ZPFTRI(3) 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

## SYNOPSIS

SUBROUTINE ZPFTRI(
TRANSR, UPLO, N, A, INFO )

CHARACTER TRANSR, UPLO

INTEGER INFO, N

COMPLEX*16 A( 0: * )

## 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.

## ARGUMENTS

TRANSR (input) CHARACTER
= 'N': The Normal TRANSR of RFP A is stored;
= 'C': The Conjugate-transpose TRANSR of RFP A is stored.
UPLO (input) CHARACTER

= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) 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 (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 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