realPOcomputational(3) real

Functions


real function sla_porcond (UPLO, N, A, LDA, AF, LDAF, CMODE, C, INFO, WORK, IWORK)
SLA_PORCOND estimates the Skeel condition number for a symmetric positive-definite matrix.
subroutine sla_porfsx_extended (PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
SLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.
real function sla_porpvgrw (UPLO, NCOLS, A, LDA, AF, LDAF, WORK)
SLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.
subroutine spocon (UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
SPOCON
subroutine spoequ (N, A, LDA, S, SCOND, AMAX, INFO)
SPOEQU
subroutine spoequb (N, A, LDA, S, SCOND, AMAX, INFO)
SPOEQUB
subroutine sporfs (UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)
SPORFS
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)
SPORFSX
subroutine spotf2 (UPLO, N, A, LDA, INFO)
SPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).
subroutine spotrf (UPLO, N, A, LDA, INFO)
SPOTRF
recursive subroutine spotrf2 (UPLO, N, A, LDA, INFO)
SPOTRF2
subroutine spotri (UPLO, N, A, LDA, INFO)
SPOTRI
subroutine spotrs (UPLO, N, NRHS, A, LDA, B, LDB, INFO)
SPOTRS

Detailed Description

This is the group of real computational functions for PO matrices

Function Documentation

real function sla_porcond (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer CMODE, real, dimension( * ) C, integer INFO, real, dimension( * ) WORK, integer, dimension( * ) IWORK)

SLA_PORCOND estimates the Skeel condition number for a symmetric positive-definite matrix.

Purpose: 

    SLA_PORCOND Estimates the Skeel condition number of  op(A) * op2(C)
    where op2 is determined by CMODE as follows
    CMODE =  1    op2(C) = C
    CMODE =  0    op2(C) = I
    CMODE = -1    op2(C) = inv(C)
    The Skeel condition number  cond(A) = norminf( |inv(A)||A| )
    is computed by computing scaling factors R such that
    diag(R)*A*op2(C) is row equilibrated and computing the standard
    infinity-norm condition number.


 

Parameters:

UPLO 

          UPLO is CHARACTER*1
       = 'U':  Upper triangle of A is stored;
       = 'L':  Lower triangle of A is stored.


N 

          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.


A 

          A is REAL array, dimension (LDA,N)
     On entry, the N-by-N matrix A.


LDA 

          LDA is INTEGER
     The leading dimension of the array A.  LDA >= max(1,N).


AF 

          AF is 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 

          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).


CMODE 

          CMODE is INTEGER
     Determines op2(C) in the formula op(A) * op2(C) as follows:
     CMODE =  1    op2(C) = C
     CMODE =  0    op2(C) = I
     CMODE = -1    op2(C) = inv(C)


C 

          C is REAL array, dimension (N)
     The vector C in the formula op(A) * op2(C).


INFO 

          INFO is INTEGER
       = 0:  Successful exit.
     i > 0:  The ith argument is invalid.


WORK 

          WORK is REAL array, dimension (3*N).
     Workspace.


IWORK 

          IWORK is INTEGER array, dimension (N).
     Workspace.


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

subroutine sla_porfsx_extended (integer PREC_TYPE, character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, logical COLEQU, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldy, * ) Y, integer LDY, real, dimension( * ) BERR_OUT, integer N_NORMS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, real, dimension( * ) RES, real, dimension(*) AYB, real, dimension( * ) DY, real, dimension( * ) Y_TAIL, real RCOND, integer ITHRESH, real RTHRESH, real DZ_UB, logical IGNORE_CWISE, integer INFO)

SLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

Purpose: 

 SLA_PORFSX_EXTENDED improves the computed solution to a system of
 linear equations by performing extra-precise iterative refinement
 and provides error bounds and backward error estimates for the solution.
 This subroutine is called by SPORFSX to perform iterative refinement.
 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. Note that this
 subroutine is only resonsible for setting the second fields of
 ERR_BNDS_NORM and ERR_BNDS_COMP.


 

Parameters:

PREC_TYPE 

          PREC_TYPE is INTEGER
     Specifies the intermediate precision to be used in refinement.
     The value is defined by ILAPREC(P) where P is a CHARACTER and
     P    = 'S':  Single
          = 'D':  Double
          = 'I':  Indigenous
          = 'X', 'E':  Extra


UPLO 

          UPLO is CHARACTER*1
       = 'U':  Upper triangle of A is stored;
       = 'L':  Lower triangle of A is stored.


N 

          N is INTEGER
     The number of linear equations, i.e., the order of the
     matrix A.  N >= 0.


NRHS 

          NRHS is INTEGER
     The number of right-hand-sides, i.e., the number of columns of the
     matrix B.


A 

          A is REAL array, dimension (LDA,N)
     On entry, the N-by-N matrix A.


LDA 

          LDA is INTEGER
     The leading dimension of the array A.  LDA >= max(1,N).


AF 

          AF is 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 

          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).


COLEQU 

          COLEQU is LOGICAL
     If .TRUE. then column equilibration was done to A before calling
     this routine. This is needed to compute the solution and error
     bounds correctly.


C 

          C is REAL array, dimension (N)
     The column scale factors for A. If COLEQU = .FALSE., C
     is not accessed. If C is input, each element of C 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 

          B is REAL array, dimension (LDB,NRHS)
     The right-hand-side matrix B.


LDB 

          LDB is INTEGER
     The leading dimension of the array B.  LDB >= max(1,N).


Y 

          Y is REAL array, dimension (LDY,NRHS)
     On entry, the solution matrix X, as computed by SPOTRS.
     On exit, the improved solution matrix Y.


LDY 

          LDY is INTEGER
     The leading dimension of the array Y.  LDY >= max(1,N).


BERR_OUT 

          BERR_OUT is REAL array, dimension (NRHS)
     On exit, BERR_OUT(j) contains the componentwise relative backward
     error for right-hand-side j from the formula
         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
     where abs(Z) is the componentwise absolute value of the matrix
     or vector Z. This is computed by SLA_LIN_BERR.


N_NORMS 

          N_NORMS is INTEGER
     Determines which error bounds to return (see ERR_BNDS_NORM
     and ERR_BNDS_COMP).
     If N_NORMS >= 1 return normwise error bounds.
     If N_NORMS >= 2 return componentwise error bounds.


ERR_BNDS_NORM 

          ERR_BNDS_NORM is 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.
     This subroutine is only responsible for setting the second field
     above.
     See Lapack Working Note 165 for further details and extra
     cautions.


ERR_BNDS_COMP 

          ERR_BNDS_COMP is 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.
     This subroutine is only responsible for setting the second field
     above.
     See Lapack Working Note 165 for further details and extra
     cautions.


RES 

          RES is REAL array, dimension (N)
     Workspace to hold the intermediate residual.


AYB 

          AYB is REAL array, dimension (N)
     Workspace. This can be the same workspace passed for Y_TAIL.


DY 

          DY is REAL array, dimension (N)
     Workspace to hold the intermediate solution.


Y_TAIL 

          Y_TAIL is REAL array, dimension (N)
     Workspace to hold the trailing bits of the intermediate solution.


RCOND 

          RCOND is 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.


ITHRESH 

          ITHRESH is INTEGER
     The maximum number of residual computations allowed for
     refinement. The default is 10. For '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.


RTHRESH 

          RTHRESH is REAL
     Determines when to stop refinement if the error estimate stops
     decreasing. Refinement will stop when the next solution no longer
     satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
     the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
     default value is 0.5. For 'aggressive' set to 0.9 to permit
     convergence on extremely ill-conditioned matrices. See LAWN 165
     for more details.


DZ_UB 

          DZ_UB is REAL
     Determines when to start considering componentwise convergence.
     Componentwise convergence is only considered after each component
     of the solution Y is stable, which we definte as the relative
     change in each component being less than DZ_UB. The default value
     is 0.25, requiring the first bit to be stable. See LAWN 165 for
     more details.


IGNORE_CWISE 

          IGNORE_CWISE is LOGICAL
     If .TRUE. then ignore componentwise convergence. Default value
     is .FALSE..


INFO 

          INFO is INTEGER
       = 0:  Successful exit.
       < 0:  if INFO = -i, the ith argument to SPOTRS had an illegal
             value


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

real function sla_porpvgrw (character*1 UPLO, integer NCOLS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, real, dimension( * ) WORK)

SLA_PORPVGRW computes the reciprocal pivot growth factor norm(A)/norm(U) for a symmetric or Hermitian positive-definite matrix.

Purpose: 

 SLA_PORPVGRW computes the reciprocal pivot growth factor
 norm(A)/norm(U). The "max absolute element" norm is used. If this is
 much less than 1, 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.


 

Parameters:

UPLO 

          UPLO is CHARACTER*1
       = 'U':  Upper triangle of A is stored;
       = 'L':  Lower triangle of A is stored.


NCOLS 

          NCOLS is INTEGER
     The number of columns of the matrix A. NCOLS >= 0.


A 

          A is REAL array, dimension (LDA,N)
     On entry, the N-by-N matrix A.


LDA 

          LDA is INTEGER
     The leading dimension of the array A.  LDA >= max(1,N).


AF 

          AF is 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 

          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).


WORK 

          WORK is REAL array, dimension (2*N)


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

subroutine spocon (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real ANORM, real RCOND, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SPOCON

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))).


 

Parameters:

UPLO 

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N 

          N is INTEGER
          The order of the matrix A.  N >= 0.


A 

          A is 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 

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).


ANORM 

          ANORM is REAL
          The 1-norm (or infinity-norm) of the symmetric matrix A.


RCOND 

          RCOND is 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 

          WORK is REAL array, dimension (3*N)


IWORK 

          IWORK is INTEGER array, dimension (N)


INFO 

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine spoequ (integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)

SPOEQU

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.


 

Parameters:

N 

          N is INTEGER
          The order of the matrix A.  N >= 0.


A 

          A is 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 

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).


S 

          S is REAL array, dimension (N)
          If INFO = 0, S contains the scale factors for A.


SCOND 

          SCOND is 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 

          AMAX is 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 

          INFO is 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.


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine spoequb (integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( * ) S, real SCOND, real AMAX, integer INFO)

SPOEQUB

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.


 

Parameters:

N 

          N is INTEGER
          The order of the matrix A.  N >= 0.


A 

          A is 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 

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).


S 

          S is REAL array, dimension (N)
          If INFO = 0, S contains the scale factors for A.


SCOND 

          SCOND is 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 

          AMAX is 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 

          INFO is 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.


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine sporfs (character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SPORFS

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.


 

Parameters:

UPLO 

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N 

          N is INTEGER
          The order of the matrix A.  N >= 0.


NRHS 

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices B and X.  NRHS >= 0.


A 

          A is 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 

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).


AF 

          AF is 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 

          LDAF is INTEGER
          The leading dimension of the array AF.  LDAF >= max(1,N).


B 

          B is REAL array, dimension (LDB,NRHS)
          The right hand side matrix B.


LDB 

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).


X 

          X is REAL array, dimension (LDX,NRHS)
          On entry, the solution matrix X, as computed by SPOTRS.
          On exit, the improved solution matrix X.


LDX 

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(1,N).


FERR 

          FERR is 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 

          BERR is 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 

          WORK is REAL array, dimension (3*N)


IWORK 

          IWORK is INTEGER array, dimension (N)


INFO 

          INFO is 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.


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine sporfsx (character UPLO, character EQUED, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, real, dimension( * ) S, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) BERR, integer N_ERR_BNDS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, real, dimension( * ) PARAMS, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SPORFSX

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.


  

     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.

Parameters:

UPLO 

          UPLO is CHARACTER*1
       = 'U':  Upper triangle of A is stored;
       = 'L':  Lower triangle of A is stored.


EQUED 

          EQUED is 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 

          N is INTEGER
     The order of the matrix A.  N >= 0.


NRHS 

          NRHS is INTEGER
     The number of right hand sides, i.e., the number of columns
     of the matrices B and X.  NRHS >= 0.


A 

          A is 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 

          LDA is INTEGER
     The leading dimension of the array A.  LDA >= max(1,N).


AF 

          AF is 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 

          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).


S 

          S is 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 

          B is REAL array, dimension (LDB,NRHS)
     The right hand side matrix B.


LDB 

          LDB is INTEGER
     The leading dimension of the array B.  LDB >= max(1,N).


X 

          X is REAL array, dimension (LDX,NRHS)
     On entry, the solution matrix X, as computed by SGETRS.
     On exit, the improved solution matrix X.


LDX 

          LDX is INTEGER
     The leading dimension of the array X.  LDX >= max(1,N).


RCOND 

          RCOND is 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 

          BERR is 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 

          N_ERR_BNDS is 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 

          ERR_BNDS_NORM is 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 

          ERR_BNDS_COMP is 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 

          NPARAMS is INTEGER
     Specifies the number of parameters set in PARAMS.  If .LE. 0, the
     PARAMS array is never referenced and default values are used.


PARAMS 

          PARAMS is 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 

          WORK is REAL array, dimension (4*N)


IWORK 

          IWORK is INTEGER array, dimension (N)


INFO 

          INFO is 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.


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

April 2012

subroutine spotf2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO)

SPOTF2 computes the Cholesky factorization of a symmetric/Hermitian positive definite matrix (unblocked algorithm).

Purpose: 

 SPOTF2 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 unblocked version of the algorithm, calling Level 2 BLAS.


 

Parameters:

UPLO 

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular


N 

          N is INTEGER
          The order of the matrix A.  N >= 0.


A 

          A is 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 

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).


INFO 

          INFO is 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.


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

subroutine spotrf (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO)

SPOTRF

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.


 

Parameters:

UPLO 

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N 

          N is INTEGER
          The order of the matrix A.  N >= 0.


A 

          A is 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 

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).


INFO 

          INFO is 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.


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2015

recursive subroutine spotrf2 (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO)

SPOTRF2

Purpose: 

 SPOTRF2 computes the Cholesky factorization of a real symmetric
 positive definite matrix A using the recursive algorithm.
 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 recursive version of the algorithm. It divides
 the matrix into four submatrices:
        [  A11 | A12  ]  where A11 is n1 by n1 and A22 is n2 by n2
    A = [ -----|----- ]  with n1 = n/2
        [  A21 | A22  ]       n2 = n-n1
 The subroutine calls itself to factor A11. Update and scale A21
 or A12, update A22 then call itself to factor A22.


 

Parameters:

UPLO 

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N 

          N is INTEGER
          The order of the matrix A.  N >= 0.


A 

          A is 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 

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).


INFO 

          INFO is 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.


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2015

subroutine spotri (character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, integer INFO)

SPOTRI

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.


 

Parameters:

UPLO 

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N 

          N is INTEGER
          The order of the matrix A.  N >= 0.


A 

          A is 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 

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).


INFO 

          INFO is 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.


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

subroutine spotrs (character UPLO, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer INFO)

SPOTRS

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.


 

Parameters:

UPLO 

          UPLO is CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.


N 

          N is INTEGER
          The order of the matrix A.  N >= 0.


NRHS 

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrix B.  NRHS >= 0.


A 

          A is 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 

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).


B 

          B is REAL array, dimension (LDB,NRHS)
          On entry, the right hand side matrix B.
          On exit, the solution matrix X.


LDB 

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).


INFO 

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Author

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