table of contents
pftri(3) | Library Functions Manual | pftri(3) |
NAME¶
pftri - pftri: triangular inverse
SYNOPSIS¶
Functions¶
subroutine CPFTRI (transr, uplo, n, a, info)
CPFTRI subroutine DPFTRI (transr, uplo, n, a, info)
DPFTRI subroutine SPFTRI (transr, uplo, n, a, info)
SPFTRI subroutine ZPFTRI (transr, uplo, n, a, info)
ZPFTRI
Detailed Description¶
Function Documentation¶
subroutine CPFTRI (character transr, character uplo, integer n, complex, dimension( 0: * ) a, integer info)¶
CPFTRI
Purpose:
!> !> CPFTRI computes the inverse of a complex Hermitian positive definite !> matrix A using the Cholesky factorization A = U**H*U or A = L*L**H !> computed by CPFTRF. !>
Parameters
!> 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 A is stored; !> = 'L': Lower triangle of 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, the Hermitian inverse of the original matrix, in the !> same storage format. !>
INFO
!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: if INFO = i, the (i,i) element of the factor U or L is !> zero, and the inverse could not be computed. !>
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
!> !> 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 !>
Definition at line 211 of file cpftri.f.
subroutine DPFTRI (character transr, character uplo, integer n, double precision, dimension( 0: * ) a, integer info)¶
DPFTRI
Purpose:
!> !> DPFTRI computes the inverse of a (real) symmetric positive definite !> matrix A using the Cholesky factorization A = U**T*U or A = L*L**T !> computed by DPFTRF. !>
Parameters
!> 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 A is stored; !> = 'L': Lower triangle of 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, the symmetric inverse of the original matrix, in the !> same storage format. !>
INFO
!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: if INFO = i, the (i,i) element of the factor U or L is !> zero, and the inverse could not be computed. !>
Author
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 190 of file dpftri.f.
subroutine SPFTRI (character transr, character uplo, integer n, real, dimension( 0: * ) a, integer info)¶
SPFTRI
Purpose:
!> !> SPFTRI computes the inverse of a real (symmetric) positive definite !> matrix A using the Cholesky factorization A = U**T*U or A = L*L**T !> computed by SPFTRF. !>
Parameters
!> 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 A is stored; !> = 'L': Lower triangle of 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, the symmetric inverse of the original matrix, in the !> same storage format. !>
INFO
!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: if INFO = i, the (i,i) element of the factor U or L is !> zero, and the inverse could not be computed. !>
Author
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 190 of file spftri.f.
subroutine ZPFTRI (character transr, character uplo, integer n, complex*16, dimension( 0: * ) a, integer info)¶
ZPFTRI
Purpose:
!> !> ZPFTRI computes the inverse of a complex Hermitian positive definite !> matrix A using the Cholesky factorization A = U**H*U or A = L*L**H !> computed by ZPFTRF. !>
Parameters
!> 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 A is stored; !> = 'L': Lower triangle of 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, the Hermitian inverse of the original matrix, in the !> same storage format. !>
INFO
!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> > 0: if INFO = i, the (i,i) element of the factor U or L is !> zero, and the inverse could not be computed. !>
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
!> !> 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 !>
Definition at line 211 of file zpftri.f.
Author¶
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