.TH "tftri" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME tftri \- tftri: triangular inverse, RFP .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBctftri\fP (transr, uplo, diag, n, a, info)" .br .RI "\fBCTFTRI\fP " .ti -1c .RI "subroutine \fBdtftri\fP (transr, uplo, diag, n, a, info)" .br .RI "\fBDTFTRI\fP " .ti -1c .RI "subroutine \fBstftri\fP (transr, uplo, diag, n, a, info)" .br .RI "\fBSTFTRI\fP " .ti -1c .RI "subroutine \fBztftri\fP (transr, uplo, diag, n, a, info)" .br .RI "\fBZTFTRI\fP " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine ctftri (character transr, character uplo, character diag, integer n, complex, dimension( 0: * ) a, integer info)" .PP \fBCTFTRI\fP .PP \fBPurpose:\fP .RS 4 .PP .nf CTFTRI computes the inverse of a triangular matrix A stored in RFP format\&. This is a Level 3 BLAS version of the algorithm\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANSR\fP .PP .nf TRANSR is CHARACTER*1 = 'N': The Normal TRANSR of RFP A is stored; = 'C': The Conjugate-transpose TRANSR of RFP A is stored\&. .fi .PP .br \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular\&. .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX array, dimension ( N*(N+1)/2 ); On entry, the triangular 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 nt 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 (triangular) inverse of the original matrix, in the same storage format\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, A(i,i) is exactly zero\&. The triangular matrix is singular and its inverse can not be computed\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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 .fi .PP .RE .PP .PP Definition at line \fB220\fP of file \fBctftri\&.f\fP\&. .SS "subroutine dtftri (character transr, character uplo, character diag, integer n, double precision, dimension( 0: * ) a, integer info)" .PP \fBDTFTRI\fP .PP \fBPurpose:\fP .RS 4 .PP .nf DTFTRI computes the inverse of a triangular matrix A stored in RFP format\&. This is a Level 3 BLAS version of the algorithm\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANSR\fP .PP .nf TRANSR is CHARACTER*1 = 'N': The Normal TRANSR of RFP A is stored; = 'T': The Transpose TRANSR of RFP A is stored\&. .fi .PP .br \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular\&. .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIA\fP .PP .nf A is DOUBLE PRECISION array, dimension (0:nt-1); nt=N*(N+1)/2\&. On entry, the triangular factor of a Hermitian Positive Definite 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 nt 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 (triangular) inverse of the original matrix, in the same storage format\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, A(i,i) is exactly zero\&. The triangular matrix is singular and its inverse can not be computed\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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 .fi .PP .RE .PP .PP Definition at line \fB200\fP of file \fBdtftri\&.f\fP\&. .SS "subroutine stftri (character transr, character uplo, character diag, integer n, real, dimension( 0: * ) a, integer info)" .PP \fBSTFTRI\fP .PP \fBPurpose:\fP .RS 4 .PP .nf STFTRI computes the inverse of a triangular matrix A stored in RFP format\&. This is a Level 3 BLAS version of the algorithm\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANSR\fP .PP .nf TRANSR is CHARACTER*1 = 'N': The Normal TRANSR of RFP A is stored; = 'T': The Transpose TRANSR of RFP A is stored\&. .fi .PP .br \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular\&. .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension (NT); NT=N*(N+1)/2\&. On entry, the triangular factor of a Hermitian Positive Definite 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 nt 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 (triangular) inverse of the original matrix, in the same storage format\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, A(i,i) is exactly zero\&. The triangular matrix is singular and its inverse can not be computed\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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 .fi .PP .RE .PP .PP Definition at line \fB200\fP of file \fBstftri\&.f\fP\&. .SS "subroutine ztftri (character transr, character uplo, character diag, integer n, complex*16, dimension( 0: * ) a, integer info)" .PP \fBZTFTRI\fP .PP \fBPurpose:\fP .RS 4 .PP .nf ZTFTRI computes the inverse of a triangular matrix A stored in RFP format\&. This is a Level 3 BLAS version of the algorithm\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fITRANSR\fP .PP .nf TRANSR is CHARACTER*1 = 'N': The Normal TRANSR of RFP A is stored; = 'C': The Conjugate-transpose TRANSR of RFP A is stored\&. .fi .PP .br \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': A is upper triangular; = 'L': A is lower triangular\&. .fi .PP .br \fIDIAG\fP .PP .nf DIAG is CHARACTER*1 = 'N': A is non-unit triangular; = 'U': A is unit triangular\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX*16 array, dimension ( N*(N+1)/2 ); On entry, the triangular 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 nt 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 (triangular) inverse of the original matrix, in the same storage format\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, A(i,i) is exactly zero\&. The triangular matrix is singular and its inverse can not be computed\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf 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 .fi .PP .RE .PP .PP Definition at line \fB220\fP of file \fBztftri\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.