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NAME

        ZGBSV  - compute 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
 

SYNOPSIS

        SUBROUTINE ZGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )
 
            INTEGER       INFO, KL, KU, LDAB, LDB, N, NRHS
 
            INTEGER       IPIV( * )
 
            COMPLEX*16    AB( LDAB, * ), B( LDB, * )
 

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 decomposi‐
        tion 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 triangu‐
        lar matrices with KL subdiagonals, and U is upper triangular with KL+KU
        superdiagonals.   The factored form of A is then used to solve the sys‐
        tem 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) 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 superdiag‐
                onals 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) 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     (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.
        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 ele‐
        ments of U because of fill-in resulting from the row interchanges.
 

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