NAG Library Function Document

nag_real_cholesky_skyline (f01mcc)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:    $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 1$.

2:    $\mathbf{a}\left[\mathbf{lal}\right]$ – const doubleInput

On entry: the elements within the envelope of the lower triangle of the positive definite symmetric matrix $A$, taken in row by row order. The following code assigns the matrix elements within the envelope to the correct elements of the array

 k=0;
 for(i=0; i<n; ++i)
   for(j=i-row[i]+1; j<=i; ++j)
    a[k++]=matrix[i][j];

See also Section 9

3:    $\mathbf{lal}$ – IntegerInput

On entry: the smaller of the dimensions of the arrays a and al as declared in the function from which nag_real_cholesky_skyline (f01mcc) is called.

Constraint: $\mathbf{lal}\ge \mathbf{row}\left[0\right]+\mathbf{row}\left[1\right]+\cdots +\mathbf{row}\left[n-1\right]$.

4:    $\mathbf{row}\left[\mathbf{n}\right]$ – IntegerInput

On entry: $\mathbf{row}\left[i\right]$ must contain the width of row $i$ of the matrix $A$, i.e., the number of elements between the first (left-most) nonzero element and the element on the diagonal, inclusive.

Constraint: $1\le \mathbf{row}\left[\mathit{i}\right]\le \mathit{i}+1$, for $\mathit{i}=0,1,\dots ,n-1$.

5:    $\mathbf{al}\left[\mathbf{lal}\right]$ – doubleOutput

On exit: the elements within the envelope of the lower triangular matrix $L$, taken in row by row order. The envelope of $L$ is identical to that of the lower triangle of $A$. The unit diagonal elements of $L$ are stored explicitly. See also Section 9

6:    $\mathbf{d}\left[\mathbf{n}\right]$ – doubleOutput

On exit: the diagonal elements of the diagonal matrix $D$. Note that the determinant of $A$ is equal to the product of these diagonal elements. If the value of the determinant is required it should not be determined by forming the product explicitly, because of the possibility of overflow or underflow. The logarithm of the determinant may safely be formed from the sum of the logarithms of the diagonal elements.

7:    $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_2_INT_ARG_GT

On entry, $\mathbf{row}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$ while $i=〈\mathit{\text{value}}〉$. These arguments must satisfy $\mathbf{row}\left[i\right]\le i+1$.

NE_2_INT_ARG_LT

On entry, $\mathbf{lal}=〈\mathit{\text{value}}〉$ while $\mathbf{row}\left[0\right]+\cdots +\mathbf{row}\left[n-1\right]=〈\mathit{\text{value}}〉$. These arguments must satisfy $\mathbf{lal}\ge \mathbf{row}\left[0\right]+\cdots +\mathbf{row}\left[n-1\right]$.

NE_INT_ARG_LT

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 1$.

On entry, $\mathbf{row}\left[〈\mathit{\text{value}}〉\right]$ must not be less than 1: $\mathbf{row}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.

NE_NOT_POS_DEF

The matrix is not positive definite, possibly due to rounding errors.

NE_NOT_POS_DEF_FACT

The matrix is not positive definite, possibly due to rounding errors. The factorization has been completed but may be very inaccurate.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f01mcce.c)

10.2

Program Data

Program Data (f01mcce.d)

10.3

Program Results

Program Results (f01mcce.r)