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

# cgbsv(3lapack)

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

## NAME

CGBSV - 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

## SYNOPSIS

- SUBROUTINE CGBSV(

  N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO )

- INTEGER

  INFO, KL, KU, LDAB, LDB, N, NRHS

- INTEGER

  IPIV( \* )

- COMPLEX

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

## PURPOSE

CGBSV 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 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) COMPLEX 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) 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, 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.