DPFTRF(3) LAPACK routine of NEC Numeric Library Collection DPFTRF(3)
NAME
DPFTRF
SYNOPSIS
SUBROUTINE DPFTRF (TRANSR, UPLO, N, A, INFO)
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.
ARGUMENTS
TRANSR (input)
TRANSR is CHARACTER*1
= 'N': The Normal TRANSR of RFP A is stored;
= 'T': The Transpose TRANSR of RFP A is stored.
UPLO (input)
UPLO is CHARACTER*1
= 'U': Upper triangle of RFP A is stored;
= 'L': Lower triangle of RFP A is stored.
N (input)
N is INTEGER
The order of the matrix A. N >= 0.
A (input/output)
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 (output)
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the leading minor of order i is not
positive definite, and the factorization could not be
completed.
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
LAPACK routine 31 October 2017 DPFTRF(3)