# pbtf2 - Man Page

pbtf2: triangular factor panel, level 2

## Synopsis

### Functions

subroutine **cpbtf2** (uplo, n, kd, ab, ldab, info)**CPBTF2** computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

subroutine **dpbtf2** (uplo, n, kd, ab, ldab, info)**DPBTF2** computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

subroutine **spbtf2** (uplo, n, kd, ab, ldab, info)**SPBTF2** computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

subroutine **zpbtf2** (uplo, n, kd, ab, ldab, info)**ZPBTF2** computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

## Detailed Description

## Function Documentation

### subroutine cpbtf2 (character uplo, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, integer info)

**CPBTF2** computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

**Purpose:**

CPBTF2 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, U**H is the conjugate transpose of U, and L is lower triangular. This is the unblocked version of the algorithm, calling Level 2 BLAS.

**Parameters***UPLO*UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular

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

*KD*KD is INTEGER The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals 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 = -k, the k-th argument had an illegal value > 0: if INFO = k, the leading principal minor of order k 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.

Definition at line **141** of file **cpbtf2.f**.

### subroutine dpbtf2 (character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, integer info)

**DPBTF2** computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

**Purpose:**

DPBTF2 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, U**T is the transpose of U, and L is lower triangular. This is the unblocked version of the algorithm, calling Level 2 BLAS.

**Parameters***UPLO*UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular

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

*KD*KD is INTEGER The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals 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 = -k, the k-th argument had an illegal value > 0: if INFO = k, the leading principal minor of order k 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.

Definition at line **141** of file **dpbtf2.f**.

### subroutine spbtf2 (character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, integer info)

**SPBTF2** computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

**Purpose:**

SPBTF2 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, U**T is the transpose of U, and L is lower triangular. This is the unblocked version of the algorithm, calling Level 2 BLAS.

**Parameters***UPLO*UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular

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

*KD*KD is INTEGER The number of super-diagonals of the matrix A if UPLO = 'U', or the number of sub-diagonals 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 = -k, the k-th argument had an illegal value > 0: if INFO = k, the leading principal minor of order k 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.

Definition at line **141** of file **spbtf2.f**.

### subroutine zpbtf2 (character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, integer info)

**ZPBTF2** computes the Cholesky factorization of a symmetric/Hermitian positive definite band matrix (unblocked algorithm).

**Purpose:**

ZPBTF2 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, U**H is the conjugate transpose of U, and L is lower triangular. This is the unblocked version of the algorithm, calling Level 2 BLAS.

**Parameters***UPLO*UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the Hermitian matrix A is stored: = 'U': Upper triangular = 'L': Lower triangular

*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:**

Definition at line **141** of file **zpbtf2.f**.

## Author

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