REAL routines for symmetric or Hermitian positive definite matrix
spocon
USAGE:
rcond, info = NumRu::Lapack.spocon( uplo, a, anorm, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPOCON( UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO )
* Purpose
* =======
*
* SPOCON 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 SPOTRF.
*
* 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) REAL 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 SPOTRF.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* ANORM (input) REAL
* The 1-norm (or infinity-norm) of the symmetric matrix A.
*
* RCOND (output) REAL
* 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) REAL 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
*
* =====================================================================
*
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spoequ
USAGE:
s, scond, amax, info = NumRu::Lapack.spoequ( a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPOEQU( N, A, LDA, S, SCOND, AMAX, INFO )
* Purpose
* =======
*
* SPOEQU 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) REAL 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) REAL array, dimension (N)
* If INFO = 0, S contains the scale factors for A.
*
* SCOND (output) REAL
* 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) REAL
* 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.
*
* =====================================================================
*
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spoequb
USAGE:
s, scond, amax, info = NumRu::Lapack.spoequb( a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPOEQUB( N, A, LDA, S, SCOND, AMAX, INFO )
* Purpose
* =======
*
* SPOEQU 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) REAL 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) REAL array, dimension (N)
* If INFO = 0, S contains the scale factors for A.
*
* SCOND (output) REAL
* 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) REAL
* 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.
*
* =====================================================================
*
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sporfs
USAGE:
ferr, berr, info, x = NumRu::Lapack.sporfs( uplo, a, af, b, x, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPORFS( UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
* Purpose
* =======
*
* SPORFS 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) REAL 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) REAL 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 SPOTRF.
*
* LDAF (input) INTEGER
* The leading dimension of the array AF. LDAF >= max(1,N).
*
* B (input) REAL array, dimension (LDB,NRHS)
* The right hand side matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (input/output) REAL array, dimension (LDX,NRHS)
* On entry, the solution matrix X, as computed by SPOTRS.
* On exit, the improved solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* FERR (output) REAL 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) REAL 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) REAL 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.
*
* =====================================================================
*
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sporfsx
USAGE:
rcond, berr, err_bnds_norm, err_bnds_comp, info, s, x, params = NumRu::Lapack.sporfsx( uplo, equed, a, af, s, b, x, params, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPORFSX( 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
* =======
*
* SPORFSX 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) REAL 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) REAL 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 SPOTRF.
*
* LDAF (input) INTEGER
* The leading dimension of the array AF. LDAF >= max(1,N).
*
* S (input or output) REAL array, dimension (N)
* The row scale factors for A. If EQUED = 'Y', A is multiplied on
* the left and right by diag(S). S is an input argument if FACT =
* 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
* = 'Y', each element of S must be positive. If S is output, each
* element of S is a power of the radix. If S is input, each element
* of S should be a power of the radix to ensure a reliable solution
* and error estimates. Scaling by powers of the radix does not cause
* rounding errors unless the result underflows or overflows.
* Rounding errors during scaling lead to refining with a matrix that
* is not equivalent to the input matrix, producing error estimates
* that may not be reliable.
*
* B (input) REAL array, dimension (LDB,NRHS)
* The right hand side matrix B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (input/output) REAL array, dimension (LDX,NRHS)
* On entry, the solution matrix X, as computed by SGETRS.
* On exit, the improved solution matrix X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* RCOND (output) REAL
* Reciprocal scaled condition number. This is an estimate of the
* reciprocal Skeel condition number of the matrix A after
* equilibration (if done). If this is less than the machine
* precision (in particular, if it is zero), the matrix is singular
* to working precision. Note that the error may still be small even
* if this number is very small and the matrix appears ill-
* conditioned.
*
* BERR (output) REAL 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) REAL array, dimension (NRHS, N_ERR_BNDS)
* For each right-hand side, this array contains information about
* various error bounds and condition numbers corresponding to the
* normwise relative error, which is defined as follows:
*
* Normwise relative error in the ith solution vector:
* max_j (abs(XTRUE(j,i) - X(j,i)))
* ------------------------------
* max_j abs(X(j,i))
*
* The array is indexed by the type of error information as described
* below. There currently are up to three pieces of information
* returned.
*
* The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
* right-hand side.
*
* The second index in ERR_BNDS_NORM(:,err) contains the following
* three fields:
* err = 1 "Trust/don't trust" boolean. Trust the answer if the
* reciprocal condition number is less than the threshold
* sqrt(n) * slamch('Epsilon').
*
* err = 2 "Guaranteed" error bound: The estimated forward error,
* almost certainly within a factor of 10 of the true error
* so long as the next entry is greater than the threshold
* sqrt(n) * slamch('Epsilon'). This error bound should only
* be trusted if the previous boolean is true.
*
* err = 3 Reciprocal condition number: Estimated normwise
* reciprocal condition number. Compared with the threshold
* sqrt(n) * slamch('Epsilon') to determine if the error
* estimate is "guaranteed". These reciprocal condition
* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
* appropriately scaled matrix Z.
* Let Z = S*A, where S scales each row by a power of the
* radix so all absolute row sums of Z are approximately 1.
*
* See Lapack Working Note 165 for further details and extra
* cautions.
*
* ERR_BNDS_COMP (output) REAL array, dimension (NRHS, N_ERR_BNDS)
* For each right-hand side, this array contains information about
* various error bounds and condition numbers corresponding to the
* componentwise relative error, which is defined as follows:
*
* Componentwise relative error in the ith solution vector:
* abs(XTRUE(j,i) - X(j,i))
* max_j ----------------------
* abs(X(j,i))
*
* The array is indexed by the right-hand side i (on which the
* componentwise relative error depends), and the type of error
* information as described below. There currently are up to three
* pieces of information returned for each right-hand side. If
* componentwise accuracy is not requested (PARAMS(3) = 0.0), then
* ERR_BNDS_COMP is not accessed. If N_ERR_BNDS .LT. 3, then at most
* the first (:,N_ERR_BNDS) entries are returned.
*
* The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
* right-hand side.
*
* The second index in ERR_BNDS_COMP(:,err) contains the following
* three fields:
* err = 1 "Trust/don't trust" boolean. Trust the answer if the
* reciprocal condition number is less than the threshold
* sqrt(n) * slamch('Epsilon').
*
* err = 2 "Guaranteed" error bound: The estimated forward error,
* almost certainly within a factor of 10 of the true error
* so long as the next entry is greater than the threshold
* sqrt(n) * slamch('Epsilon'). This error bound should only
* be trusted if the previous boolean is true.
*
* err = 3 Reciprocal condition number: Estimated componentwise
* reciprocal condition number. Compared with the threshold
* sqrt(n) * slamch('Epsilon') to determine if the error
* estimate is "guaranteed". These reciprocal condition
* numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
* appropriately scaled matrix Z.
* Let Z = S*(A*diag(x)), where x is the solution for the
* current right-hand side and S scales each row of
* A*diag(x) by a power of the radix so all absolute row
* sums of Z are approximately 1.
*
* 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) REAL 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: 1.0
* = 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) REAL 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.
*
* ==================================================================
*
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sposv
USAGE:
info, a, b = NumRu::Lapack.sposv( uplo, a, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPOSV( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
* Purpose
* =======
*
* SPOSV 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) REAL 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) REAL 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 SPOTRF, SPOTRS, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
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sposvx
USAGE:
x, rcond, ferr, berr, info, a, af, equed, s, b = NumRu::Lapack.sposvx( fact, uplo, a, af, equed, s, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPOSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO )
* Purpose
* =======
*
* SPOSVX 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) REAL array, dimension (LDA,N)
* On entry, the symmetric matrix A, except if FACT = 'F' and
* EQUED = 'Y', then A must contain the equilibrated matrix
* diag(S)*A*diag(S). If UPLO = 'U', the leading
* N-by-N upper triangular part of A contains the upper
* triangular part of the matrix A, and the strictly lower
* triangular part of A is not referenced. If UPLO = 'L', the
* leading N-by-N lower triangular part of A contains the lower
* triangular part of the matrix A, and the strictly upper
* triangular part of A is not referenced. A is not modified if
* FACT = 'F' or 'N', or if FACT = 'E' and EQUED = 'N' on exit.
*
* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
* diag(S)*A*diag(S).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* AF (input or output) REAL array, dimension (LDAF,N)
* If FACT = 'F', then AF is an input argument and on entry
* contains the triangular factor U or L from the Cholesky
* factorization A = U**T*U or A = L*L**T, in the same storage
* format as A. If EQUED .ne. 'N', then AF is the factored form
* of the equilibrated matrix diag(S)*A*diag(S).
*
* If FACT = 'N', then AF is an output argument and on exit
* returns the triangular factor U or L from the Cholesky
* factorization A = U**T*U or A = L*L**T of the original
* matrix A.
*
* If FACT = 'E', then AF is an output argument and on exit
* returns the triangular factor U or L from the Cholesky
* factorization A = U**T*U or A = L*L**T of the equilibrated
* matrix A (see the description of A for the form of the
* equilibrated matrix).
*
* LDAF (input) INTEGER
* The leading dimension of the array AF. LDAF >= max(1,N).
*
* EQUED (input or output) CHARACTER*1
* Specifies the form of equilibration that was done.
* = 'N': No equilibration (always true if FACT = 'N').
* = 'Y': 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) REAL 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) REAL array, dimension (LDB,NRHS)
* On entry, the N-by-NRHS right hand side matrix B.
* On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
* B is overwritten by diag(S) * B.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (output) REAL array, dimension (LDX,NRHS)
* If INFO = 0 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) REAL
* 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) REAL 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) REAL 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) REAL 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.
*
* =====================================================================
*
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sposvxx
USAGE:
x, rcond, rpvgrw, berr, err_bnds_norm, err_bnds_comp, info, a, af, equed, s, b, params = NumRu::Lapack.sposvxx( fact, uplo, a, af, equed, s, b, params, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPOSVXX( 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
* =======
*
* SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T
* to compute the solution to a real system of linear equations
* A * X = B, where A is an N-by-N symmetric positive definite matrix
* and X and B are N-by-NRHS matrices.
*
* If requested, both normwise and maximum componentwise error bounds
* are returned. SPOSVXX will return a solution with a tiny
* guaranteed error (O(eps) where eps is the working machine
* precision) unless the matrix is very ill-conditioned, in which
* case a warning is returned. Relevant condition numbers also are
* calculated and returned.
*
* SPOSVXX accepts user-provided factorizations and equilibration
* factors; see the definitions of the FACT and EQUED options.
* Solving with refinement and using a factorization from a previous
* SPOSVXX call will also produce a solution with either O(eps)
* errors or warnings, but we cannot make that claim for general
* user-provided factorizations and equilibration factors if they
* differ from what SPOSVXX would itself produce.
*
* Description
* ===========
*
* The following steps are performed:
*
* 1. If FACT = 'E', real scaling factors are computed to equilibrate
* the system:
*
* diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
*
* Whether or not the system will be equilibrated depends on the
* scaling of the matrix A, but if equilibration is used, A is
* overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
*
* 2. If FACT = 'N' or 'E', the Cholesky decomposition is used to
* factor the matrix A (after equilibration if FACT = 'E') as
* A = U**T* U, if UPLO = 'U', or
* A = L * L**T, if UPLO = 'L',
* where U is an upper triangular matrix and L is a lower triangular
* matrix.
*
* 3. If the leading i-by-i principal minor is not positive definite,
* then the routine returns with INFO = i. Otherwise, the factored
* form of A is used to estimate the condition number of the matrix
* A (see argument RCOND). If the reciprocal of the condition number
* is less than machine precision, the routine still goes on to solve
* for X and compute error bounds as described below.
*
* 4. The system of equations is solved for X using the factored form
* of A.
*
* 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
* the routine will use iterative refinement to try to get a small
* error and error bounds. Refinement calculates the residual to at
* least twice the working precision.
*
* 6. If equilibration was used, the matrix X is premultiplied by
* diag(S) so that it solves the original system before
* equilibration.
*
* Arguments
* =========
*
* Some optional parameters are bundled in the PARAMS array. These
* settings determine how refinement is performed, but often the
* defaults are acceptable. If the defaults are acceptable, users
* can pass NPARAMS = 0 which prevents the source code from accessing
* the PARAMS argument.
*
* FACT (input) CHARACTER*1
* Specifies whether or not the factored form of the matrix A is
* supplied on entry, and if not, whether the matrix A should be
* equilibrated before it is factored.
* = 'F': On entry, AF contains the factored form of A.
* If EQUED is not 'N', the matrix A has been
* equilibrated with scaling factors given by S.
* A and AF are not modified.
* = 'N': The matrix A will be copied to AF and factored.
* = 'E': The matrix A will be equilibrated if necessary, then
* copied to AF and factored.
*
* UPLO (input) CHARACTER*1
* = 'U': Upper triangle of A is stored;
* = 'L': Lower triangle of A is stored.
*
* N (input) INTEGER
* The number of linear equations, i.e., the order of the
* matrix A. N >= 0.
*
* NRHS (input) INTEGER
* The number of right hand sides, i.e., the number of columns
* of the matrices B and X. NRHS >= 0.
*
* A (input/output) REAL array, dimension (LDA,N)
* On entry, the symmetric matrix A, except if FACT = 'F' and EQUED =
* 'Y', then A must contain the equilibrated matrix
* diag(S)*A*diag(S). If UPLO = 'U', the leading N-by-N upper
* triangular part of A contains the upper triangular part of the
* matrix A, and the strictly lower triangular part of A is not
* referenced. If UPLO = 'L', the leading N-by-N lower triangular
* part of A contains the lower triangular part of the matrix A, and
* the strictly upper triangular part of A is not referenced. A is
* not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED =
* 'N' on exit.
*
* On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
* diag(S)*A*diag(S).
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* AF (input or output) REAL array, dimension (LDAF,N)
* If FACT = 'F', then AF is an input argument and on entry
* contains the triangular factor U or L from the Cholesky
* factorization A = U**T*U or A = L*L**T, in the same storage
* format as A. If EQUED .ne. 'N', then AF is the factored
* form of the equilibrated matrix diag(S)*A*diag(S).
*
* If FACT = 'N', then AF is an output argument and on exit
* returns the triangular factor U or L from the Cholesky
* factorization A = U**T*U or A = L*L**T of the original
* matrix A.
*
* If FACT = 'E', then AF is an output argument and on exit
* returns the triangular factor U or L from the Cholesky
* factorization A = U**T*U or A = L*L**T of the equilibrated
* matrix A (see the description of A for the form of the
* equilibrated matrix).
*
* LDAF (input) INTEGER
* The leading dimension of the array AF. LDAF >= max(1,N).
*
* EQUED (input or output) CHARACTER*1
* Specifies the form of equilibration that was done.
* = 'N': No equilibration (always true if FACT = 'N').
* = 'Y': Both row and column equilibration, i.e., A has been
* replaced by diag(S) * A * diag(S).
* EQUED is an input argument if FACT = 'F'; otherwise, it is an
* output argument.
*
* S (input or output) REAL array, dimension (N)
* The row scale factors for A. If EQUED = 'Y', A is multiplied on
* the left and right by diag(S). S is an input argument if FACT =
* 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
* = 'Y', each element of S must be positive. If S is output, each
* element of S is a power of the radix. If S is input, each element
* of S should be a power of the radix to ensure a reliable solution
* and error estimates. Scaling by powers of the radix does not cause
* rounding errors unless the result underflows or overflows.
* Rounding errors during scaling lead to refining with a matrix that
* is not equivalent to the input matrix, producing error estimates
* that may not be reliable.
*
* B (input/output) REAL array, dimension (LDB,NRHS)
* On entry, the N-by-NRHS right hand side matrix B.
* On exit,
* if EQUED = 'N', B is not modified;
* if EQUED = 'Y', B is overwritten by diag(S)*B;
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= max(1,N).
*
* X (output) REAL array, dimension (LDX,NRHS)
* If INFO = 0, the N-by-NRHS solution matrix X to the original
* system of equations. Note that A and B are modified on exit if
* EQUED .ne. 'N', and the solution to the equilibrated system is
* inv(diag(S))*X.
*
* LDX (input) INTEGER
* The leading dimension of the array X. LDX >= max(1,N).
*
* RCOND (output) REAL
* Reciprocal scaled condition number. This is an estimate of the
* reciprocal Skeel condition number of the matrix A after
* equilibration (if done). If this is less than the machine
* precision (in particular, if it is zero), the matrix is singular
* to working precision. Note that the error may still be small even
* if this number is very small and the matrix appears ill-
* conditioned.
*
* RPVGRW (output) REAL
* Reciprocal pivot growth. On exit, this contains the reciprocal
* pivot growth factor norm(A)/norm(U). The "max absolute element"
* norm is used. If this is much less than 1, then the stability of
* the LU factorization of the (equilibrated) matrix A could be poor.
* This also means that the solution X, estimated condition numbers,
* and error bounds could be unreliable. If factorization fails with
* 0 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
* has been completed, but the factor U is exactly singular, so
* the solution and error bounds could not be computed. RCOND = 0
* is returned.
* = N+J: The solution corresponding to the Jth right-hand side is
* not guaranteed. The solutions corresponding to other right-
* hand sides K with K > J may not be guaranteed as well, but
* only the first such right-hand side is reported. If a small
* componentwise error is not requested (PARAMS(3) = 0.0) then
* the Jth right-hand side is the first with a normwise error
* bound that is not guaranteed (the smallest J such
* that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
* the Jth right-hand side is the first with either a normwise or
* componentwise error bound that is not guaranteed (the smallest
* J such that either ERR_BNDS_NORM(J,1) = 0.0 or
* ERR_BNDS_COMP(J,1) = 0.0). See the definition of
* ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
* about all of the right-hand sides check ERR_BNDS_NORM or
* ERR_BNDS_COMP.
*
* ==================================================================
*
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spotf2
USAGE:
info, a = NumRu::Lapack.spotf2( uplo, a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPOTF2( UPLO, N, A, LDA, INFO )
* Purpose
* =======
*
* SPOTF2 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) REAL 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.
*
* =====================================================================
*
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spotrf
USAGE:
info, a = NumRu::Lapack.spotrf( uplo, a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPOTRF( UPLO, N, A, LDA, INFO )
* Purpose
* =======
*
* SPOTRF 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) REAL 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.
*
* =====================================================================
*
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spotri
USAGE:
info, a = NumRu::Lapack.spotri( uplo, a, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPOTRI( UPLO, N, A, LDA, INFO )
* Purpose
* =======
*
* SPOTRI 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 SPOTRF.
*
* 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) REAL 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
* SPOTRF.
* 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 SLAUUM, STRTRI, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
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spotrs
USAGE:
info, b = NumRu::Lapack.spotrs( uplo, a, b, [:usage => usage, :help => help])
FORTRAN MANUAL
SUBROUTINE SPOTRS( UPLO, N, NRHS, A, LDA, B, LDB, INFO )
* Purpose
* =======
*
* SPOTRS 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 SPOTRF.
*
* 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) REAL 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 SPOTRF.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,N).
*
* B (input/output) REAL 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
*
* =====================================================================
*
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