NAG Library Function Document

nag_nearest_correlation_bounded (g02abc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by $\mathbf{order}=\mathrm{Nag\_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint: $\mathbf{order}=\mathrm{Nag\_RowMajor}$ or Nag_ColMajor.

2:     g[$\mathbf{pdg}\times \mathbf{n}$] – doubleInput/Output

Note: the $\left(i,j\right)$th element of the matrix $G$ is stored in

• $\mathbf{g}\left[\left(j-1\right)\times \mathbf{pdg}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{g}\left[\left(i-1\right)\times \mathbf{pdg}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On entry: $G$, the initial matrix.

On exit: a symmetric matrix $\frac{1}{2}\left(G+G^{\mathrm{T}}\right)$ with the diagonal set to $I$.

3:     pdg – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array g.

Constraint: $\mathbf{pdg}\ge \mathbf{n}$.

4:     n – IntegerInput

On entry: the order of the matrix $G$.

Constraint: $\mathbf{n}>0$.

5:     opt – Nag_NearCorr_ProbTypeInput

On entry: indicates the problem to be solved.

$\mathbf{opt}=\mathrm{Nag\_LowerBound}$

The lower bound problem is solved.

$\mathbf{opt}=\mathrm{Nag\_WeightedNorm}$

The weighted norm problem is solved.

$\mathbf{opt}=\mathrm{Nag\_Both}$

Both problems are solved.

Constraint: $\mathbf{opt}=\mathrm{Nag\_LowerBound}$, $\mathrm{Nag\_WeightedNorm}$ or $\mathrm{Nag\_Both}$.

6:     alpha – doubleInput

On entry: the value of $\alpha$.

If $\mathbf{opt}=\mathrm{Nag\_WeightedNorm}$, alpha need not be set.

Constraint: $0.0<\mathbf{alpha}<1.0$.

7:     w[n] – doubleInput/Output

On entry: the square roots of the diagonal elements of $W$, that is the diagonal of $W^{\frac{1}{2}}$.

If $\mathbf{opt}=\mathrm{Nag\_LowerBound}$, w need not be set.

On exit: if $\mathbf{opt}=\mathrm{Nag\_WeightedNorm}$ or $\mathrm{Nag\_Both}$, the array is scaled so $0<\mathbf{w}\left[\mathit{i}-1\right]\le 1$, for $\mathit{i}=1,2,\dots ,n$.

Constraint: $\mathbf{w}\left[\mathit{i}-1\right]>0.0$, for $\mathit{i}=1,2,\dots ,n$.

8:     errtol – doubleInput

On entry: the termination tolerance for the Newton iteration. If $\mathbf{errtol}\le 0.0$ then $n\times \sqrt{\mathit{machine precision}}$ is used.

9:     maxits – IntegerInput

On entry: specifies the maximum number of iterations to be used by the iterative scheme to solve the linear algebraic equations at each Newton step.

If $\mathbf{maxits}\le 0$, $2\times \mathbf{n}$ is used.

10:   maxit – IntegerInput

On entry: specifies the maximum number of Newton iterations.

If $\mathbf{maxit}\le 0$, $200$ is used.

11:   x[$\mathbf{pdx}\times \mathbf{n}$] – doubleOutput

Note: the $\left(i,j\right)$th element of the matrix $X$ is stored in

• $\mathbf{x}\left[\left(j-1\right)\times \mathbf{pdx}+i-1\right]$ when $\mathbf{order}=\mathrm{Nag\_ColMajor}$;
• $\mathbf{x}\left[\left(i-1\right)\times \mathbf{pdx}+j-1\right]$ when $\mathbf{order}=\mathrm{Nag\_RowMajor}$.

On exit: contains the nearest correlation matrix.

12:   pdx – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array x.

Constraint: $\mathbf{pdx}\ge \mathbf{n}$.

13:   iter – Integer *Output

On exit: the number of Newton steps taken.

14:   feval – Integer *Output

On exit: the number of function evaluations of the dual problem.

15:   nrmgrd – double *Output

On exit: the norm of the gradient of the last Newton step.

16:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_CONVERGENCE

Machine precision is limiting convergence. In this instance the returned matrix $X$ may be useful.

Newton iteration fails to converge in $〈\mathit{\text{value}}〉$ iterations. Increase maxit or check the call to the function.

NE_EIGENPROBLEM

Failure to solve intermediate eigenproblem.

NE_INT

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}>0$.

NE_INT_2

On entry, $\mathbf{pdg}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdg}\ge \mathbf{n}$.

On entry, $\mathbf{pdx}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{pdx}\ge \mathbf{n}$

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

NE_REAL

On entry, $\mathbf{alpha}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<\mathbf{alpha}<1.0$.

NE_WEIGHTS_NOT_POSITIVE

On entry, all elements of w were not positive.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (g02abce.c)

9.2  Program Data

Program Data (g02abce.d)

9.3  Program Results

Program Results (g02abce.r)