, the approximation is based on expansions proposed by Cody and Thatcher Jr. (1969). Precautions are taken to maintain good relative accuracy in the vicinity of

x_{0} \approx - 0.372507 \dots

, which corresponds to a simple zero of Ei(

- x

s13aaf guards against producing underflows and overflows by using the argument

x_{hi}

; see the Users' Note for your implementation for the value of

x_{hi}

. To guard against overflow, if

x < - x_{hi}

the routine terminates and returns the negative of the largest representable machine number. To guard against underflow, if

x > x_{hi}

the result is set directly to zero.

4 References

NIST Digital Library of Mathematical Functions

Cody W J and Thatcher Jr. H C (1969) Rational Chebyshev approximations for the exponential integral Ei

(x)

Math. Comp. 23 289–303

5 Arguments

1: $x$ – Real (Kind=nag_wp) Input: On entry: the argument $x$ of the function.

Constraint: $- x_{hi} \leq x < 0.0$ or $x > 0.0$ .
2: $ifail$ – 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 $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $x = 0.0$ and the function is infinite.

$ifail = 2$: The evaluation has been abandoned due to the likelihood of overflow. The argument $x < - x_{hi}$ .

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Unless stated otherwise, it is assumed that

x > 0

δ

and

ε

are the relative errors in argument and result respectively, then in principle,

|ε| ≃ |\frac{e^{- x}}{E_{1} (x)} \times δ|

so the relative error in the argument is amplified in the result by at least a factor

e^{- x} / E_{1} (x)

. The equality should hold if

δ

is greater than the machine precision (

δ

due to data errors etc.) but if

δ

is simply a result of round-off in the machine representation, it is possible that an extra figure may be lost in internal calculation and round-off.

The behaviour of this amplification factor is shown in the following graph:

Figure 1

It should be noted that, for absolutely small

x

, the amplification factor tends to zero and eventually the error in the result will be limited by machine precision.

For absolutely large

x

ε \sim x δ = Δ,

the absolute error in the argument.

For

x < 0

, empirical tests have shown that the maximum relative error is a loss of approximately

1

decimal place.

8 Parallelism and Performance

s13aaf is not threaded in any implementation.

9 Further Comments

None.

10 Example

The following program reads values of the argument

x

from a file, evaluates the function at each value of

x

and prints the results.

s13aa: FL CL CPP AD

NAG FL Interfaces13aaf (integral_​exp)

▸▿ 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 FL Interface
s13aaf (integral_exp)