realOTHERsolve(3)

NAME
realOTHERsolve

SYNOPSIS

  Functions
subroutine sgglse (M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, INFO)
SGGLSE solves overdetermined or underdetermined systems for OTHER matrices
subroutine spbsv (UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
SPBSV computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine spbsvx (FACT, UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
SPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine sppsv (UPLO, N, NRHS, AP, B, LDB, INFO)
SPPSV computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine sppsvx (FACT, UPLO, N, NRHS, AP, AFP, EQUED, S, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
SPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine sspsv (UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
SSPSV computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine sspsvx (FACT, UPLO, N, NRHS, AP, AFP, IPIV, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO)
SSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Detailed Description

This is the group of real Other Solve routines

Function Documentation

  subroutine sgglse (integer M, integer N, integer P, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real, dimension( * ) C, real, dimension( * ) D, real, dimension( * ) X, real, dimension( * ) WORK, integer LWORK, integer INFO)

SGGLSE solves overdetermined or underdetermined systems for OTHER matrices

Purpose:

     SGGLSE solves the linear equality-constrained least squares (LSE)
     problem:
             minimize || c - A*x ||_2   subject to   B*x = d
     where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
     M-vector, and d is a given P-vector. It is assumed that
     P <= N <= M+P, and
              rank(B) = P and  rank( (A) ) = N.
                                   ( (B) )
     These conditions ensure that the LSE problem has a unique solution,
     which is obtained using a generalized RQ factorization of the
     matrices (B, A) given by
        B = (0 R)*Q,   A = Z*T*Q.

Parameters:
    M

              M is INTEGER
              The number of rows of the matrix A.  M >= 0.

    N

              N is INTEGER
              The number of columns of the matrices A and B. N >= 0.

    P

              P is INTEGER
              The number of rows of the matrix B. 0 <= P <= N <= M+P.

    A

              A is REAL array, dimension (LDA,N)
              On entry, the M-by-N matrix A.
              On exit, the elements on and above the diagonal of the array
              contain the min(M,N)-by-N upper trapezoidal matrix T.

    LDA

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

    B

              B is REAL array, dimension (LDB,N)
              On entry, the P-by-N matrix B.
              On exit, the upper triangle of the subarray B(1:P,N-P+1:N)
              contains the P-by-P upper triangular matrix R.

    LDB

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

    C

              C is REAL array, dimension (M)
              On entry, C contains the right hand side vector for the
              least squares part of the LSE problem.
              On exit, the residual sum of squares for the solution
              is given by the sum of squares of elements N-P+1 to M of
              vector C.

    D

              D is REAL array, dimension (P)
              On entry, D contains the right hand side vector for the
              constrained equation.
              On exit, D is destroyed.

    X

              X is REAL array, dimension (N)
              On exit, X is the solution of the LSE problem.

    WORK

              WORK is REAL array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK. LWORK >= max(1,M+N+P).
              For optimum performance LWORK >= P+min(M,N)+max(M,N)*NB,
              where NB is an upper bound for the optimal blocksizes for
              SGEQRF, SGERQF, SORMQR and SORMRQ.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the optimal size of the WORK array, returns
              this value as the first entry of the WORK array, and no error
              message related to LWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit.
              < 0:  if INFO = -i, the i-th argument had an illegal value.
              = 1:  the upper triangular factor R associated with B in the
                    generalized RQ factorization of the pair (B, A) is
                    singular, so that rank(B) < P; the least squares
                    solution could not be computed.
              = 2:  the (N-P) by (N-P) part of the upper trapezoidal factor
                    T associated with A in the generalized RQ factorization
                    of the pair (B, A) is singular, so that
                    rank( (A) ) < N; the least squares solution could not
                        ( (B) )
                    be computed.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Definition at line 182 of file sgglse.f.

  subroutine spbsv (character UPLO, integer N, integer KD, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldb, * ) B, integer LDB, integer INFO)

SPBSV computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

     SPBSV computes the solution to a real system of linear equations
        A * X = B,
     where A is an N-by-N symmetric positive definite band 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 band matrix, and L is a lower
     triangular band matrix, with the same number of superdiagonals or
     subdiagonals as A.  The factored form of A is then used to solve the
     system of equations A * X = B.

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.

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 0.

    NRHS

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

    AB

              AB is REAL array, dimension (LDAB,N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array.  The
              j-th column of A is stored in the j-th column of the array AB
              as follows:
              if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
              See below for further details.
              On exit, if INFO = 0, the triangular factor U or L from the
              Cholesky factorization A = U**T*U or A = L*L**T of the band
              matrix A, in the same storage format as A.

    LDAB

              LDAB is INTEGER
              The leading dimension of the array AB.  LDAB >= KD+1.

    B

              B is 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

              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
              > 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.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Further Details:

      The band storage scheme is illustrated by the following example, when
      N = 6, KD = 2, and UPLO = 'U':
      On entry:                       On exit:
          *    *   a13  a24  a35  a46      *    *   u13  u24  u35  u46
          *   a12  a23  a34  a45  a56      *   u12  u23  u34  u45  u56
         a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55  u66
      Similarly, if UPLO = 'L' the format of A is as follows:
      On entry:                       On exit:
         a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55  l66
         a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65   *
         a31  a42  a53  a64   *    *      l31  l42  l53  l64   *    *
      Array elements marked * are not used by the routine.

Definition at line 166 of file spbsv.f.

  subroutine spbsvx (character FACT, character UPLO, integer N, integer KD, integer NRHS, real, dimension( ldab, * ) AB, integer LDAB, real, dimension( ldafb, * ) AFB, integer LDAFB, character EQUED, real, dimension( * ) S, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

     SPBSVX 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 band 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 band matrix, and L is a lower
        triangular band 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.

Parameters:
    FACT

              FACT is 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, AFB contains the factored form of A.
                      If EQUED = 'Y', the matrix A has been equilibrated
                      with scaling factors given by S.  AB and AFB will not
                      be modified.
              = 'N':  The matrix A will be copied to AFB and factored.
              = 'E':  The matrix A will be equilibrated if necessary, then
                      copied to AFB and factored.

    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.

    KD

              KD is INTEGER
              The number of superdiagonals of the matrix A if UPLO = 'U',
              or the number of subdiagonals if UPLO = 'L'.  KD >= 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.

    AB

              AB is REAL array, dimension (LDAB,N)
              On entry, the upper or lower triangle of the symmetric band
              matrix A, stored in the first KD+1 rows of the array, except
              if FACT = 'F' and EQUED = 'Y', then A must contain the
              equilibrated matrix diag(S)*A*diag(S).  The j-th column of A
              is stored in the j-th column of the array AB as follows:
              if UPLO = 'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
              if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(N,j+KD).
              See below for further details.
              On exit, if FACT = 'E' and EQUED = 'Y', A is overwritten by
              diag(S)*A*diag(S).

    LDAB

              LDAB is INTEGER
              The leading dimension of the array A.  LDAB >= KD+1.

    AFB

              AFB is REAL array, dimension (LDAFB,N)
              If FACT = 'F', then AFB 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 of the band matrix
              A, in the same storage format as A (see AB).  If EQUED = 'Y',
              then AFB is the factored form of the equilibrated matrix A.
              If FACT = 'N', then AFB 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.
              If FACT = 'E', then AFB 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).

    LDAFB

              LDAFB is INTEGER
              The leading dimension of the array AFB.  LDAFB >= KD+1.

    EQUED

              EQUED is 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

              S is 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

              B is 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

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

    X

              X is 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

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

    RCOND

              RCOND is 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

              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
              > 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.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    April 2012

Further Details:

      The band storage scheme is illustrated by the following example, when
      N = 6, KD = 2, and UPLO = 'U':
      Two-dimensional storage of the symmetric matrix A:
         a11  a12  a13
              a22  a23  a24
                   a33  a34  a35
                        a44  a45  a46
                             a55  a56
         (aij=conjg(aji))         a66
      Band storage of the upper triangle of A:
          *    *   a13  a24  a35  a46
          *   a12  a23  a34  a45  a56
         a11  a22  a33  a44  a55  a66
      Similarly, if UPLO = 'L' the format of A is as follows:
         a11  a22  a33  a44  a55  a66
         a21  a32  a43  a54  a65   *
         a31  a42  a53  a64   *    *
      Array elements marked * are not used by the routine.

Definition at line 345 of file spbsvx.f.

  subroutine sppsv (character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real, dimension( ldb, * ) B, integer LDB, integer INFO)

SPPSV computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

     SPPSV computes the solution to a real system of linear equations
        A * X = B,
     where A is an N-by-N symmetric positive definite matrix stored in
     packed format 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.

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.

    NRHS

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

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
              See below for further details.
              On exit, if INFO = 0, the factor U or L from the Cholesky
              factorization A = U**T*U or A = L*L**T, in the same storage
              format as A.

    B

              B is 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

              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
              > 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.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Further Details:

      The packed storage scheme is illustrated by the following example
      when N = 4, UPLO = 'U':
      Two-dimensional storage of the symmetric matrix A:
         a11 a12 a13 a14
             a22 a23 a24
                 a33 a34     (aij = conjg(aji))
                     a44
      Packed storage of the upper triangle of A:
      AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

Definition at line 146 of file sppsv.f.

  subroutine sppsvx (character FACT, character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real, dimension( * ) AFP, character EQUED, real, dimension( * ) S, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SPPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

     SPPSVX 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 stored in
     packed format 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.

Parameters:
    FACT

              FACT is 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, AFP contains the factored form of A.
                      If EQUED = 'Y', the matrix A has been equilibrated
                      with scaling factors given by S.  AP and AFP will not
                      be modified.
              = 'N':  The matrix A will be copied to AFP and factored.
              = 'E':  The matrix A will be equilibrated if necessary, then
                      copied to AFP and factored.

    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 matrices B and X.  NRHS >= 0.

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array, except if FACT = 'F'
              and EQUED = 'Y', then A must contain the equilibrated matrix
              diag(S)*A*diag(S).  The j-th column of A is stored in the
              array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
              See below for further details.  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).

    AFP

              AFP is REAL array, dimension (N*(N+1)/2)
              If FACT = 'F', then AFP 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 AFP is the factored
              form of the equilibrated matrix A.
              If FACT = 'N', then AFP 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 AFP 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 AP for the form of the
              equilibrated matrix).

    EQUED

              EQUED is 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

              S is 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

              B is 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

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

    X

              X is 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

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

    RCOND

              RCOND is 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

              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
              > 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.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    April 2012

Further Details:

      The packed storage scheme is illustrated by the following example
      when N = 4, UPLO = 'U':
      Two-dimensional storage of the symmetric matrix A:
         a11 a12 a13 a14
             a22 a23 a24
                 a33 a34     (aij = conjg(aji))
                     a44
      Packed storage of the upper triangle of A:
      AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

Definition at line 313 of file sppsvx.f.

  subroutine sspsv (character UPLO, integer N, integer NRHS, real, dimension( * ) AP, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, integer INFO)

SSPSV computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

     SSPSV computes the solution to a real system of linear equations
        A * X = B,
     where A is an N-by-N symmetric matrix stored in packed format and X
     and B are N-by-NRHS matrices.
     The diagonal pivoting method is used to factor A as
        A = U * D * U**T,  if UPLO = 'U', or
        A = L * D * L**T,  if UPLO = 'L',
     where U (or L) is a product of permutation and unit upper (lower)
     triangular matrices, D is symmetric and block diagonal with 1-by-1
     and 2-by-2 diagonal blocks.  The factored form of A is then used to
     solve the system of equations A * X = B.

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.

    NRHS

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

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
              See below for further details.
              On exit, the block diagonal matrix D and the multipliers used
              to obtain the factor U or L from the factorization
              A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as
              a packed triangular matrix in the same storage format as A.

    IPIV

              IPIV is INTEGER array, dimension (N)
              Details of the interchanges and the block structure of D, as
              determined by SSPTRF.  If IPIV(k) > 0, then rows and columns
              k and IPIV(k) were interchanged, and D(k,k) is a 1-by-1
              diagonal block.  If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0,
              then rows and columns k-1 and -IPIV(k) were interchanged and
              D(k-1:k,k-1:k) is a 2-by-2 diagonal block.  If UPLO = 'L' and
              IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and
              -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
              diagonal block.

    B

              B is 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

              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
              > 0:  if INFO = i, D(i,i) is exactly zero.  The factorization
                    has been completed, but the block diagonal matrix D is
                    exactly singular, so the solution could not be
                    computed.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

Further Details:

      The packed storage scheme is illustrated by the following example
      when N = 4, UPLO = 'U':
      Two-dimensional storage of the symmetric matrix A:
         a11 a12 a13 a14
             a22 a23 a24
                 a33 a34     (aij = aji)
                     a44
      Packed storage of the upper triangle of A:
      AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

Definition at line 164 of file sspsv.f.

  subroutine sspsvx (character FACT, character UPLO, integer N, integer NRHS, real, dimension( * ) AP, real, dimension( * ) AFP, integer, dimension( * ) IPIV, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx, * ) X, integer LDX, real RCOND, real, dimension( * ) FERR, real, dimension( * ) BERR, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

SSPSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Purpose:

     SSPSVX uses the diagonal pivoting factorization A = U*D*U**T or
     A = L*D*L**T to compute the solution to a real system of linear
     equations A * X = B, where A is an N-by-N symmetric matrix stored
     in packed format 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 = 'N', the diagonal pivoting method is used to factor A as
           A = U * D * U**T,  if UPLO = 'U', or
           A = L * D * L**T,  if UPLO = 'L',
        where U (or L) is a product of permutation and unit upper (lower)
        triangular matrices and D is symmetric and block diagonal with
        1-by-1 and 2-by-2 diagonal blocks.
     2. If some D(i,i)=0, so that D is exactly singular, then the routine
        returns with INFO = i. Otherwise, the factored form of A is used
        to estimate the condition number of the matrix A.  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.
     3. The system of equations is solved for X using the factored form
        of A.
     4. Iterative refinement is applied to improve the computed solution
        matrix and calculate error bounds and backward error estimates
        for it.

Parameters:
    FACT

              FACT is CHARACTER*1
              Specifies whether or not the factored form of A has been
              supplied on entry.
              = 'F':  On entry, AFP and IPIV contain the factored form of
                      A.  AP, AFP and IPIV will not be modified.
              = 'N':  The matrix A will be copied to AFP and factored.

    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 matrices B and X.  NRHS >= 0.

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              The upper or lower triangle of the symmetric matrix A, packed
              columnwise in a linear array.  The j-th column of A is stored
              in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              See below for further details.

    AFP

              AFP is REAL array, dimension (N*(N+1)/2)
              If FACT = 'F', then AFP is an input argument and on entry
              contains the block diagonal matrix D and the multipliers used
              to obtain the factor U or L from the factorization
              A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as
              a packed triangular matrix in the same storage format as A.
              If FACT = 'N', then AFP is an output argument and on exit
              contains the block diagonal matrix D and the multipliers used
              to obtain the factor U or L from the factorization
              A = U*D*U**T or A = L*D*L**T as computed by SSPTRF, stored as
              a packed triangular matrix in the same storage format as A.

    IPIV

              IPIV is INTEGER array, dimension (N)
              If FACT = 'F', then IPIV is an input argument and on entry
              contains details of the interchanges and the block structure
              of D, as determined by SSPTRF.
              If IPIV(k) > 0, then rows and columns k and IPIV(k) were
              interchanged and D(k,k) is a 1-by-1 diagonal block.
              If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
              columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
              is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
              IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
              interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
              If FACT = 'N', then IPIV is an output argument and on exit
              contains details of the interchanges and the block structure
              of D, as determined by SSPTRF.

    B

              B is REAL array, dimension (LDB,NRHS)
              The N-by-NRHS 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)
              If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

    LDX

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

    RCOND

              RCOND is REAL
              The estimate of the reciprocal condition number of the matrix
              A.  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

              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
              > 0:  if INFO = i, and i is
                    <= N:  D(i,i) is exactly zero.  The factorization
                           has been completed but the factor D is exactly
                           singular, so the solution and error bounds could
                           not be computed. RCOND = 0 is returned.
                    = N+1: D 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.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    April 2012

Further Details:

      The packed storage scheme is illustrated by the following example
      when N = 4, UPLO = 'U':
      Two-dimensional storage of the symmetric matrix A:
         a11 a12 a13 a14
             a22 a23 a24
                 a33 a34     (aij = aji)
                     a44
      Packed storage of the upper triangle of A:
      AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]

Definition at line 278 of file sspsvx.f.

Author

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