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linalg: Cholesky factorization (#840)
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‎doc/specs/stdlib_linalg.md‎

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{!example/linalg/example_svdvals.f90!}
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```
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## `cholesky` - Compute the Cholesky factorization of a rank-2 square array (matrix)
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### Status
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Experimental
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### Description
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This subroutine computes the Cholesky factorization of a `real` or `complex` rank-2 square array (matrix),
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\( A = L \cdot L^T \), or \( A = U^T \cdot U \). \( A \) is symmetric or complex Hermitian, and \( L \),
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\( U \) are lower- or upper-triangular, respectively.
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The solver is based on LAPACK's `*POTRF` backends.
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### Syntax
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`call ` [[stdlib_linalg(module):cholesky(interface)]] `(a, c, lower [, other_zeroed] [, err])`
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### Class
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Subroutine
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### Arguments
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`a`: Shall be a rank-2 square `real` or `complex` array containing the coefficient matrix of size `[n,n]`. It is an `intent(inout)` argument, but returns unchanged if the argument `c` is present.
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`c` (optional): Shall be a rank-2 square `real` or `complex` of the same size and kind as `a`. It is an `intent(out)` argument, that returns the triangular Cholesky matrix `L` or `U`.
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`lower`: Shall be an input `logical` flag. If `.true.`, the lower triangular decomposition \( A = L \cdot L^T \) will be performed. If `.false.`, the upper decomposition \( A = U^T \cdot U \) will be performed.
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`other_zeroed` (optional): Shall be an input `logical` flag. If `.true.`, the unused part of the output matrix will contain zeroes. Otherwise, it will not be accessed. This saves cpu time. By default, `other_zeroed=.true.`. It is an `intent(in)` argument.
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`err` (optional): Shall be a `type(linalg_state_type)` value. It is an `intent(out)` argument.
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### Return values
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The factorized matrix is returned in-place overwriting `a` if no other arguments are provided.
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Otherwise, it can be provided as a second argument `c`. In this case, `a` is not overwritten.
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The `logical` flag `lower` determines whether the lower- or the upper-triangular factorization should be performed.
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Results are returned on the applicable triangular region of the output matrix, while the unused triangular region
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is filled by zeroes by default. Optional argument `other_zeroed`, if `.false.` allows the expert user to avoid zeroing the unused part;
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however, in this case, the unused region of the matrix is not accessed and will usually contain invalid values.
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Raises `LINALG_ERROR` if the underlying process did not converge.
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Raises `LINALG_VALUE_ERROR` if the matrix or any of the output arrays invalid/incompatible sizes.
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Exceptions trigger an `error stop`, unless argument `err` is present.
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### Example
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```fortran
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{!example/linalg/example_cholesky.f90!}
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```
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## `chol` - Compute the Cholesky factorization of a rank-2 square array (matrix)
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### Status
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Experimental
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### Description
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This is a `pure` functional interface for the Cholesky factorization of a `real` or
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`complex` rank-2 square array (matrix) computed as \( A = L \cdot L^T \), or \( A = U^T \cdot U \).
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\( A \) is symmetric or complex Hermitian, and \( L \), \( U \) are lower- or upper-triangular, respectively.
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The solver is based on LAPACK's `*POTRF` backends.
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Result matrix `c` has the same size and kind as `a`, and returns the lower or upper triangular factor.
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### Syntax
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`c = ` [[stdlib_linalg(module):chol(interface)]] `(a, lower [, other_zeroed])`
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### Arguments
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`a`: Shall be a rank-2 square `real` or `complex` array containing the coefficient matrix of size `[n,n]`. It is an `intent(inout)` argument, but returns unchanged if argument `c` is present.
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`lower`: Shall be an input `logical` flag. If `.true.`, the lower triangular decomposition \( A = L \cdot L^T \) will be performed. If `.false.`, the upper decomposition \( A = U^T \cdot U \) will be performed.
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`other_zeroed` (optional): Shall be an input `logical` flag. If `.true.`, the unused part of the output matrix will contain zeroes. Otherwise, it will not be accessed. This saves cpu time. By default, `other_zeroed=.true.`. It is an `intent(in)` argument.
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### Return values
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Returns a rank-2 array `c` of the same size and kind as `a`, that contains the triangular Cholesky matrix `L` or `U`.
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Raises `LINALG_ERROR` if the underlying process did not converge.
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Raises `LINALG_VALUE_ERROR` if the matrix or any of the output arrays invalid/incompatible sizes.
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Exceptions trigger an `error stop`, unless argument `err` is present.
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### Example
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```fortran
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{!example/linalg/example_chol.f90!}
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```
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## `.inv.` - Inverse operator of a square matrix
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### Status
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For fine-grained error control in case of singular matrices prefer the `subroutine` and the `function`
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interfaces.
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### Example
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```fortran

‎example/linalg/CMakeLists.txt‎

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ADD_EXAMPLE(svdvals)
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ADD_EXAMPLE(determinant)
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ADD_EXAMPLE(determinant2)
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ADD_EXAMPLE(cholesky)
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ADD_EXAMPLE(chol)

‎example/linalg/example_chol.f90‎

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! Cholesky factorization: function interface
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program example_chol
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use stdlib_linalg, only: chol
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implicit none
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real, allocatable, dimension(:,:) :: A,L,U
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! Set real matrix
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A = reshape( [ [6, 15, 55], &
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[15, 55, 225], &
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[55, 225, 979] ], [3,3] )
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! Decompose (lower)
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L = chol(A, lower=.true.)
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! Compare decomposition
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print *, maxval(abs(A-matmul(L,transpose(L))))
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! Decompose (upper)
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U = chol(A, lower=.false.)
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! Compare decomposition
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print *, maxval(abs(A-matmul(transpose(U),U)))
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end program example_chol

‎example/linalg/example_cholesky.f90‎

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! Cholesky factorization: subroutine interface
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program example_cholesky
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use stdlib_linalg, only: cholesky
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implicit none
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real, dimension(3,3) :: A,L,U
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! Set real matrix
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A = reshape( [ [6, 15, 55], &
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[15, 55, 225], &
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[55, 225, 979] ], [3,3] )
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! Decompose (lower)
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call cholesky(A, L, lower=.true.)
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! Compare decomposition
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print *, maxval(abs(A-matmul(L,transpose(L))))
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! Decompose (upper)
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call cholesky(A, U, lower=.false.)
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! Compare decomposition
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print *, maxval(abs(A-matmul(transpose(U),U)))
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end program example_cholesky

‎src/CMakeLists.txt‎

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stdlib_linalg_inverse.fypp
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stdlib_linalg_state.fypp
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stdlib_linalg_svd.fypp
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stdlib_linalg_cholesky.fypp
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stdlib_optval.fypp
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stdlib_selection.fypp
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stdlib_sorting.fypp

‎src/stdlib_linalg.fypp‎

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implicit none
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private
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public :: chol
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public :: cholesky
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public :: det
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public :: operator(.det.)
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public :: diag
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! Export linalg error handling
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public :: linalg_state_type, linalg_error_handling
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interface chol
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!! version: experimental
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!!
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!! Computes the Cholesky factorization \( A = L \cdot L^T \), or \( A = U^T \cdot U \).
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!! ([Specification](../page/specs/stdlib_linalg.html#chol-compute-the-cholesky-factorization-of-a-rank-2-square-array-matrix))
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!!
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!!### Summary
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!! Pure function interface for computing the Cholesky triangular factors.
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!!
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!!### Description
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!!
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!! This interface provides methods for computing the lower- or upper- triangular matrix from the
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!! Cholesky factorization of a `real` symmetric or `complex` Hermitian matrix.
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!! Supported data types include `real` and `complex`.
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!!
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!!@note The solution is based on LAPACK's `*POTRF` methods.
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!!
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#:for rk,rt,ri in RC_KINDS_TYPES
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pure module function stdlib_linalg_${ri}$_cholesky_fun(a,lower,other_zeroed) result(c)
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!> Input matrix a[m,n]
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${rt}$, intent(in) :: a(:,:)
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!> [optional] is the lower or upper triangular factor required? Default = lower
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logical(lk), optional, intent(in) :: lower
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!> [optional] should the unused half of the return matrix be zeroed out? Default: yes
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logical(lk), optional, intent(in) :: other_zeroed
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!> Output matrix with Cholesky factors c[n,n]
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${rt}$ :: c(size(a,1),size(a,2))
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end function stdlib_linalg_${ri}$_cholesky_fun
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#:endfor
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end interface chol
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interface cholesky
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!! version: experimental
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!!
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!! Computes the Cholesky factorization \( A = L \cdot L^T \), or \( A = U^T \cdot U \).
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!! ([Specification](../page/specs/stdlib_linalg.html#cholesky-compute-the-cholesky-factorization-of-a-rank-2-square-array-matrix))
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!!
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!!### Summary
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!! Pure subroutine interface for computing the Cholesky triangular factors.
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!!
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!!### Description
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!!
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!! This interface provides methods for computing the lower- or upper- triangular matrix from the
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!! Cholesky factorization of a `real` symmetric or `complex` Hermitian matrix.
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!! Supported data types include `real` and `complex`.
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!! The factorization is computed in-place if only one matrix argument is present; or returned into
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!! a second matrix argument, if present. The `lower` `logical` flag allows to select between upper or
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!! lower factorization; the `other_zeroed` optional `logical` flag allows to choose whether the unused
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!! part of the triangular matrix should be filled with zeroes.
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!!
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!!@note The solution is based on LAPACK's `*POTRF` methods.
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!!
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#:for rk,rt,ri in RC_KINDS_TYPES
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pure module subroutine stdlib_linalg_${ri}$_cholesky_inplace(a,lower,other_zeroed,err)
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!> Input matrix a[m,n]
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${rt}$, intent(inout), target :: a(:,:)
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!> [optional] is the lower or upper triangular factor required? Default = lower
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logical(lk), optional, intent(in) :: lower
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!> [optional] should the unused half of the return matrix be zeroed out? Default: yes
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logical(lk), optional, intent(in) :: other_zeroed
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!> [optional] state return flag. On error if not requested, the code will stop
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type(linalg_state_type), optional, intent(out) :: err
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end subroutine stdlib_linalg_${ri}$_cholesky_inplace
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pure module subroutine stdlib_linalg_${ri}$_cholesky(a,c,lower,other_zeroed,err)
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!> Input matrix a[n,n]
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${rt}$, intent(in) :: a(:,:)
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!> Output matrix with Cholesky factors c[n,n]
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${rt}$, intent(out) :: c(:,:)
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!> [optional] is the lower or upper triangular factor required? Default = lower
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logical(lk), optional, intent(in) :: lower
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!> [optional] should the unused half of the return matrix be zeroed out? Default: yes
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logical(lk), optional, intent(in) :: other_zeroed
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!> [optional] state return flag. On error if not requested, the code will stop
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type(linalg_state_type), optional, intent(out) :: err
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end subroutine stdlib_linalg_${ri}$_cholesky
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#:endfor
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end interface cholesky
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interface diag
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!! version: experimental
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!!

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