tftri(3)

📦

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💅 tftri(3) — triangular inverse, RFP, but make it make sense ✨

okay so bestie, welcome to tftri(3) from the Library Functions Manual 🫶. This is the RFP triangular-inverse situation, and we're keeping every single technical detail while making it feel like a group chat. No cap.

✨ NAME ✨

tftri — tftri: triangular inverse, RFP 💖

🌈 SYNOPSIS 🌈

💫 Functions 💫

subroutine ctftri (transr, uplo, diag, n, a, info)
    CTFTRI
subroutine dtftri (transr, uplo, diag, n, a, info)
    DTFTRI
subroutine stftri (transr, uplo, diag, n, a, info)
    STFTRI
subroutine ztftri (transr, uplo, diag, n, a, info)
    ZTFTRI

🦋 Detailed Description 🦋

This page is where the tftri crew lives: CTFTRI, DTFTRI, STFTRI, and ZTFTRI are all listed up top in the synopsis. The function-by-function documentation starts right below, and yes, we're keeping all the !>-worthy details. 💅

🔥 Function Documentation 🔥

👑 `subroutine ctftri (character transr, character uplo, character diag, integer n, complex, dimension( 0: * ) a, integer info)` 👑

CTFTRI

Purpose: CTFTRI computes the inverse of a triangular matrix A stored in RFP format. This is a Level 3 BLAS version of the algorithm. 💖

✨ 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': A is upper triangular; = 'L': A is lower triangular. 💖
DIAG — DIAG is CHARACTER*1; = 'N': A is non-unit triangular; = 'U': A is unit triangular. 🌸
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 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. 😭
INFO — 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. 💀

🌟 Author 🌟

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

📚 Further Details 📚

Okay so first up: Standard Packed Format when N is even. Example time: 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

Next we consider Standard Packed Format when N is odd. Example: 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 220 of file ctftri.f.

🌟 `subroutine dtftri (character transr, character uplo, character diag, integer n, double precision, dimension( 0: * ) a, integer info)` 🌟

DTFTRI

Purpose: DTFTRI computes the inverse of a triangular matrix A stored in RFP format. This is a Level 3 BLAS version of the algorithm. 🔥

✨ 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': A is upper triangular; = 'L': A is lower triangular. 💖
DIAG — DIAG is CHARACTER*1; = 'N': A is non-unit triangular; = 'U': A is unit triangular. 🌸
N — N is INTEGER; the order of the matrix A. N >= 0. 🎀
A — 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. 😭
INFO — 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. 💀

🌟 Author 🌟

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

🎀 Further Details: 🎀

okay so before we meet the next two flavours, we HAVE to talk about RFP layout, because it's literally giving "where did the data go" and you need it, bestie 💅

Case one: N is even. Picture N = 6. 🎀

  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

(that's AP is Upper on the left and AP is Lower on the right, in case the vibe wasn't clear ✨)

Now 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'.

So RFP A comes out looking like this (left is UPLO = 'U', right is UPLO = 'L') 🦋:

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. So you get, no cap 🌸:

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

Case two: N is odd. 🔥 Picture N = 5.

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

(same deal — AP is Upper left, AP is Lower right)

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'.

And RFP A looks like 💜:

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, so you get ⭐:

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 200 of file dtftri.f. 🫶

✨ `stftri` — the single-precision one ✨

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

STFTRI 🎀

💖 Purpose

STFTRI computes the inverse of a triangular matrix A stored in RFP format.
This is a Level 3 BLAS version of the algorithm.

🌈 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': A is upper triangular;
- = 'L': A is lower triangular.
DIAG — DIAG is CHARACTER*1
- = 'N': A is non-unit triangular;
- = 'U': A is unit triangular.
N — N is INTEGER. The order of the matrix A. N >= 0.
A — 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.
INFO — 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.

👑 Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🧚 Further Details

okay we're doing the RFP tour again, because yes, it's that important, bestie ✨

We first consider Rectangular Full Packed (RFP) Format when N is even. We give an example where N = 6.

  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

(AP is Upper on the left, AP is Lower on the right 🦋)

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'.

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 💅:

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.

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'.

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 🌈:

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 200 of file stftri.f. 🫶

💀 `ztftri` — the big complex one 💀

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

ZTFTRI 🔥

🌟 Purpose

ZTFTRI computes the inverse of a triangular matrix A stored in RFP format.
This is a Level 3 BLAS version of the algorithm.

🍓 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.

heads up bestie — complex version swaps the 'T' for a 'C', that's the conjugate-transpose, not a typo, do NOT come for me 🙈

UPLO — UPLO is CHARACTER*1
- = 'U': A is upper triangular;
- = 'L': A is lower triangular.
DIAG — DIAG is CHARACTER*1
- = 'N': A is non-unit triangular;
- = 'U': A is unit triangular.
N — N is INTEGER. The order of the matrix A. N >= 0.
A — it's A is COMPLEX*16 array, dimension (NT)... and then we get into the whole RFP storage saga, which, iykyk, is the exact same layout described up top 👀

A is a COMPLEX*16 array, dimension ( N*(N+1)/2 ). On entry, it's 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, it's the (triangular) inverse of the original matrix, in the same storage format. 💖

INFO is an INTEGER:

= 0: successful exit 🎉
< 0: if INFO = -i, the i-th argument had an illegal value (so double-check your inputs, bestie)
> 0: if INFO = i, A(i,i) is exactly zero. The triangular matrix is singular and its inverse can not be computed. 🚨 (like, you literally can't invert it, no cap)

👑 Author 👑

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

✨ Further Details ✨

Okay so first we 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 220 of file ztftri.f.

Author (auto-generated)

Generated automatically by Doxygen for LAPACK from the source code.

💖 that's the whole page, bestie! this is from LAPACK Version 3.12.0, page tftri(3). catch you in the next manpage! 💅🎀

📦 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

tftri(3)

💅 tftri(3) — triangular inverse, RFP, but make it make sense ✨

✨ NAME ✨

🌈 SYNOPSIS 🌈

💫 Functions 💫

🦋 Detailed Description 🦋

🔥 Function Documentation 🔥

👑 subroutine ctftri (character transr, character uplo, character diag, integer n, complex, dimension( 0: * ) a, integer info) 👑

✨ Parameters ✨

🌟 Author 🌟

📚 Further Details 📚

🌟 subroutine dtftri (character transr, character uplo, character diag, integer n, double precision, dimension( 0: * ) a, integer info) 🌟

✨ Parameters ✨

🌟 Author 🌟

🎀 Further Details: 🎀

✨ stftri — the single-precision one ✨

💖 Purpose

🌈 Parameters

👑 Author

🧚 Further Details

💀 ztftri — the big complex one 💀

🌟 Purpose

🍓 Parameters

👑 Author 👑

✨ Further Details ✨

Author (auto-generated)

👑 `subroutine ctftri (character transr, character uplo, character diag, integer n, complex, dimension( 0: * ) a, integer info)` 👑

🌟 `subroutine dtftri (character transr, character uplo, character diag, integer n, double precision, dimension( 0: * ) a, integer info)` 🌟

✨ `stftri` — the single-precision one ✨

💀 `ztftri` — the big complex one 💀