spftri (l) - Linux Manuals

spftri: computes the inverse of a real (symmetric) positive definite matrix A using the Cholesky factorization A = U**TU or A = LL**T computed by SPFTRF

Command to display spftri manual in Linux: $ man l spftri

NAME

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

SYNOPSIS

SUBROUTINE SPFTRI(: TRANSR, UPLO, N, A, INFO )
: CHARACTER TRANSR, UPLO
: INTEGER INFO, N
: REAL A( 0: * )

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.

ARGUMENTS

TRANSR (input) CHARACTER: = aqNaq: The Normal TRANSR of RFP A is stored;
= aqTaq: The Transpose TRANSR of RFP A is stored.
UPLO (input) CHARACTER: = aqUaq: Upper triangle of A is stored;
= aqLaq: Lower triangle of A is stored.
N (input) INTEGER: The order of the matrix A. N >= 0.
A (input/output) 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 = aqNaq
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 = aqTaq then RFP is the transpose of RFP A as defined when TRANSR = aqNaq. The contents of RFP A are defined by UPLO as follows: If UPLO = aqUaq the RFP A contains the nt elements of upper packed A. If UPLO = aqLaq the RFP A contains the elements of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = aqTaq. When TRANSR is aqNaq 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 (output) 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.

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 = aqNaq. RFP holds AP as follows:
For UPLO = aqUaq 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 = aqLaq 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 = aqNaq.

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 = aqTaq. 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 first 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 = aqNaq. RFP holds AP as follows:
For UPLO = aqUaq 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 = aqLaq 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 = aqNaq.

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 = aqTaq. 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