---
name: "cpbsv"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/cpbsv/3lapack"
---

# cpbsv(3lapack)

computes the solution to a complex system of linear equations A \* X = B,

## NAME

CPBSV - computes the solution to a complex system of linear equations  A \* X = B,

## SYNOPSIS

- SUBROUTINE CPBSV(

  UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO )

- CHARACTER

  UPLO

- INTEGER

  INFO, KD, LDAB, LDB, N, NRHS

- COMPLEX

  AB( LDAB, \* ), B( LDB, \* )

## PURPOSE

CPBSV computes the solution to a complex system of linear equations\
    A \* X = B, where A is an N-by-N Hermitian positive definite band matrix and X and B are N-by-NRHS matrices.\
 The Cholesky decomposition is used to factor A as\
\
    A = U\*\*H \* U,  if UPLO = 'U', or\
\
    A = L \* L\*\*H,  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.\

## ARGUMENTS

- UPLO (input) CHARACTER\*1

  \= 'U':  Upper triangle of A is stored;\
   \= 'L':  Lower triangle of A is stored.

- N (input) INTEGER

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

- KD (input) INTEGER

  The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'.  KD \>= 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 array, dimension (LDAB,N)

  On entry, the upper or lower triangle of the Hermitian 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\*\*H\*U or A = L\*L\*\*H of the band matrix A, in the same storage format as A.

- LDAB (input) INTEGER

  The leading dimension of the array AB.  LDAB \>= KD+1.

- B (input/output) 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 (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, 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.

## FURTHER DETAILS

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.\