# pbtrf - Man Page

pbtrf: triangular factor

## Synopsis

### Functions

subroutine **cpbtrf** (uplo, n, kd, ab, ldab, info)**CPBTRF**

subroutine **dpbtrf** (uplo, n, kd, ab, ldab, info)**DPBTRF**

subroutine **spbtrf** (uplo, n, kd, ab, ldab, info)**SPBTRF**

subroutine **zpbtrf** (uplo, n, kd, ab, ldab, info)**ZPBTRF**

## Detailed Description

## Function Documentation

### subroutine cpbtrf (character uplo, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, integer info)

**CPBTRF**

**Purpose:**

CPBTRF computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. The factorization has the form A = U**H * U, if UPLO = 'U', or A = L * L**H, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular.

**Parameters***UPLO*UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*N is INTEGER 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.

*AB*AB is 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). 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*LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.

*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 principal minor of order i is not positive, and the factorization could not be completed.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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.

**Contributors:**Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

Definition at line **141** of file **cpbtrf.f**.

### subroutine dpbtrf (character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, integer info)

**DPBTRF**

**Purpose:**

DPBTRF 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.

**Parameters***UPLO*UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*N is INTEGER 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.

*AB*AB is DOUBLE PRECISION 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*LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.

*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 principal minor of order i is not positive, and the factorization could not be completed.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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.

**Contributors:**Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

Definition at line **141** of file **dpbtrf.f**.

### subroutine spbtrf (character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, integer info)

**SPBTRF**

**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.

**Parameters***UPLO*UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*N is INTEGER 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.

*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). 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.

*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 principal minor of order i is not positive, and the factorization could not be completed.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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.

**Contributors:**Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

Definition at line **141** of file **spbtrf.f**.

### subroutine zpbtrf (character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, integer info)

**ZPBTRF**

**Purpose:**

ZPBTRF computes the Cholesky factorization of a complex Hermitian positive definite band matrix A. The factorization has the form A = U**H * U, if UPLO = 'U', or A = L * L**H, if UPLO = 'L', where U is an upper triangular matrix and L is lower triangular.

**Parameters***UPLO*UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.

*N*N is INTEGER The order of the matrix A. N >= 0.

*KD**AB*AB is COMPLEX*16 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). 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*LDAB is INTEGER The leading dimension of the array AB. LDAB >= KD+1.

*INFO***Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Further Details:**

**Contributors:**Peter Mayes and Giuseppe Radicati, IBM ECSEC, Rome, March 23, 1989

Definition at line **141** of file **zpbtrf.f**.

## Author

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