NAG Library Function Document

1Purpose

nag_quad_md_numth_coeff_2prime (d01gzc) calculates the optimal coefficients for use by nag_quad_md_numth_vec (d01gdc), when the number of points is the product of two primes.

2Specification

 #include #include
 void nag_quad_md_numth_coeff_2prime (Integer ndim, Integer np1, Integer np2, double vk[], NagError *fail)

3Description

Korobov (1963) gives a procedure for calculating optimal coefficients for $p$-point integration over the $n$-cube ${\left[0,1\right]}^{n}$, when the number of points is
 $p=p1p2$ (1)
where ${p}_{1}$ and ${p}_{2}$ are distinct prime numbers.
The advantage of this procedure is that if ${p}_{1}$ is chosen to be the nearest prime integer to ${p}_{2}^{2}$, then the number of elementary operations required to compute the rule is of the order of ${p}^{4/3}$ which grows less rapidly than the number of operations required by nag_quad_md_numth_coeff_prime (d01gyc). The associated error is likely to be larger although it may be the only practical alternative for high values of $p$.

4References

Korobov N M (1963) Number Theoretic Methods in Approximate Analysis Fizmatgiz, Moscow

5Arguments

1:    $\mathbf{ndim}$IntegerInput
On entry: $n$, the number of dimensions of the integral.
Constraint: ${\mathbf{ndim}}\ge 1$.
2:    $\mathbf{np1}$IntegerInput
On entry: the larger prime factor ${p}_{1}$ of the number of points in the integration rule.
Constraint: ${\mathbf{np1}}$ must be a prime number $\text{}\ge 5$.
3:    $\mathbf{np2}$IntegerInput
On entry: the smaller prime factor ${p}_{2}$ of the number of points in the integration rule. For maximum efficiency, ${p}_{2}^{2}$ should be close to ${p}_{1}$.
Constraint: ${\mathbf{np2}}$ must be a prime number such that ${\mathbf{np1}}>{\mathbf{np2}}\ge 2$.
4:    $\mathbf{vk}\left[{\mathbf{ndim}}\right]$doubleOutput
On exit: the $n$ optimal coefficients.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ACCURACY
The machine precision is insufficient to perform the computation exactly. Try reducing np1 or np2: ${\mathbf{np1}}=〈\mathit{\text{value}}〉$ and ${\mathbf{np2}}=〈\mathit{\text{value}}〉$.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{ndim}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ndim}}\ge 1$.
On entry, ${\mathbf{np1}}=〈\mathit{\text{value}}〉$.
Constraint: np1 must be a prime number.
On entry, ${\mathbf{np1}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{np1}}\ge 5$.
On entry, ${\mathbf{np2}}=〈\mathit{\text{value}}〉$.
Constraint: np2 must be a prime number.
On entry, ${\mathbf{np2}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{np2}}\ge 2$.
NE_INT_2
On entry, ${\mathbf{np1}}×{\mathbf{np2}}$ exceeds largest machine integer. ${\mathbf{np1}}=〈\mathit{\text{value}}〉$ and ${\mathbf{np2}}=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{np1}}=〈\mathit{\text{value}}〉$ and ${\mathbf{np2}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{np1}}>{\mathbf{np2}}$.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7Accuracy

The optimal coefficients are returned as exact integers (though stored in a double array).

8Parallelism and Performance

The time taken by nag_quad_md_numth_coeff_2prime (d01gzc) grows at least as fast as ${\left({p}_{1}{p}_{2}\right)}^{4/3}$. (See Section 3.)

10Example

This example calculates the Korobov optimal coefficients where the number of dimensons is $4$ and the number of points is the product of the two prime numbers, $89$ and $11$.

10.1Program Text

Program Text (d01gzce.c)

None.

10.3Program Results

Program Results (d01gzce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017