pbsv(3)

NAME
pbsv - pbsv: factor and solve

SYNOPSIS

  Functions

subroutine cpbsv (uplo, n, kd, nrhs, ab, ldab, b, ldb, info)
CPBSV computes the solution to system of linear equations A * X = B for OTHER matrices
subroutine dpbsv (uplo, n, kd, nrhs, ab, ldab, b, ldb, info)
DPBSV computes the solution to system of linear equations A * X = B 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 zpbsv (uplo, n, kd, nrhs, ab, ldab, b, ldb, info)
ZPBSV computes the solution to system of linear equations A * X = B for OTHER matrices

Detailed Description

Function Documentation

  subroutine cpbsv (character uplo, integer n, integer kd, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldb, * ) b, integer ldb, integer info)

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

Purpose:

    !>
    !> CPBSV computes 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.
    !>
    !> The Cholesky decomposition is used to factor A 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, 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 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.  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**H*U or A = L*L**H 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 COMPLEX 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 principal minor of order i
    !>                of A is not positive, 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.

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.
    !>

  subroutine dpbsv (character uplo, integer n, integer kd, integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( ldb, * ) b, integer ldb, integer info)

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

Purpose:

    !>
    !> DPBSV 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 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.  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 DOUBLE PRECISION array, dimension (LDB,NRHS)
    !>          On entry, the N-by-NRHS right hand side matrix B.
    !>          On exit, if INFO = 0, the N-by-NRHS solution matrix X.
    !>

    LDB

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

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.
    !>

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

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.
    !>

  subroutine zpbsv (character uplo, integer n, integer kd, integer nrhs, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

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

Purpose:

    !>
    !> ZPBSV computes 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.
    !>
    !> The Cholesky decomposition is used to factor A 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, 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 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.  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**H *U or A = L*L**H 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 COMPLEX*16 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 principal minor of order i
    !>                of A is not positive, 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.

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.
    !>

Author

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