linalg::solve_cholesky Interface Reference

Solves a system of Cholesky factored equations. More...

Public Member Functions
	solve_cholesky_mtx
	solve_cholesky_mtx_cmplx
	solve_cholesky_vec
	solve_cholesky_vec_cmplx

Detailed Description

Solves a system of Cholesky factored equations.

Syntax

subroutine solve_cholesky(logical upper, real(real64) a(:,:), real(real64) b(:,:), optional class(errors) err)
subroutine solve_cholesky(logical upper, complex(real64) a(:,:), complex(real64) b(:,:), optional class(errors) err)
subroutine solve_cholesky(logical upper, real(real64) a(:,:), real(real64) b(:), optional class(errors) err)
subroutine solve_cholesky(logical upper, complex(real64) a(:,:), complex(real64) b(:), optional class(errors) err)

Parameters

[in]	upper	Set to true if the original matrix \( A \) was factored such that \( A = U^T U \); else, set to false if the factorization of \( A \) was \( A = L L^T \).
[in]	a	The N-by-N Cholesky factored matrix as returned by cholesky_factor.
[in,out]	b	On input, the N-by-NRHS right-hand-side matrix B. On output, the solution matrix X.
[in,out]	err	An optional errors-based object that if provided can be used to retrieve information relating to any errors encountered during execution. If not provided, a default implementation of the errors class is used internally to provide error handling. Possible errors and warning messages that may be encountered are as follows. LA_ARRAY_SIZE_ERROR: Occurs if any of the input array sizes are incorrect.

Notes

This routine utilizes the LAPACK routine DPOTRS (ZPOTRS in the complex case).

Usage

The following example illustrates the solution of a positive-definite system of equations via Cholesky factorization.

program example
    use iso_fortran_env, only : real64, int32
    use linalg
    implicit none
 
    ! Variables
    real(real64) :: a(3, 3), b(3), bu(3)
    integer(int32) :: i
 
    ! Build the 3-by-3 positive-definite matrix A.
    !     | 4   12   -16 |
    ! A = | 12  37   -43 |
    !     |-16 -43    98 |
    a = reshape([4.0d0, 12.0d0, -16.0d0, 12.0d0, 37.0d0, -43.0d0, -16.0d0, &
        -43.0d0, 98.0d0], [3, 3])
 
    ! Build the 3-element array B
    !     | 5 |
    ! b = | 1 |
    !     | 3 |
    b = [5.0d0, 1.0d0, 3.0d0]
 
    ! Make a copy of B for later use - not necessary, but just for example to
    ! illustrate the long or manual method of solving a Cholesky factored system
    bu = b
 
    ! Compute the Cholesky factorization of A considering only the upper
    ! triangular portion of A (the default configuration).
    call cholesky_factor(a)
 
    ! Compute the solution
    call solve_cholesky(.true., a, b)
 
    ! Display the results
    print '(A)', "Cholesky Solution: X = "
    print '(F8.4)', (b(i), i = 1, size(b))
 
    ! The solution could also be computed manually noting the Cholesky
    ! factorization causes A = U**T * U.  Then U**T * U * X = B.
 
    ! Step 1 would then be to solve the problem U**T * Y = B, for Y.
    call solve_triangular_system(.true., .true., .true., a, bu)
 
    ! Now, solve the problem U * X = Y, for X
    call solve_triangular_system(.true., .false., .true., a, bu)
 
    ! Display the results
    print '(A)', "Cholesky Solution (Manual Approach): X = "
    print '(F8.4)', (bu(i), i = 1, size(bu))
end program

The above program produces the following output.

Cholesky Solution: X =
 239.5833
 -65.6667
 10.3333
Cholesky Solution (Manual Approach): X =
 239.5833
 -65.6667
 10.3333

Definition at line 2663 of file linalg.f90.

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The documentation for this interface was generated from the following file:

• D:/Code/linalg/src/linalg.f90