pftri#

Functions

void spftri(
    const char*          transr,
    const char*          uplo,
    const INT            n,
          f32*  restrict A,
          INT*           info
);
void spftri(const char *transr, const char *uplo, const INT n, f32 *restrict A, INT *info)#

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.

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='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

Parameters

in
transr

  • 'N': The Normal TRANSR of RFP A is stored

  • 'T': The Transpose TRANSR of RFP A is stored

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

in
n

The order of the matrix A. n>=0.

inout
A

Array of dimension n*(n+1)/2. On entry, the Cholesky factor in RFP format. On exit, the inverse of the original matrix.

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, the (i,i) element of the factor is zero

Functions

void dpftri(
    const char*          transr,
    const char*          uplo,
    const INT            n,
          f64*  restrict A,
          INT*           info
);
void dpftri(const char *transr, const char *uplo, const INT n, f64 *restrict A, INT *info)#

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.

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='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

Parameters

in
transr

  • 'N': The Normal TRANSR of RFP A is stored

  • 'T': The Transpose TRANSR of RFP A is stored

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

in
n

The order of the matrix A. n>=0.

inout
A

Array of dimension n*(n+1)/2. On entry, the Cholesky factor in RFP format. On exit, the inverse of the original matrix.

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, the (i,i) element of the factor is zero

Functions

void cpftri(
    const char*          transr,
    const char*          uplo,
    const INT            n,
          c64*  restrict A,
          INT*           info
);
void cpftri(const char *transr, const char *uplo, const INT n, c64 *restrict A, INT *info)#

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.

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='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 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 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

Parameters

in
transr

  • 'N': The Normal TRANSR of RFP A is stored

  • 'C': The Conjugate-transpose TRANSR of RFP A is stored

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

in
n

The order of the matrix A. n>=0.

inout
A

Complex array of dimension n*(n+1)/2. On entry, the Cholesky factor in RFP format. On exit, the inverse of the original matrix.

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, the (i,i) element of the factor is zero

Functions

void zpftri(
    const char*          transr,
    const char*          uplo,
    const INT            n,
          c128* restrict A,
          INT*           info
);
void zpftri(const char *transr, const char *uplo, const INT n, c128 *restrict A, INT *info)#

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.

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='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 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 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

Parameters

in
transr

  • 'N': The Normal TRANSR of RFP A is stored

  • 'C': The Conjugate-transpose TRANSR of RFP A is stored

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

in
n

The order of the matrix A. n>=0.

inout
A

Complex array of dimension n*(n+1)/2. On entry, the Cholesky factor in RFP format. On exit, the inverse of the original matrix.

out
info

  • info=0: successful exit

  • info<0: if info=-i, the i-th argument had an illegal value

  • info>0: if info=i, the (i,i) element of the factor is zero