gbsv(3)

NAME
gbsv - gbsv: factor and solve

SYNOPSIS

  Functions

subroutine cgbsv (n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
CGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)
subroutine dgbsv (n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
DGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)
subroutine sgbsv (n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
SGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)
subroutine zgbsv (n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
ZGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)

Detailed Description

Function Documentation

  subroutine cgbsv (integer n, integer kl, integer ku, integer nrhs, complex, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, complex, dimension( ldb, * ) b, integer ldb, integer info)

CGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)

Purpose:

    !>
    !> CGBSV computes the solution to a complex system of linear equations
    !> A * X = B, where A is a band matrix of order N with KL subdiagonals
    !> and KU superdiagonals, and X and B are N-by-NRHS matrices.
    !>
    !> The LU decomposition with partial pivoting and row interchanges is
    !> used to factor A as A = L * U, where L is a product of permutation
    !> and unit lower triangular matrices with KL subdiagonals, and U is
    !> upper triangular with KL+KU superdiagonals.  The factored form of A
    !> is then used to solve the system of equations A * X = B.
    !>

Parameters
    N

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

    KL

    !>          KL is INTEGER
    !>          The number of subdiagonals within the band of A.  KL >= 0.
    !>

    KU

    !>          KU is INTEGER
    !>          The number of superdiagonals within the band of A.  KU >= 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 matrix A in band storage, in rows KL+1 to
    !>          2*KL+KU+1; rows 1 to KL of the array need not be set.
    !>          The j-th column of A is stored in the j-th column of the
    !>          array AB as follows:
    !>          AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
    !>          On exit, details of the factorization: U is stored as an
    !>          upper triangular band matrix with KL+KU superdiagonals in
    !>          rows 1 to KL+KU+1, and the multipliers used during the
    !>          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
    !>          See below for further details.
    !>

    LDAB

    !>          LDAB is INTEGER
    !>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          The pivot indices that define the permutation matrix P;
    !>          row i of the matrix was interchanged with row IPIV(i).
    !>

    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, U(i,i) is exactly zero.  The factorization
    !>                has been completed, but the factor U is exactly
    !>                singular, 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
    !>  M = N = 6, KL = 2, KU = 1:
    !>
    !>  On entry:                       On exit:
    !>
    !>      *    *    *    +    +    +       *    *    *   u14  u25  u36
    !>      *    *    +    +    +    +       *    *   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
    !>     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
    !>     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
    !>
    !>  Array elements marked * are not used by the routine; elements marked
    !>  + need not be set on entry, but are required by the routine to store
    !>  elements of U because of fill-in resulting from the row interchanges.
    !>

  subroutine dgbsv (integer n, integer kl, integer ku, integer nrhs, double precision, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, double precision, dimension( ldb, * ) b, integer ldb, integer info)

DGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)

Purpose:

    !>
    !> DGBSV computes the solution to a real system of linear equations
    !> A * X = B, where A is a band matrix of order N with KL subdiagonals
    !> and KU superdiagonals, and X and B are N-by-NRHS matrices.
    !>
    !> The LU decomposition with partial pivoting and row interchanges is
    !> used to factor A as A = L * U, where L is a product of permutation
    !> and unit lower triangular matrices with KL subdiagonals, and U is
    !> upper triangular with KL+KU superdiagonals.  The factored form of A
    !> is then used to solve the system of equations A * X = B.
    !>

Parameters
    N

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

    KL

    !>          KL is INTEGER
    !>          The number of subdiagonals within the band of A.  KL >= 0.
    !>

    KU

    !>          KU is INTEGER
    !>          The number of superdiagonals within the band of A.  KU >= 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 matrix A in band storage, in rows KL+1 to
    !>          2*KL+KU+1; rows 1 to KL of the array need not be set.
    !>          The j-th column of A is stored in the j-th column of the
    !>          array AB as follows:
    !>          AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
    !>          On exit, details of the factorization: U is stored as an
    !>          upper triangular band matrix with KL+KU superdiagonals in
    !>          rows 1 to KL+KU+1, and the multipliers used during the
    !>          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
    !>          See below for further details.
    !>

    LDAB

    !>          LDAB is INTEGER
    !>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          The pivot indices that define the permutation matrix P;
    !>          row i of the matrix was interchanged with row IPIV(i).
    !>

    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, U(i,i) is exactly zero.  The factorization
    !>                has been completed, but the factor U is exactly
    !>                singular, 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
    !>  M = N = 6, KL = 2, KU = 1:
    !>
    !>  On entry:                       On exit:
    !>
    !>      *    *    *    +    +    +       *    *    *   u14  u25  u36
    !>      *    *    +    +    +    +       *    *   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
    !>     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
    !>     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
    !>
    !>  Array elements marked * are not used by the routine; elements marked
    !>  + need not be set on entry, but are required by the routine to store
    !>  elements of U because of fill-in resulting from the row interchanges.
    !>

  subroutine sgbsv (integer n, integer kl, integer ku, integer nrhs, real, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, real, dimension( ldb, * ) b, integer ldb, integer info)

SGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)

Purpose:

    !>
    !> SGBSV computes the solution to a real system of linear equations
    !> A * X = B, where A is a band matrix of order N with KL subdiagonals
    !> and KU superdiagonals, and X and B are N-by-NRHS matrices.
    !>
    !> The LU decomposition with partial pivoting and row interchanges is
    !> used to factor A as A = L * U, where L is a product of permutation
    !> and unit lower triangular matrices with KL subdiagonals, and U is
    !> upper triangular with KL+KU superdiagonals.  The factored form of A
    !> is then used to solve the system of equations A * X = B.
    !>

Parameters
    N

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

    KL

    !>          KL is INTEGER
    !>          The number of subdiagonals within the band of A.  KL >= 0.
    !>

    KU

    !>          KU is INTEGER
    !>          The number of superdiagonals within the band of A.  KU >= 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 matrix A in band storage, in rows KL+1 to
    !>          2*KL+KU+1; rows 1 to KL of the array need not be set.
    !>          The j-th column of A is stored in the j-th column of the
    !>          array AB as follows:
    !>          AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
    !>          On exit, details of the factorization: U is stored as an
    !>          upper triangular band matrix with KL+KU superdiagonals in
    !>          rows 1 to KL+KU+1, and the multipliers used during the
    !>          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
    !>          See below for further details.
    !>

    LDAB

    !>          LDAB is INTEGER
    !>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          The pivot indices that define the permutation matrix P;
    !>          row i of the matrix was interchanged with row IPIV(i).
    !>

    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, U(i,i) is exactly zero.  The factorization
    !>                has been completed, but the factor U is exactly
    !>                singular, 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
    !>  M = N = 6, KL = 2, KU = 1:
    !>
    !>  On entry:                       On exit:
    !>
    !>      *    *    *    +    +    +       *    *    *   u14  u25  u36
    !>      *    *    +    +    +    +       *    *   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
    !>     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
    !>     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
    !>
    !>  Array elements marked * are not used by the routine; elements marked
    !>  + need not be set on entry, but are required by the routine to store
    !>  elements of U because of fill-in resulting from the row interchanges.
    !>

  subroutine zgbsv (integer n, integer kl, integer ku, integer nrhs, complex*16, dimension( ldab, * ) ab, integer ldab, integer, dimension( * ) ipiv, complex*16, dimension( ldb, * ) b, integer ldb, integer info)

ZGBSV computes the solution to system of linear equations A * X = B for GB matrices (simple driver)

Purpose:

    !>
    !> ZGBSV computes the solution to a complex system of linear equations
    !> A * X = B, where A is a band matrix of order N with KL subdiagonals
    !> and KU superdiagonals, and X and B are N-by-NRHS matrices.
    !>
    !> The LU decomposition with partial pivoting and row interchanges is
    !> used to factor A as A = L * U, where L is a product of permutation
    !> and unit lower triangular matrices with KL subdiagonals, and U is
    !> upper triangular with KL+KU superdiagonals.  The factored form of A
    !> is then used to solve the system of equations A * X = B.
    !>

Parameters
    N

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

    KL

    !>          KL is INTEGER
    !>          The number of subdiagonals within the band of A.  KL >= 0.
    !>

    KU

    !>          KU is INTEGER
    !>          The number of superdiagonals within the band of A.  KU >= 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 matrix A in band storage, in rows KL+1 to
    !>          2*KL+KU+1; rows 1 to KL of the array need not be set.
    !>          The j-th column of A is stored in the j-th column of the
    !>          array AB as follows:
    !>          AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
    !>          On exit, details of the factorization: U is stored as an
    !>          upper triangular band matrix with KL+KU superdiagonals in
    !>          rows 1 to KL+KU+1, and the multipliers used during the
    !>          factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
    !>          See below for further details.
    !>

    LDAB

    !>          LDAB is INTEGER
    !>          The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          The pivot indices that define the permutation matrix P;
    !>          row i of the matrix was interchanged with row IPIV(i).
    !>

    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, U(i,i) is exactly zero.  The factorization
    !>                has been completed, but the factor U is exactly
    !>                singular, 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
    !>  M = N = 6, KL = 2, KU = 1:
    !>
    !>  On entry:                       On exit:
    !>
    !>      *    *    *    +    +    +       *    *    *   u14  u25  u36
    !>      *    *    +    +    +    +       *    *   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
    !>     a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65   *
    !>     a31  a42  a53  a64   *    *      m31  m42  m53  m64   *    *
    !>
    !>  Array elements marked * are not used by the routine; elements marked
    !>  + need not be set on entry, but are required by the routine to store
    !>  elements of U because of fill-in resulting from the row interchanges.
    !>

Author

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