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

# spbtrf(3lapack)

computes the Cholesky factorization of a real symmetric positive definite band matrix A

## NAME

SPBTRF - computes the Cholesky factorization of a real symmetric positive definite band matrix A

## SYNOPSIS

- SUBROUTINE SPBTRF(

  UPLO, N, KD, AB, LDAB, INFO )

- CHARACTER

  UPLO

- INTEGER

  INFO, KD, LDAB, N

- REAL

  AB( LDAB, \* )

## PURPOSE

SPBTRF computes the Cholesky factorization of a real symmetric positive definite band matrix A. The factorization has the form\
\
    A = U\*\*T \* U,  if UPLO = 'U', or\
\
    A = L  \* L\*\*T,  if UPLO = 'L',\
 where U is an upper triangular matrix and L is lower triangular.

## ARGUMENTS

- UPLO (input) CHARACTER\*1

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

- N (input) INTEGER

  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.

- AB (input/output) 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). 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 (input) INTEGER

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

- 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 is not positive definite, and the factorization could not be completed.

## 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.\
 Contributed by\
 Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989