CHOL more...
Computes the Cholesky factorization of a positive-definite, symmetric or self-adjoint matrix.
Function Return Value
Matrix containing the Cholesky factorization of A. The factor is upper triangular, RTR = A. (Output)
Required Argument
A — Matrix to be factored. This argument must be a rank-2 or rank-3 array that contains a positive-definite, symmetric or self-adjoint matrix. It may be real, double, complex, double complex. (Input)
For rank-3 arrays each rank-2 array (for fixed third subscript) is a positive-definite, symmetric or self-adjoint matrix. In this case, the output is a rank-3 array of Cholesky factors for the individual problems.
Optional Arguments, Packaged Options
This function uses LIN_SOL_SELF (See Chapter 1, “Linear Systems”), using the appropriate options to obtain the Cholesky factorization.
The option and derived type names are given in the following tables:
Option Names for CHOL
Option Value
Use_lin_sol_gen_only
4
Use_lin_sol_lsq_only
5
Name of Unallocated Option Array to Use for Setting Options
Use
Derived Type
?_chol_options(:)
Use when setting options for calls hereafter.
?_options
?_chol_options_once(:)
Use when setting options for next call only.
?_options
For a description on how to use these options, see Matrix Optional Data Changes. See LIN_SOL_SELF in Chapter 1, “Linear Systems” for the specific options for these routines.
FORTRAN 90 Interface
CHOL(A)
Description
Computes the Cholesky factorization of a positive-definite, symmetric or self-adjoint matrix, A. The factor is upper triangular, RTR = A.
Examples
Dense Matrix Example (operator_ex06.f90)

use linear_operators
implicit none
! This is the equivalent of Example 2 for LIN_SOL_SELF using operators
! and functions.
integer, parameter :: m=64, n=32
real(kind(1e0)) :: one=1e0, zero=0e0, err
real(kind(1e0)) A(n,n), b(n), C(m,n), d(m), cov(n,n), x(n)
! Generate a random rectangular matrix and right-hand side.
C = rand(C); d=rand(d)
! Form the normal equations for the rectangular system.
A = C .tx. C; b = C .tx. d
COV = .i. CHOL(A); COV = COV .xt. COV
! Compute the least-squares solution.
x = C .ix. d
! Compare with solution obtained using the inverse matrix.
err = norm(x - (COV .x. b))/norm(cov)
! Scale the inverse to obtain the sample covariance matrix.
COV = sum((d - (C .x. x))**2)/(m-n) * COV
! Check the results.
if (err <= sqrt(epsilon(one))) then
write (*,*) 'Example 2 for LIN_SOL_SELF (operators) is correct.'
end if
end
Parallel Example (parallel_ex06.f90)

use linear_operators
use mpi_setup_int

implicit none

! This is the equivalent of Parallel Example 6 for box data types, operators ! and functions.

integer, parameter :: m=64, n=32, nr=4
real(kind(1e0)) :: one=1e0, zero=0e0, err(nr)
real(kind(1e0)), dimension(m,n,nr) :: C, d(m,1,nr)
real(kind(1e0)), dimension(n,n,nr) :: A, cov
real(kind(1e0)), dimension(n,1,nr) :: b, x

! Setup for MPI:
mp_nprocs=mp_setup()
! Generate a random rectangular matrix and right-hand side.
if(mp_rank == 0) then
C = rand(C); d=rand(d)
endif

! Form the normal equations for the rectangular system.
A = C .tx. C; b = C .tx. d
COV = .i. CHOL(A); COV = COV .xt. COV

! Compute the least-squares solution.
x = C .ix. d

! Compare with solution obtained using the inverse matrix.
err = norm(x - (COV .x. b))/norm(cov)

! Check the results.
if (ALL(err <= sqrt(epsilon(one))) .and. mp_rank == 0) &
write (*,*) 'Parallel Example 6 is correct.'
! See to any eror messages and quit MPI
mp_nprocs=mp_setup('Final')

end