table of contents
| pftrf(3) | Library Functions Manual | pftrf(3) |
NAME¶
pftrf - pftrf: triangular factor
SYNOPSIS¶
Functions¶
subroutine CPFTRF (transr, uplo, n, a, info)
CPFTRF subroutine DPFTRF (transr, uplo, n, a, info)
DPFTRF subroutine SPFTRF (transr, uplo, n, a, info)
SPFTRF subroutine ZPFTRF (transr, uplo, n, a, info)
ZPFTRF
Detailed Description¶
Function Documentation¶
subroutine CPFTRF (character transr, character uplo, integer n, complex, dimension( 0: * ) a, integer info)¶
CPFTRF
Purpose:
!> !> CPFTRF computes the Cholesky factorization of a complex Hermitian !> positive definite 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. !> !> This is the block version of the algorithm, calling Level 3 BLAS. !>
Parameters
TRANSR
!> TRANSR is CHARACTER*1 !> = 'N': The Normal TRANSR of RFP A is stored; !> = 'C': The Conjugate-transpose TRANSR of RFP A is stored. !>
UPLO
!> UPLO is CHARACTER*1 !> = 'U': Upper triangle of RFP A is stored; !> = 'L': Lower triangle of RFP A is stored. !>
N
!> N is INTEGER !> The order of the matrix A. N >= 0. !>
A
!> A is COMPLEX array, dimension ( N*(N+1)/2 ); !> On entry, the Hermitian matrix A in RFP format. RFP format is !> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' !> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is !> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is !> the Conjugate-transpose of RFP A as defined when !> TRANSR = 'N'. The contents of RFP A are defined by UPLO as !> follows: If UPLO = 'U' the RFP A contains the nt elements of !> upper packed A. If UPLO = 'L' the RFP A contains the elements !> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = !> 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N !> is odd. See the Note below for more details. !> !> On exit, if INFO = 0, the factor U or L from the Cholesky !> factorization RFP A = U**H*U or RFP A = L*L**H. !>
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. !> !> Further Notes on RFP Format: !> ============================ !> !> We first consider Standard Packed Format when N is even. !> We give an example where N = 6. !> !> AP is Upper AP is Lower !> !> 00 01 02 03 04 05 00 !> 11 12 13 14 15 10 11 !> 22 23 24 25 20 21 22 !> 33 34 35 30 31 32 33 !> 44 45 40 41 42 43 44 !> 55 50 51 52 53 54 55 !> !> Let TRANSR = 'N'. RFP holds AP as follows: !> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last !> three columns of AP upper. The lower triangle A(4:6,0:2) consists of !> conjugate-transpose of the first three columns of AP upper. !> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first !> three columns of AP lower. The upper triangle A(0:2,0:2) consists of !> conjugate-transpose of the last three columns of AP lower. !> To denote conjugate we place -- above the element. This covers the !> case N even and TRANSR = 'N'. !> !> RFP A RFP A !> !> -- -- -- !> 03 04 05 33 43 53 !> -- -- !> 13 14 15 00 44 54 !> -- !> 23 24 25 10 11 55 !> !> 33 34 35 20 21 22 !> -- !> 00 44 45 30 31 32 !> -- -- !> 01 11 55 40 41 42 !> -- -- -- !> 02 12 22 50 51 52 !> !> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- !> transpose of RFP A above. One therefore gets: !> !> RFP A RFP A !> !> -- -- -- -- -- -- -- -- -- -- !> 03 13 23 33 00 01 02 33 00 10 20 30 40 50 !> -- -- -- -- -- -- -- -- -- -- !> 04 14 24 34 44 11 12 43 44 11 21 31 41 51 !> -- -- -- -- -- -- -- -- -- -- !> 05 15 25 35 45 55 22 53 54 55 22 32 42 52 !> !> We next consider Standard Packed Format when N is odd. !> We give an example where N = 5. !> !> AP is Upper AP is Lower !> !> 00 01 02 03 04 00 !> 11 12 13 14 10 11 !> 22 23 24 20 21 22 !> 33 34 30 31 32 33 !> 44 40 41 42 43 44 !> !> Let TRANSR = 'N'. RFP holds AP as follows: !> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last !> three columns of AP upper. The lower triangle A(3:4,0:1) consists of !> conjugate-transpose of the first two columns of AP upper. !> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first !> three columns of AP lower. The upper triangle A(0:1,1:2) consists of !> conjugate-transpose of the last two columns of AP lower. !> To denote conjugate we place -- above the element. This covers the !> case N odd and TRANSR = 'N'. !> !> RFP A RFP A !> !> -- -- !> 02 03 04 00 33 43 !> -- !> 12 13 14 10 11 44 !> !> 22 23 24 20 21 22 !> -- !> 00 33 34 30 31 32 !> -- -- !> 01 11 44 40 41 42 !> !> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- !> transpose of RFP A above. One therefore gets: !> !> RFP A RFP A !> !> -- -- -- -- -- -- -- -- -- !> 02 12 22 00 01 00 10 20 30 40 50 !> -- -- -- -- -- -- -- -- -- !> 03 13 23 33 11 33 11 21 31 41 51 !> -- -- -- -- -- -- -- -- -- !> 04 14 24 34 44 43 44 22 32 42 52 !>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Definition at line 210 of file cpftrf.f.
subroutine DPFTRF (character transr, character uplo, integer n, double precision, dimension( 0: * ) a, integer info)¶
DPFTRF
Purpose:
!> !> DPFTRF computes the Cholesky factorization of a real symmetric !> positive definite 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. !> !> This is the block version of the algorithm, calling Level 3 BLAS. !>
Parameters
TRANSR
!> TRANSR is CHARACTER*1 !> = 'N': The Normal TRANSR of RFP A is stored; !> = 'T': The Transpose TRANSR of RFP A is stored. !>
UPLO
!> UPLO is CHARACTER*1 !> = 'U': Upper triangle of RFP A is stored; !> = 'L': Lower triangle of RFP A is stored. !>
N
!> N is INTEGER !> The order of the matrix A. N >= 0. !>
A
!> A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ); !> On entry, the symmetric matrix A in RFP format. RFP format is !> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' !> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is !> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is !> the transpose of RFP A as defined when !> TRANSR = 'N'. The contents of RFP A are defined by UPLO as !> follows: If UPLO = 'U' the RFP A contains the NT elements of !> upper packed A. If UPLO = 'L' the RFP A contains the elements !> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = !> 'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N !> is odd. See the Note below for more details. !> !> On exit, if INFO = 0, the factor U or L from the Cholesky !> factorization RFP A = U**T*U or RFP A = L*L**T. !>
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:
!> !> We first consider Rectangular Full Packed (RFP) Format when N is !> even. We give an example where N = 6. !> !> AP is Upper AP is Lower !> !> 00 01 02 03 04 05 00 !> 11 12 13 14 15 10 11 !> 22 23 24 25 20 21 22 !> 33 34 35 30 31 32 33 !> 44 45 40 41 42 43 44 !> 55 50 51 52 53 54 55 !> !> !> Let TRANSR = 'N'. RFP holds AP as follows: !> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last !> three columns of AP upper. The lower triangle A(4:6,0:2) consists of !> the transpose of the first three columns of AP upper. !> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first !> three columns of AP lower. The upper triangle A(0:2,0:2) consists of !> the transpose of the last three columns of AP lower. !> This covers the case N even and TRANSR = 'N'. !> !> RFP A RFP A !> !> 03 04 05 33 43 53 !> 13 14 15 00 44 54 !> 23 24 25 10 11 55 !> 33 34 35 20 21 22 !> 00 44 45 30 31 32 !> 01 11 55 40 41 42 !> 02 12 22 50 51 52 !> !> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the !> transpose of RFP A above. One therefore gets: !> !> !> RFP A RFP A !> !> 03 13 23 33 00 01 02 33 00 10 20 30 40 50 !> 04 14 24 34 44 11 12 43 44 11 21 31 41 51 !> 05 15 25 35 45 55 22 53 54 55 22 32 42 52 !> !> !> We then consider Rectangular Full Packed (RFP) Format when N is !> odd. We give an example where N = 5. !> !> AP is Upper AP is Lower !> !> 00 01 02 03 04 00 !> 11 12 13 14 10 11 !> 22 23 24 20 21 22 !> 33 34 30 31 32 33 !> 44 40 41 42 43 44 !> !> !> Let TRANSR = 'N'. RFP holds AP as follows: !> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last !> three columns of AP upper. The lower triangle A(3:4,0:1) consists of !> the transpose of the first two columns of AP upper. !> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first !> three columns of AP lower. The upper triangle A(0:1,1:2) consists of !> the transpose of the last two columns of AP lower. !> This covers the case N odd and TRANSR = 'N'. !> !> RFP A RFP A !> !> 02 03 04 00 33 43 !> 12 13 14 10 11 44 !> 22 23 24 20 21 22 !> 00 33 34 30 31 32 !> 01 11 44 40 41 42 !> !> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the !> transpose of RFP A above. One therefore gets: !> !> RFP A RFP A !> !> 02 12 22 00 01 00 10 20 30 40 50 !> 03 13 23 33 11 33 11 21 31 41 51 !> 04 14 24 34 44 43 44 22 32 42 52 !>
Definition at line 197 of file dpftrf.f.
subroutine SPFTRF (character transr, character uplo, integer n, real, dimension( 0: * ) a, integer info)¶
SPFTRF
Purpose:
!> !> SPFTRF computes the Cholesky factorization of a real symmetric !> positive definite 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. !> !> This is the block version of the algorithm, calling Level 3 BLAS. !>
Parameters
TRANSR
!> TRANSR is CHARACTER*1 !> = 'N': The Normal TRANSR of RFP A is stored; !> = 'T': The Transpose TRANSR of RFP A is stored. !>
UPLO
!> UPLO is CHARACTER*1 !> = 'U': Upper triangle of RFP A is stored; !> = 'L': Lower triangle of RFP A is stored. !>
N
!> N is INTEGER !> The order of the matrix A. N >= 0. !>
A
!> A is REAL array, dimension ( N*(N+1)/2 ); !> On entry, the symmetric matrix A in RFP format. RFP format is !> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' !> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is !> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is !> the transpose of RFP A as defined when !> TRANSR = 'N'. The contents of RFP A are defined by UPLO as !> follows: If UPLO = 'U' the RFP A contains the NT elements of !> upper packed A. If UPLO = 'L' the RFP A contains the elements !> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = !> 'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N !> is odd. See the Note below for more details. !> !> On exit, if INFO = 0, the factor U or L from the Cholesky !> factorization RFP A = U**T*U or RFP A = L*L**T. !>
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:
!> !> We first consider Rectangular Full Packed (RFP) Format when N is !> even. We give an example where N = 6. !> !> AP is Upper AP is Lower !> !> 00 01 02 03 04 05 00 !> 11 12 13 14 15 10 11 !> 22 23 24 25 20 21 22 !> 33 34 35 30 31 32 33 !> 44 45 40 41 42 43 44 !> 55 50 51 52 53 54 55 !> !> !> Let TRANSR = 'N'. RFP holds AP as follows: !> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last !> three columns of AP upper. The lower triangle A(4:6,0:2) consists of !> the transpose of the first three columns of AP upper. !> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first !> three columns of AP lower. The upper triangle A(0:2,0:2) consists of !> the transpose of the last three columns of AP lower. !> This covers the case N even and TRANSR = 'N'. !> !> RFP A RFP A !> !> 03 04 05 33 43 53 !> 13 14 15 00 44 54 !> 23 24 25 10 11 55 !> 33 34 35 20 21 22 !> 00 44 45 30 31 32 !> 01 11 55 40 41 42 !> 02 12 22 50 51 52 !> !> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the !> transpose of RFP A above. One therefore gets: !> !> !> RFP A RFP A !> !> 03 13 23 33 00 01 02 33 00 10 20 30 40 50 !> 04 14 24 34 44 11 12 43 44 11 21 31 41 51 !> 05 15 25 35 45 55 22 53 54 55 22 32 42 52 !> !> !> We then consider Rectangular Full Packed (RFP) Format when N is !> odd. We give an example where N = 5. !> !> AP is Upper AP is Lower !> !> 00 01 02 03 04 00 !> 11 12 13 14 10 11 !> 22 23 24 20 21 22 !> 33 34 30 31 32 33 !> 44 40 41 42 43 44 !> !> !> Let TRANSR = 'N'. RFP holds AP as follows: !> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last !> three columns of AP upper. The lower triangle A(3:4,0:1) consists of !> the transpose of the first two columns of AP upper. !> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first !> three columns of AP lower. The upper triangle A(0:1,1:2) consists of !> the transpose of the last two columns of AP lower. !> This covers the case N odd and TRANSR = 'N'. !> !> RFP A RFP A !> !> 02 03 04 00 33 43 !> 12 13 14 10 11 44 !> 22 23 24 20 21 22 !> 00 33 34 30 31 32 !> 01 11 44 40 41 42 !> !> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the !> transpose of RFP A above. One therefore gets: !> !> RFP A RFP A !> !> 02 12 22 00 01 00 10 20 30 40 50 !> 03 13 23 33 11 33 11 21 31 41 51 !> 04 14 24 34 44 43 44 22 32 42 52 !>
Definition at line 197 of file spftrf.f.
subroutine ZPFTRF (character transr, character uplo, integer n, complex*16, dimension( 0: * ) a, integer info)¶
ZPFTRF
Purpose:
!> !> ZPFTRF computes the Cholesky factorization of a complex Hermitian !> positive definite 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. !> !> This is the block version of the algorithm, calling Level 3 BLAS. !>
Parameters
TRANSR
!> TRANSR is CHARACTER*1 !> = 'N': The Normal TRANSR of RFP A is stored; !> = 'C': The Conjugate-transpose TRANSR of RFP A is stored. !>
UPLO
!> UPLO is CHARACTER*1 !> = 'U': Upper triangle of RFP A is stored; !> = 'L': Lower triangle of RFP A is stored. !>
N
!> N is INTEGER !> The order of the matrix A. N >= 0. !>
A
!> A is COMPLEX*16 array, dimension ( N*(N+1)/2 ); !> On entry, the Hermitian matrix A in RFP format. RFP format is !> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' !> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is !> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'C' then RFP is !> the Conjugate-transpose of RFP A as defined when !> TRANSR = 'N'. The contents of RFP A are defined by UPLO as !> follows: If UPLO = 'U' the RFP A contains the nt elements of !> upper packed A. If UPLO = 'L' the RFP A contains the elements !> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = !> 'C'. When TRANSR is 'N' the LDA is N+1 when N is even and N !> is odd. See the Note below for more details. !> !> On exit, if INFO = 0, the factor U or L from the Cholesky !> factorization RFP A = U**H*U or RFP A = L*L**H. !>
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. !> !> Further Notes on RFP Format: !> ============================ !> !> We first consider Standard Packed Format when N is even. !> We give an example where N = 6. !> !> AP is Upper AP is Lower !> !> 00 01 02 03 04 05 00 !> 11 12 13 14 15 10 11 !> 22 23 24 25 20 21 22 !> 33 34 35 30 31 32 33 !> 44 45 40 41 42 43 44 !> 55 50 51 52 53 54 55 !> !> Let TRANSR = 'N'. RFP holds AP as follows: !> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last !> three columns of AP upper. The lower triangle A(4:6,0:2) consists of !> conjugate-transpose of the first three columns of AP upper. !> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first !> three columns of AP lower. The upper triangle A(0:2,0:2) consists of !> conjugate-transpose of the last three columns of AP lower. !> To denote conjugate we place -- above the element. This covers the !> case N even and TRANSR = 'N'. !> !> RFP A RFP A !> !> -- -- -- !> 03 04 05 33 43 53 !> -- -- !> 13 14 15 00 44 54 !> -- !> 23 24 25 10 11 55 !> !> 33 34 35 20 21 22 !> -- !> 00 44 45 30 31 32 !> -- -- !> 01 11 55 40 41 42 !> -- -- -- !> 02 12 22 50 51 52 !> !> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- !> transpose of RFP A above. One therefore gets: !> !> RFP A RFP A !> !> -- -- -- -- -- -- -- -- -- -- !> 03 13 23 33 00 01 02 33 00 10 20 30 40 50 !> -- -- -- -- -- -- -- -- -- -- !> 04 14 24 34 44 11 12 43 44 11 21 31 41 51 !> -- -- -- -- -- -- -- -- -- -- !> 05 15 25 35 45 55 22 53 54 55 22 32 42 52 !> !> We next consider Standard Packed Format when N is odd. !> We give an example where N = 5. !> !> AP is Upper AP is Lower !> !> 00 01 02 03 04 00 !> 11 12 13 14 10 11 !> 22 23 24 20 21 22 !> 33 34 30 31 32 33 !> 44 40 41 42 43 44 !> !> Let TRANSR = 'N'. RFP holds AP as follows: !> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last !> three columns of AP upper. The lower triangle A(3:4,0:1) consists of !> conjugate-transpose of the first two columns of AP upper. !> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first !> three columns of AP lower. The upper triangle A(0:1,1:2) consists of !> conjugate-transpose of the last two columns of AP lower. !> To denote conjugate we place -- above the element. This covers the !> case N odd and TRANSR = 'N'. !> !> RFP A RFP A !> !> -- -- !> 02 03 04 00 33 43 !> -- !> 12 13 14 10 11 44 !> !> 22 23 24 20 21 22 !> -- !> 00 33 34 30 31 32 !> -- -- !> 01 11 44 40 41 42 !> !> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate- !> transpose of RFP A above. One therefore gets: !> !> RFP A RFP A !> !> -- -- -- -- -- -- -- -- -- !> 02 12 22 00 01 00 10 20 30 40 50 !> -- -- -- -- -- -- -- -- -- !> 03 13 23 33 11 33 11 21 31 41 51 !> -- -- -- -- -- -- -- -- -- !> 04 14 24 34 44 43 44 22 32 42 52 !>
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Definition at line 210 of file zpftrf.f.
Author¶
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