0: if INFO = k, the leading minor of order k is not * positive definite, and the factorization could not be * completed. * * ===================================================================== *go to the page top

USAGE: info, a = NumRu::Lapack.dpotrf( uplo, a, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DPOTRF( UPLO, N, A, LDA, INFO ) * Purpose * ======= * * DPOTRF computes the Cholesky factorization of a real symmetric * positive definite matrix A. * * The factorization has the form * 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 lower triangular. * * This is the block version of the algorithm, calling Level 3 BLAS. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, 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 INFO = 0, the factor U or L from the Cholesky * factorization A = U**T*U or A = L*L**T. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, the leading minor of order i is not * positive definite, and the factorization could not be * completed. * * ===================================================================== *go to the page top

USAGE: info, a = NumRu::Lapack.dpotri( uplo, a, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DPOTRI( UPLO, N, A, LDA, INFO ) * Purpose * ======= * * DPOTRI computes the inverse of a real symmetric positive definite * matrix A using the Cholesky factorization A = U**T*U or A = L*L**T * computed by DPOTRF. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) * On entry, the triangular factor U or L from the Cholesky * factorization A = U**T*U or A = L*L**T, as computed by * DPOTRF. * On exit, the upper or lower triangle of the (symmetric) * inverse of A, overwriting the input factor U or L. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * > 0: if INFO = i, the (i,i) element of the factor U or L is * zero, and the inverse could not be computed. * * ===================================================================== * * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL DLAUUM, DTRTRI, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX * ..go to the page top

USAGE: info, b = NumRu::Lapack.dpotrs( uplo, a, b, [:usage => usage, :help => help]) FORTRAN MANUAL SUBROUTINE DPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO ) * Purpose * ======= * * DPOTRS solves a system of linear equations A*X = B with a symmetric * positive definite matrix A using the Cholesky factorization * A = U**T*U or A = L*L**T computed by DPOTRF. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * NRHS (input) INTEGER * The number of right hand sides, i.e., the number of columns * of the matrix B. NRHS >= 0. * * A (input) DOUBLE PRECISION array, dimension (LDA,N) * The triangular factor U or L from the Cholesky factorization * A = U**T*U or A = L*L**T, as computed by DPOTRF. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) * On entry, the right hand side matrix B. * On exit, the solution matrix X. * * LDB (input) INTEGER * The leading dimension of the array B. LDB >= max(1,N). * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * ===================================================================== *go to the page top

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