.Modelica.Math.Matrices.LAPACK.dgbsv

Information

Lapack documentation
    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.

    Arguments
    =========

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

    KL      (input) INTEGER
            The number of subdiagonals within the band of A.  KL >= 0.

    KU      (input) INTEGER
            The number of superdiagonals within the band of A.  KU >= 0.

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

    AB      (input/output) 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    (input) INTEGER
            The leading dimension of the array AB.  LDAB >= 2*KL+KU+1.

    IPIV    (output) 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       (input/output) 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     (input) INTEGER
            The leading dimension of the array B.  LDB >= max(1,N).

    INFO    (output) 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.

    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.

Interface

pure function dgbsv
  extends Modelica.Icons.Function;
  input Integer n "Number of equations";
  input Integer kLower "Number of lower bands";
  input Integer kUpper "Number of upper bands";
  input Real A[2*kLower + kUpper + 1, n];
  input Real B[n, :];
  output Real X[n, size(B, 2)] = B;
  output Integer info;
end dgbsv;