pftri(3)

📦

version 3.12.1 · 9 versions

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3.12.1 · ubuntu stonking
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3.12.1 · debian trixie

🎀 pftri(3) 🎀

okay so you're reading pftri(3) from the Library Functions Manual, bestie, and it's literally the "triangular inverse" page — four whole routines that undo a Cholesky factorization. buckle up 💖

✨ NAME ✨

pftri — pftri: triangular inverse

no cap, that's the whole name section. the manpage authors were feeling minimal that day 🫶

🦋 SYNOPSIS 🦋

🌸 Functions

four flavors of the exact same vibe, one per precision, and they all take the same five arguments:

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

cpftri — the complex one 💅
dpftri — the double precision one
spftri — the single precision one
zpftri — the double complex one

💜 Detailed Description 💜

okay so this section is literally empty in the original. like, nothing. blank. the icon that leaves the group chat on read 😭 the actual content is all down in Function Documentation, so keep scrolling bestie.

👑 FUNCTION DOCUMENTATION 👑

🌈 `subroutine cpftri` 🌈

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. ✨

so like, CPFTRF does the factoring, and this one flips it inside out. iconic teamwork 🫶

🧚 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 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.

⚠️ heads up: if INFO > 0 you literally do not have an inverse. the factorization hit a zero on the diagonal and it's giving "nope". check INFO fr fr 🙈

💖 Author

Univ. of Tennessee

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` 🌈

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 — 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.

⚠️ same deal as before but for the real ones: INFO > 0 means a zero landed on the (i,i) of U or L and the inverse could not be computed. not slay. 💀

💖 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🦋 Further Details:

okay so the further details for dpftri start here and keep going — that's literally where this part of the page runs out, so hold that thought 🎀

okay so we're still in dpftri's Further Details and it's literally the whole RFP lore. Buckle up, bestie, because this is giving "matrix storage but make it fashion" 💅. No cap, every index matters.

First up: Rectangular Full Packed (RFP) Format when N is even. The example is 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 like this:

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

Then we consider RFP Format when N is odd. The example is 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.

✨ `spftri` — the single-precision sister

   subroutine spftri (character transr, character uplo, integer n, real,
       dimension( 0: * ) a, integer info)

The name is SPFTRI. She's the real single-precision version of the inverse-computing situation.

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. Obsessed, honestly.

🧾 Parameters

TRANSR — CHARACTER*1
- = 'N': The Normal TRANSR of RFP A is stored;
- = 'T': The Transpose TRANSR of RFP A is stored.
UPLO — CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — INTEGER
- The order of the matrix A. N >= 0.
A — 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 — 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

yes, this is the exact same RFP dissertation as above — the manpage repeats it for spftri, and we do not skip. periodt.

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.

🦋 `zpftri` — complex Hermitian edition

   subroutine zpftri (character transr, character uplo, integer n, complex*16,
       dimension( 0: * ) a, integer info)

The name is ZPFTRI. She's the complex double-precision girlie, and she's Hermitian, so adjust your wig accordingly.

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. The **H is giving conjugate-transpose realness, iykyk.

🧾 Parameters

TRANSR — CHARACTER*1
- = 'N': The Normal TRANSR of RFP A is stored;
- = 'C': The Conjugate-transpose TRANSR of RFP A is stored.
UPLO — CHARACTER*1
- = 'U': Upper triangle of A is stored;
- = 'L': Lower triangle of A is stored.
N — INTEGER
- The order of the matrix A. N >= 0.

`A`

and now for the A parameter for ZPFTRI, which is exactly where the page cuts off — hold that thought, bestie, because part 3 is picking it up right here. 🫶

okay so A is a COMPLEX*16 array, dimension ( N*(N+1)/2 ) 💅

On entry, it’s the Hermitian matrix A in RFP format. RFP format is described by TRANSR, UPLO, and N like this: 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, it’s the Hermitian inverse of the original matrix, in the same storage format. slay ✨

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 let’s get into the ✨further details✨ because this storage format is a whole vibe and you deserve to understand it 💅

First up: Standard Packed Format when N is even. Example with 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

When TRANSR = 'N', RFP holds AP like this:

For UPLO = 'U', the upper trapezoid A(0:5,0:2) is literally the last three columns of AP upper. The lower triangle A(4:6,0:2) is the conjugate-transpose of the first three columns of AP upper.
For UPLO = 'L', the lower trapezoid A(1:6,0:2) is the first three columns of AP lower. The upper triangle A(0:2,0:2) is the conjugate-transpose of the last three columns of AP lower.

To show conjugate we put -- 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. So you get:

          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

Next: Standard Packed Format when N is odd. Example with 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

When TRANSR = 'N', RFP holds AP like this:

For UPLO = 'U', the upper trapezoid A(0:4,0:2) is the last three columns of AP upper. The lower triangle A(3:4,0:1) is the conjugate-transpose of the first two columns of AP upper.
For UPLO = 'L', the lower trapezoid A(0:4,0:2) is the first three columns of AP lower. The upper triangle A(0:1,1:2) is the conjugate-transpose of the last two columns of AP lower.

To show conjugate we put -- 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. So you get:

          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

and that’s the whole page, bestie! this is from LAPACK version 3.12.0, page pftri(3), so you know it’s official ✨💖

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

pftri(3)

🎀 pftri(3) 🎀

✨ NAME ✨

🦋 SYNOPSIS 🦋

🌸 Functions

💜 Detailed Description 💜

👑 FUNCTION DOCUMENTATION 👑

🌈 subroutine cpftri 🌈

⭐ Purpose

🧚 Parameters

💖 Author

🦋 Further Details:

🌈 subroutine dpftri 🌈

⭐ Purpose

🧚 Parameters

💖 Author

🦋 Further Details:

✨ spftri — the single-precision sister

🧾 Parameters

💫 Further Details

🦋 zpftri — complex Hermitian edition

🧾 Parameters

A

INFO 💖

Author 👑

Further Details 🌸

🌈 `subroutine cpftri` 🌈

🌈 `subroutine dpftri` 🌈

✨ `spftri` — the single-precision sister

🦋 `zpftri` — complex Hermitian edition

`A`