nag_quad_1d_gauss_wgen (d01tcc) : NAG Library, Mark 24

3 Description

nag_quad_1d_gauss_wgen (d01tcc) returns the weights

w_{i}

and abscissae

x_{i}

for use in the summation

S = \sum_{i = 1}^{n} w_{i} f (x_{i})

which approximates a definite integral (see Davis and Rabinowitz (1975) or Stroud and Secrest (1966)). The following types are provided:

(a)

Gauss–Legendre

S ≃ \int_{a}^{b} f (x) d x,   exact for ​ f (x) = P_{2 n - 1} (x) .

Constraint:

b > a

(b)

Gauss–Jacobi

normal weights:

S ≃ \int_{a}^{b} {(b - x)}^{c} {(x - a)}^{d} f (x) d x,   exact for ​ f (x) = P_{2 n - 1} (x),

adjusted weights:

S ≃ \int_{a}^{b} f (x) d x,   exact for ​ f (x) = {(b - x)}^{c} {(x - a)}^{d} P_{2 n - 1} (x) .

Constraint:

c > - 1

d > - 1

b > a

(c)

Exponential Gauss

normal weights:

S ≃ \int_{a}^{b} {|x - \frac{a + b}{2}|}^{c} f (x) d x,   exact for ​ f (x) = P_{2 n - 1} (x),

adjusted weights:

S ≃ \int_{a}^{b} f (x) d x,   exact for ​ f (x) = {|x - \frac{a + b}{2}|}^{c} P_{2 n - 1} (x) .

Constraint:

c > - 1

b > a

(d)

Gauss–Laguerre

normal weights:

\begin{array}{l} S & ≃ \int_{a}^{\infty} {|x - a|}^{c} e^{- b x} f (x) d x (b > 0), \\ ≃ \int_{- \infty}^{a} {|x - a|}^{c} e^{- b x} f (x) d x (b < 0),   exact for ​ f (x) = P_{2 n - 1} (x), \end{array}

adjusted weights:

\begin{array}{l} S & ≃ \int_{a}^{\infty} f (x) d x (b > 0), \\ ≃ \int_{- \infty}^{a} f (x) d x (b < 0),   exact for ​ f (x) = {|x - a|}^{c} e^{- b x} P_{2 n - 1} (x) . \end{array}

Constraint:

c > - 1

b \neq 0

(e)

Gauss–Hermite

normal weights:

S ≃ \int_{- \infty}^{+ \infty} {|x - a|}^{c} e^{- b {(x - a)}^{2}} f (x) d x,   exact for ​ f (x) = P_{2 n - 1} (x),

adjusted weights:

S ≃ \int_{- \infty}^{+ \infty} f (x) d x,   exact for ​ f (x) = {|x - a|}^{c} e^{- b {(x - a)}^{2}} P_{2 n - 1} (x) .

Constraint:

c > - 1

b > 0

(f)

Rational Gauss

normal weights:

\begin{array}{l} S & ≃ \int_{a}^{\infty} \frac{{|x - a|}^{c}}{{|x + b|}^{d}} f (x) d x (a + b > 0), \\ ≃ \int_{- \infty}^{a} \frac{{|x - a|}^{c}}{{|x + b|}^{d}} f (x) d x (a + b < 0),   exact for ​ f (x) = P_{2 n - 1} (\frac{1}{x + b}), \end{array}

adjusted weights:

\begin{array}{l} S & ≃ \int_{a}^{\infty} f (x) d x (a + b > 0), \\ ≃ \int_{- \infty}^{a} f (x) d x (a + b < 0),   exact for ​ f (x) = \frac{{|x - a|}^{c}}{{|x + b|}^{d}} P_{2 n - 1} (\frac{1}{x + b}) . \end{array}

Constraint:

c > - 1

d > c + 1

a + b \neq 0

In the above formulae,

P_{2 n - 1} (x)

stands for any polynomial of degree

2 n - 1

or less in

x

The method used to calculate the abscissae involves finding the eigenvalues of the appropriate tridiagonal matrix (see Golub and Welsch (1969)). The weights are then determined by the formula

w_{i} = {\{\sum_{j = 0}^{n - 1} P_{j}^{*} {(x_{i})}^{2}\}}^{- 1}

where

P_{j}^{*} (x)

is the

j

th orthogonal polynomial with respect to the weight function over the appropriate interval.

The weights and abscissae produced by nag_quad_1d_gauss_wgen (d01tcc) may be passed to nag_quad_md_gauss (d01fbc), which will evaluate the summations in one or more dimensions.

5 Arguments

1: quad_type – Nag_QuadTypeInput

On entry: indicates the type of quadrature rule.

$quad_type = Nag_Quad_Gauss_Legendre$: Gauss–Legendre, with normal weights.
$quad_type = Nag_Quad_Gauss_Jacobi$: Gauss–Jacobi, with normal weights.
$quad_type = Nag_Quad_Gauss_Jacobi_Adjusted$: Gauss–Jacobi, with adjusted weights.
$quad_type = Nag_Quad_Gauss_Exponential$: Exponential Gauss, with normal weights.
$quad_type = Nag_Quad_Gauss_Exponential_Adjusted$: Exponential Gauss, with adjusted weights.
$quad_type = Nag_Quad_Gauss_Laguerre$: Gauss–Laguerre, with normal weights.
$quad_type = Nag_Quad_Gauss_Laguerre_Adjusted$: Gauss–Laguerre, with adjusted weights.
$quad_type = Nag_Quad_Gauss_Hermite$: Gauss–Hermite, with normal weights.
$quad_type = Nag_Quad_Gauss_Hermite_Adjusted$: Gauss–Hermite, with adjusted weights.
$quad_type = Nag_Quad_Gauss_Rational$: Rational Gauss, with normal weights.
$quad_type = Nag_Quad_Gauss_Rational_Adjusted$: Rational Gauss, with adjusted weights.

Constraint:

quad_type = Nag_Quad_Gauss_Legendre

Nag_Quad_Gauss_Jacobi

Nag_Quad_Gauss_Jacobi_Adjusted

Nag_Quad_Gauss_Exponential

Nag_Quad_Gauss_Exponential_Adjusted

Nag_Quad_Gauss_Laguerre

Nag_Quad_Gauss_Laguerre_Adjusted

Nag_Quad_Gauss_Hermite

Nag_Quad_Gauss_Hermite_Adjusted

Nag_Quad_Gauss_Rational

Nag_Quad_Gauss_Rational_Adjusted

2: a – doubleInput

3: b – doubleInput

4: c – doubleInput

5: d – doubleInput

On entry: the parameters

a

b

c

and

d

which occur in the quadrature formulae. c is not used if

quad_type = Nag_Quad_Gauss_Legendre

; d is not used unless

quad_type = Nag_Quad_Gauss_Jacobi

Nag_Quad_Gauss_Jacobi_Adjusted

Nag_Quad_Gauss_Rational

Nag_Quad_Gauss_Rational_Adjusted

. For some rules c and d must not be too large (see Section 6).

Constraints:

if $quad_type = Nag_Quad_Gauss_Legendre$ , $a < b$ ;
if $quad_type = Nag_Quad_Gauss_Jacobi$ or $Nag_Quad_Gauss_Jacobi_Adjusted$ , $a < b$ and $c > - 1.0$ and $d > - 1.0$ ;
if $quad_type = Nag_Quad_Gauss_Exponential$ or $Nag_Quad_Gauss_Exponential_Adjusted$ , $a < b$ and $c > - 1.0$ ;
if $quad_type = Nag_Quad_Gauss_Laguerre$ or $Nag_Quad_Gauss_Laguerre_Adjusted$ , $b \neq 0.0$ and $c > - 1.0$ ;
if $quad_type = Nag_Quad_Gauss_Hermite$ or $Nag_Quad_Gauss_Hermite_Adjusted$ , $b > 0.0$ and $c > - 1.0$ ;
if $quad_type = Nag_Quad_Gauss_Rational$ or $Nag_Quad_Gauss_Rational_Adjusted$ , $a + b \neq 0.0$ and $c > - 1.0$ and $d > c + 1.0$ .

6: n – IntegerInput

On entry:

n

, the number of weights and abscissae to be returned. If

quad_type = Nag_Quad_Gauss_Exponential_Adjusted

Nag_Quad_Gauss_Hermite_Adjusted

and

c \neq 0.0

, an odd value of n may raise problems (see

fail . code =

NE_INDETERMINATE).

Constraint:

n > 0

7: weight[n] – doubleOutput

On exit: the n weights.

8: abscis[n] – doubleOutput

On exit: the n abscissae.

9: fail – NagError *Input/Output

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

6 Error Indicators and Warnings

NE_BAD_PARAM

On entry, argument

⟨value⟩

had an illegal value.

NE_CONSTRAINT

On entry, a, b, c, or d is not in the allowed range:

a = ⟨value⟩

b = ⟨value⟩

c = ⟨value⟩

d = ⟨value⟩

and

quad_type = ⟨value⟩

NE_CONVERGENCE

The algorithm for computing eigenvalues of a tridiagonal matrix has failed to converge.

NE_INDETERMINATE

Exponential Gauss or Gauss–Hermite adjusted weights with n odd and

c \neq 0.0

. Theoretically, in these cases:

for $c > 0.0$ , the central adjusted weight is infinite, and the exact function $f (x)$ is zero at the central abscissa;
for $c < 0.0$ , the central adjusted weight is zero, and the exact function $f (x)$ is infinite at the central abscissa.

In either case, the contribution of the central abscissa to the summation is indeterminate.In practice, the central weight may not have overflowed or underflowed, if there is sufficient rounding error in the value of the central abscissa.The weights and abscissa returned may be usable; you must be particularly careful not to ‘round’ the central abscissa to its true value without simultaneously ‘rounding’ the central weight to zero or

\infty

as appropriate, or the summation will suffer. It would be preferable to use normal weights, if possible.Note: remember that, when switching from normal weights to adjusted weights or vice versa, redefinition of

f (x)

is involved.

NE_INT

On entry,

n = ⟨value⟩

.
Constraint:

n > 0

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_TOO_BIG

One or more of the weights are larger than

rmax

, the largest floating point number on this computer (see nag_real_largest_number (X02ALC)):

rmax = ⟨value⟩

.
Possible solutions are to use a smaller value of

n

; or, if using adjusted weights to change to normal weights.

NE_TOO_SMALL

One or more of the weights are too small to be distinguished from zero on this machine.
The underflowing weights are returned as zero, which may be a usable approximation.
Possible solutions are to use a smaller value of

n

; or, if using normal weights, to change to adjusted weights.

8 Parallelism and Performance

nag_quad_1d_gauss_wgen (d01tcc) is not threaded by NAG in any implementation.

nag_quad_1d_gauss_wgen (d01tcc) 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 Users' Note for your implementation for any additional implementation-specific information.

10 Example

This example returns the abscissae and (adjusted) weights for the seven-point Gauss–Laguerre formula.

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_quad_1d_gauss_wgen (d01tcc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Documentnag_quad_1d_gauss_wgen (d01tcc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_quad_1d_gauss_wgen (d01tcc)