# NAG FL Interfaceg02asf (corrmat_​fixed)

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## 1Purpose

g02asf computes the nearest correlation matrix, in the Frobenius norm, while fixing elements and optionally with bounds on the eigenvalues, to a given square input matrix.

## 2Specification

Fortran Interface
 Subroutine g02asf ( g, ldg, n, h, ldh, m, x, ldx, its,
 Integer, Intent (In) :: ldg, n, h(ldh,n), ldh, maxit, m, ldx Integer, Intent (Inout) :: ifail Integer, Intent (Out) :: its Real (Kind=nag_wp), Intent (In) :: alpha, errtol Real (Kind=nag_wp), Intent (Inout) :: g(ldg,n), x(ldx,n) Real (Kind=nag_wp), Intent (Out) :: fnorm
#include <nag.h>
 void g02asf_ (double g[], const Integer *ldg, const Integer *n, const double *alpha, const Integer h[], const Integer *ldh, const double *errtol, const Integer *maxit, const Integer *m, double x[], const Integer *ldx, Integer *its, double *fnorm, Integer *ifail)
The routine may be called by the names g02asf or nagf_correg_corrmat_fixed.

## 3Description

g02asf finds the nearest correlation matrix, $X$, to a matrix, $G$, in the Frobenius norm. It uses an alternating projections algorithm with Anderson acceleration. Elements in the input matrix can be fixed by supplying the value $1$ in the corresponding element of the matrix $H$. However, note that the algorithm may fail to converge if the fixed elements do not form part of a valid correlation matrix. You can optionally specify a lower bound, $\alpha$, on the eigenvalues of the computed correlation matrix, forcing the matrix to be positive definite with $0\le \alpha <1$.

## 4References

Anderson D G (1965) Iterative Procedures for Nonlinear Integral Equations J. Assoc. Comput. Mach. 12 547–560
Higham N J and Strabić N (2016) Anderson acceleration of the alternating projections method for computing the nearest correlation matrix Numer. Algor. 72 1021–1042

## 5Arguments

1: $\mathbf{g}\left({\mathbf{ldg}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Input/Output
On entry: $\stackrel{~}{G}$, the initial matrix.
On exit: the symmetric matrix $G=\frac{1}{2}\left(\stackrel{~}{G}+{\stackrel{~}{G}}^{\mathrm{T}}\right)$ with the diagonal elements set to $1.0$.
2: $\mathbf{ldg}$Integer Input
On entry: the first dimension of the array g as declared in the (sub)program from which g02asf is called.
Constraint: ${\mathbf{ldg}}\ge {\mathbf{n}}$.
3: $\mathbf{n}$Integer Input
On entry: the order of the matrix $G$.
Constraint: ${\mathbf{n}}>0$.
4: $\mathbf{alpha}$Real (Kind=nag_wp) Input
On entry: the value of $\alpha$.
If ${\mathbf{alpha}}<0.0$, a value of $0.0$ is used.
Constraint: ${\mathbf{alpha}}<1.0$.
5: $\mathbf{h}\left({\mathbf{ldh}},{\mathbf{n}}\right)$Integer array Input
On entry: the symmetric matrix $H$. If an element of $H$ is $1$ then the corresponding element in $G$ is fixed in the output $X$. Only the strictly lower triangular part of $H$ need be set.
6: $\mathbf{ldh}$Integer Input
On entry: the first dimension of the array h as declared in the (sub)program from which g02asf is called.
Constraint: ${\mathbf{ldh}}\ge {\mathbf{n}}$.
7: $\mathbf{errtol}$Real (Kind=nag_wp) Input
On entry: the termination tolerance for the iteration.
If ${\mathbf{errtol}}\le 0.0$, is used. See Section 7 for further details.
8: $\mathbf{maxit}$Integer Input
On entry: specifies the maximum number of iterations.
If ${\mathbf{maxit}}\le 0$, a value of $200$ is used.
9: $\mathbf{m}$Integer Input
On entry: the number of previous iterates to use in the Anderson acceleration. If ${\mathbf{m}}=0$, Anderson acceleration is not used. See Section 7 for further details.
If ${\mathbf{m}}<0$, a value of $4$ is used.
Constraint: ${\mathbf{m}}\le {\mathbf{n}}×{\mathbf{n}}$.
10: $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: contains the matrix $X$.
11: $\mathbf{ldx}$Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g02asf is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
12: $\mathbf{its}$Integer Output
On exit: the number of iterations taken.
13: $\mathbf{fnorm}$Real (Kind=nag_wp) Output
On exit: the value of ${‖G-X‖}_{F}$ after the final iteration.
14: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}>0$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{ldg}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldg}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ldh}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldh}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=4$
On entry, ${\mathbf{alpha}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{alpha}}<1.0$.
${\mathbf{ifail}}=5$
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\le {\mathbf{n}}×{\mathbf{n}}$.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{ldx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=7$
Routine failed to converge in $⟨\mathit{\text{value}}⟩$ iterations.
A solution may not exist, however, try increasing maxit.
${\mathbf{ifail}}=8$
Failure during Anderson acceleration.
Consider setting ${\mathbf{m}}=0$ and recomputing.
${\mathbf{ifail}}=9$
The fixed element ${G}_{ij}$, lies outside the interval $\left[-1,1\right]$, for $i=⟨\mathit{\text{value}}⟩$ and $j=⟨\mathit{\text{value}}⟩$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

Alternating projections is an iterative process where at each iteration the new iterate, ${X}_{k}$, can be improved by using Anderson acceleration to reduce the overall number of iterations. The alternating projections algorithm terminates at the $k$th iteration when
 $‖Xk-Yk‖ F ‖Xk‖ F ≤ errtol$
where ${Y}_{k}$ is the result of the first of two projections computed at each step.
Without Anderson acceleration this algorithm is guaranteed to converge. There is no theoretical guarantee of convergence of Anderson acceleration and, therefore, when it is used, no guarantee of convergence of g02asf. However, in practice it can be seen to significantly reduce the number of alternating projection iterations. Anderson acceleration is not used when m is set to zero. See c05mdf and Higham and Strabić (2016) and Anderson (1965) for further information.

## 8Parallelism and Performance

g02asf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02asf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Arrays are internally allocated by g02asf. The total size of these arrays does not exceed $12×{n}^{2}$ real elements. All allocated memory is freed before return of g02asf.

## 10Example

This example finds the nearest correlation matrix, $X$, to the input, $G$, whilst fixing two diagonal blocks as given by $H$. The minimum eigenvalue of $X$ is stipulated to be $0.04$.
 $G = ( 1.0000 -0.0991 0.5665 -0.5653 -0.3441 -0.0991 1.0000 -0.4273 0.8474 0.4975 0.5665 -0.4273 1.0000 -0.1837 -0.0585 -0.5653 0.8474 -0.1837 1.0000 -0.2713 -0.3441 0.4975 -0.0585 -0.2713 1.0000 )$
and
 $H = ( 1 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 1 1 ) .$
Only the strictly lower half of $H$ is supplied in the example.

### 10.1Program Text

Program Text (g02asfe.f90)

### 10.2Program Data

Program Data (g02asfe.d)

### 10.3Program Results

Program Results (g02asfe.r)