table of contents
| pbstf(3) | LAPACK | pbstf(3) |
NAME¶
pbstf - pbstf: split Cholesky factor, use with hbgst
SYNOPSIS¶
Functions¶
subroutine cpbstf (uplo, n, kd, ab, ldab, info)
CPBSTF subroutine dpbstf (uplo, n, kd, ab, ldab, info)
DPBSTF subroutine spbstf (uplo, n, kd, ab, ldab, info)
SPBSTF subroutine zpbstf (uplo, n, kd, ab, ldab, info)
ZPBSTF
Detailed Description¶
Function Documentation¶
subroutine cpbstf (character uplo, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, integer info)¶
CPBSTF
Purpose:
CPBSTF computes a split Cholesky factorization of a complex
Hermitian positive definite band matrix A.
This routine is designed to be used in conjunction with CHBGST.
The factorization has the form A = S**H*S where S is a band matrix
of the same bandwidth as A and the following structure:
S = ( U )
( M L )
where U is upper triangular of order m = (n+kd)/2, and L is lower
triangular of order n-m.
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 factor S from the split Cholesky
factorization A = S**H*S. See Further Details.
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 factorization could not be completed,
because the updated element a(i,i) was negative; the
matrix A is not positive definite.
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 = 7, KD = 2:
S = ( s11 s12 s13 )
( s22 s23 s24 )
( s33 s34 )
( s44 )
( s53 s54 s55 )
( s64 s65 s66 )
( s75 s76 s77 )
If UPLO = 'U', the array AB holds:
on entry: on exit:
* * a13 a24 a35 a46 a57 * * s13 s24 s53**H s64**H s75**H
* a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
If UPLO = 'L', the array AB holds:
on entry: on exit:
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
a21 a32 a43 a54 a65 a76 * s12**H s23**H s34**H s54 s65 s76 *
a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * *
Array elements marked * are not used by the routine; s12**H denotes
conjg(s12); the diagonal elements of S are real.
subroutine dpbstf (character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, integer info)¶
DPBSTF
Purpose:
DPBSTF computes a split Cholesky factorization of a real
symmetric positive definite band matrix A.
This routine is designed to be used in conjunction with DSBGST.
The factorization has the form A = S**T*S where S is a band matrix
of the same bandwidth as A and the following structure:
S = ( U )
( M L )
where U is upper triangular of order m = (n+kd)/2, and L is lower
triangular of order n-m.
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 factor S from the split Cholesky
factorization A = S**T*S. See Further Details.
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 factorization could not be completed,
because the updated element a(i,i) was negative; the
matrix A is not positive definite.
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 = 7, KD = 2:
S = ( s11 s12 s13 )
( s22 s23 s24 )
( s33 s34 )
( s44 )
( s53 s54 s55 )
( s64 s65 s66 )
( s75 s76 s77 )
If UPLO = 'U', the array AB holds:
on entry: on exit:
* * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
* a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
If UPLO = 'L', the array AB holds:
on entry: on exit:
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 *
a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * *
Array elements marked * are not used by the routine.
subroutine spbstf (character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, integer info)¶
SPBSTF
Purpose:
SPBSTF computes a split Cholesky factorization of a real
symmetric positive definite band matrix A.
This routine is designed to be used in conjunction with SSBGST.
The factorization has the form A = S**T*S where S is a band matrix
of the same bandwidth as A and the following structure:
S = ( U )
( M L )
where U is upper triangular of order m = (n+kd)/2, and L is lower
triangular of order n-m.
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 factor S from the split Cholesky
factorization A = S**T*S. See Further Details.
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 factorization could not be completed,
because the updated element a(i,i) was negative; the
matrix A is not positive definite.
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 = 7, KD = 2:
S = ( s11 s12 s13 )
( s22 s23 s24 )
( s33 s34 )
( s44 )
( s53 s54 s55 )
( s64 s65 s66 )
( s75 s76 s77 )
If UPLO = 'U', the array AB holds:
on entry: on exit:
* * a13 a24 a35 a46 a57 * * s13 s24 s53 s64 s75
* a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54 s65 s76
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
If UPLO = 'L', the array AB holds:
on entry: on exit:
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
a21 a32 a43 a54 a65 a76 * s12 s23 s34 s54 s65 s76 *
a31 a42 a53 a64 a64 * * s13 s24 s53 s64 s75 * *
Array elements marked * are not used by the routine.
subroutine zpbstf (character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, integer info)¶
ZPBSTF
Purpose:
ZPBSTF computes a split Cholesky factorization of a complex
Hermitian positive definite band matrix A.
This routine is designed to be used in conjunction with ZHBGST.
The factorization has the form A = S**H*S where S is a band matrix
of the same bandwidth as A and the following structure:
S = ( U )
( M L )
where U is upper triangular of order m = (n+kd)/2, and L is lower
triangular of order n-m.
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*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 factor S from the split Cholesky
factorization A = S**H*S. See Further Details.
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 factorization could not be completed,
because the updated element a(i,i) was negative; the
matrix A is not positive definite.
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 = 7, KD = 2:
S = ( s11 s12 s13 )
( s22 s23 s24 )
( s33 s34 )
( s44 )
( s53 s54 s55 )
( s64 s65 s66 )
( s75 s76 s77 )
If UPLO = 'U', the array AB holds:
on entry: on exit:
* * a13 a24 a35 a46 a57 * * s13 s24 s53**H s64**H s75**H
* a12 a23 a34 a45 a56 a67 * s12 s23 s34 s54**H s65**H s76**H
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
If UPLO = 'L', the array AB holds:
on entry: on exit:
a11 a22 a33 a44 a55 a66 a77 s11 s22 s33 s44 s55 s66 s77
a21 a32 a43 a54 a65 a76 * s12**H s23**H s34**H s54 s65 s76 *
a31 a42 a53 a64 a64 * * s13**H s24**H s53 s64 s75 * *
Array elements marked * are not used by the routine; s12**H denotes
conjg(s12); the diagonal elements of S are real.
Author¶
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| Wed Feb 7 2024 11:30:40 | Version 3.12.0 |