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.

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

  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.