Man page - pftrf(3)

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Manual

pftrf

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine cpftrf (character transr, character uplo, integer n, complex,dimension( 0: * ) a, integer info)
subroutine dpftrf (character transr, character uplo, integer n, doubleprecision, dimension( 0: * ) a, integer info)
subroutine spftrf (character transr, character uplo, integer n, real,dimension( 0: * ) a, integer info)
subroutine zpftrf (character transr, character uplo, integer n, complex*16,dimension( 0: * ) a, integer info)
Author

NAME

pftrf - pftrf: triangular factor

SYNOPSIS

Functions

subroutine cpftrf (transr, uplo, n, a, info)
CPFTRF
subroutine dpftrf (transr, uplo, n, a, info)
DPFTRF
subroutine spftrf (transr, uplo, n, a, info)
SPFTRF
subroutine zpftrf (transr, uplo, n, a, info)
ZPFTRF

Detailed Description

Function Documentation

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

CPFTRF

Purpose:

CPFTRF computes the Cholesky factorization of a complex Hermitian
positive definite matrix A.

The factorization has the form
A = U**H * U, if UPLO = ’U’, or
A = L * L**H, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular.

This is the block version of the algorithm, calling Level 3 BLAS.

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 RFP A is stored;
= ’L’: Lower triangle of RFP 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, if INFO = 0, the factor U or L from the Cholesky
factorization RFP A = U**H*U or RFP A = L*L**H.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading principal minor of order i
is not positive, and the factorization could not be
completed.

Further Notes on RFP Format:
============================

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dpftrf (character transr, character uplo, integer n, doubleprecision, dimension( 0: * ) a, integer info)

DPFTRF

Purpose:

DPFTRF computes the Cholesky factorization of a real symmetric
positive definite matrix A.

The factorization has the form
A = U**T * U, if UPLO = ’U’, or
A = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular.

This is the block version of the algorithm, calling Level 3 BLAS.

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 RFP A is stored;
= ’L’: Lower triangle of RFP 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, if INFO = 0, the factor U or L from the Cholesky
factorization RFP A = U**T*U or RFP A = L*L**T.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading principal minor of order i
is not positive, and the factorization could not be
completed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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

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

SPFTRF

Purpose:

SPFTRF computes the Cholesky factorization of a real symmetric
positive definite matrix A.

The factorization has the form
A = U**T * U, if UPLO = ’U’, or
A = L * L**T, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular.

This is the block version of the algorithm, calling Level 3 BLAS.

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 RFP A is stored;
= ’L’: Lower triangle of RFP A is stored.

N

N is INTEGER
The order of the matrix A. N >= 0.

A

A is 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, if INFO = 0, the factor U or L from the Cholesky
factorization RFP A = U**T*U or RFP A = L*L**T.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading principal minor of order i
is not positive, and the factorization could not be
completed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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

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

ZPFTRF

Purpose:

ZPFTRF computes the Cholesky factorization of a complex Hermitian
positive definite matrix A.

The factorization has the form
A = U**H * U, if UPLO = ’U’, or
A = L * L**H, if UPLO = ’L’,
where U is an upper triangular matrix and L is lower triangular.

This is the block version of the algorithm, calling Level 3 BLAS.

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 RFP A is stored;
= ’L’: Lower triangle of RFP 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, if INFO = 0, the factor U or L from the Cholesky
factorization RFP A = U**H*U or RFP A = L*L**H.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading principal minor of order i
is not positive, and the factorization could not be
completed.

Further Notes on RFP Format:
============================

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Author

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