pbsvx(3)

NAME
pbsvx - pbsvx: factor and solve, expert

SYNOPSIS

  Functions

subroutine cpbsvx (fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
CPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine dpbsvx (fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, info)
DPBSVX 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 zpbsvx (fact, uplo, n, kd, nrhs, ab, ldab, afb, ldafb, equed, s, b, ldb, x, ldx, rcond, ferr, berr, work, rwork, info)
ZPBSVX computes the solution to system of linear equations A * X = B for OTHER matrices

Detailed Description

Function Documentation

  subroutine cpbsvx (character fact, character uplo, integer n, integer kd, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldafb, * ) afb, integer ldafb, character equed, real, dimension( * ) s, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldx, * ) x, integer ldx, real rcond, real, dimension( * ) ferr, real, dimension( * ) berr, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

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

Purpose:

    !>
    !> CPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
    !> compute the solution to a complex system of linear equations
    !>    A * X = B,
    !> where A is an N-by-N Hermitian 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**H * U,  if UPLO = 'U', or
    !>       A = L * L**H,  if UPLO = 'L',
    !>    where U is an upper triangular band matrix, and L is a lower
    !>    triangular band matrix.
    !>
    !> 3. If the leading principal minor of order i is not positive,
    !>    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 COMPLEX array, dimension (LDAB,N)
    !>          On entry, the upper or lower triangle of the Hermitian 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 COMPLEX 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**H*U or A = L*L**H 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**H*U or A = L*L**H.
    !>
    !>          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**H*U or A = L*L**H 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 COMPLEX 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 COMPLEX 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 COMPLEX array, dimension (2*N)
    !>

    RWORK

    !>          RWORK is REAL 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 principal minor of order i of A
    !>                       is not positive, 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.

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 Hermitian 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.
    !>

  subroutine dpbsvx (character fact, character uplo, integer n, integer kd, integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldafb, * ) afb, integer ldafb, character equed, double precision, dimension( * ) s, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

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

Purpose:

    !>
    !> DPBSVX 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 principal minor of order i is not positive,
    !>    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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)
    !>          The scale factors for A; not accessed if EQUED = 'N'.  S is
    !>          an input argument if FACT = 'F'; otherwise, S is an output
    !>          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
    !>          must be positive.
    !>

    B

    !>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
    !>          On entry, the N-by-NRHS right hand side matrix B.
    !>          On exit, if EQUED = 'N', B is not modified; if EQUED = 'Y',
    !>          B is overwritten by diag(S) * B.
    !>

    LDB

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

    X

    !>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
    !>          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X to
    !>          the original system of equations.  Note that if EQUED = 'Y',
    !>          A and B are modified on exit, and the solution to the
    !>          equilibrated system is inv(diag(S))*X.
    !>

    LDX

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

    RCOND

    !>          RCOND is DOUBLE PRECISION
    !>          The estimate of the reciprocal condition number of the matrix
    !>          A after equilibration (if done).  If RCOND is less than the
    !>          machine precision (in particular, if RCOND = 0), the matrix
    !>          is singular to working precision.  This condition is
    !>          indicated by a return code of INFO > 0.
    !>

    FERR

    !>          FERR is DOUBLE PRECISION array, dimension (NRHS)
    !>          The estimated forward error bound for each solution vector
    !>          X(j) (the j-th column of the solution matrix X).
    !>          If XTRUE is the true solution corresponding to X(j), FERR(j)
    !>          is an estimated upper bound for the magnitude of the largest
    !>          element in (X(j) - XTRUE) divided by the magnitude of the
    !>          largest element in X(j).  The estimate is as reliable as
    !>          the estimate for RCOND, and is almost always a slight
    !>          overestimate of the true error.
    !>

    BERR

    !>          BERR is DOUBLE PRECISION array, dimension (NRHS)
    !>          The componentwise relative backward error of each solution
    !>          vector X(j) (i.e., the smallest relative change in
    !>          any element of A or B that makes X(j) an exact solution).
    !>

    WORK

    !>          WORK is DOUBLE PRECISION 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 principal minor of order i of A
    !>                       is not positive, 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.

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.
    !>

  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 principal minor of order i is not positive,
    !>    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 principal minor of order i of A
    !>                       is not positive, 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.

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.
    !>

  subroutine zpbsvx (character fact, character uplo, integer n, integer kd, integer nrhs, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldafb, * ) afb, integer ldafb, character equed, double precision, dimension( * ) s, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldx, * ) x, integer ldx, double precision rcond, double precision, dimension( * ) ferr, double precision, dimension( * ) berr, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

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

Purpose:

    !>
    !> ZPBSVX uses the Cholesky factorization A = U**H*U or A = L*L**H to
    !> compute the solution to a complex system of linear equations
    !>    A * X = B,
    !> where A is an N-by-N Hermitian 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**H * U,  if UPLO = 'U', or
    !>       A = L * L**H,  if UPLO = 'L',
    !>    where U is an upper triangular band matrix, and L is a lower
    !>    triangular band matrix.
    !>
    !> 3. If the leading principal minor of order i is not positive,
    !>    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 COMPLEX*16 array, dimension (LDAB,N)
    !>          On entry, the upper or lower triangle of the Hermitian 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 COMPLEX*16 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**H *U or A = L*L**H 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**H *U or A = L*L**H.
    !>
    !>          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**H *U or A = L*L**H 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 DOUBLE PRECISION array, dimension (N)
    !>          The scale factors for A; not accessed if EQUED = 'N'.  S is
    !>          an input argument if FACT = 'F'; otherwise, S is an output
    !>          argument.  If FACT = 'F' and EQUED = 'Y', each element of S
    !>          must be positive.
    !>

    B

    !>          B is COMPLEX*16 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 COMPLEX*16 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 DOUBLE PRECISION
    !>          The estimate of the reciprocal condition number of the matrix
    !>          A after equilibration (if done).  If RCOND is less than the
    !>          machine precision (in particular, if RCOND = 0), the matrix
    !>          is singular to working precision.  This condition is
    !>          indicated by a return code of INFO > 0.
    !>

    FERR

    !>          FERR is DOUBLE PRECISION array, dimension (NRHS)
    !>          The estimated forward error bound for each solution vector
    !>          X(j) (the j-th column of the solution matrix X).
    !>          If XTRUE is the true solution corresponding to X(j), FERR(j)
    !>          is an estimated upper bound for the magnitude of the largest
    !>          element in (X(j) - XTRUE) divided by the magnitude of the
    !>          largest element in X(j).  The estimate is as reliable as
    !>          the estimate for RCOND, and is almost always a slight
    !>          overestimate of the true error.
    !>

    BERR

    !>          BERR is DOUBLE PRECISION array, dimension (NRHS)
    !>          The componentwise relative backward error of each solution
    !>          vector X(j) (i.e., the smallest relative change in
    !>          any element of A or B that makes X(j) an exact solution).
    !>

    WORK

    !>          WORK is COMPLEX*16 array, dimension (2*N)
    !>

    RWORK

    !>          RWORK is DOUBLE PRECISION 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 principal minor of order i of A
    !>                       is not positive, 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.

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 Hermitian 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.
    !>

Author

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