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.

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 minor of order i of A is not


                positive definite, so the factorization could not be


                completed, 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


  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.