ZPFTRI(3)

💖 ZPFTRI(3) 💖

okay so this is from the Library Functions Manual, and it’s the ZPFTRI page. Buckle up, bestie — it’s giving packed-format Hermitian inverse realness. 🦋

✨ NAME ✨

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/zpftri.f

That’s the source file this page is attached to, periodt. 💅

🌸 SYNOPSIS 🌸

Functions/Subroutines

subroutine ZPFTRI (transr, uplo, n, a, info)

And yeah — the routine’s name again: ZPFTRI.

🦋 Function/Subroutine Documentation 🦋

**`subroutine ZPFTRI (character transr, character uplo, integer n, complex16, dimension( 0: ) a, integer info)`**

ZPFTRI

🎀 Purpose

okay so ZPFTRI computes the inverse of a complex Hermitian positive definite matrix A using the Cholesky factorization

A = U**H*U

A = L*L**H

computed by ZPFTRF. That’s the whole main-character mission. 👑

🌟 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 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 Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🌈 Further Details 🌈

okay so here’s the RFP diagram lore, and yes, we’re keeping all of it. 💖

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