naginterfaces.library.specfun.hyperg_​confl_​real

naginterfaces.library.specfun.hyperg_confl_real(a, b, x)

hyperg_confl_real returns a value for the confluent hypergeometric function ${}_1{F}_1(a;b;x)$ with real parameters $a$ and $b$, and real argument $x$. This function is sometimes also known as Kummer’s function $M(a,b,x)$.

For full information please refer to the NAG Library document for s22ba

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s22baf.html

Parameters

afloat

The parameter $a$ of the function.

bfloat

The parameter $b$ of the function.

xfloat

The argument $x$ of the function.

Returns

mfloat

The solution $M(a,b,x)$.

Note: if overflow occurs upon completion, as indicated by $errno$ = 2, $\left|M(a,b,x)\right|$ may be assumed to be too large to be representable. $m$ will be returned as $\pm R_{max}$, where $R_{max}$ is the largest representable real number (see machine.real_largest). The sign of $m$ should match the sign of $M(a,b,x)$. If overflow occurs during a subcalculation, as indicated by $errno$ = 5, the sign may be incorrect, and the true value of $M(a,b,x)$ may or may not be greater than $R_{max}$. In either case it is advisable to subsequently use hyperg_confl_real_scaled().

Raises

NagValueError

(errno $4$)

All approximations have completed, and the final residual estimate indicates no accuracy can be guaranteed.

Relative residual $=⟨value⟩$.

(errno $5$)

Overflow occurred in a subcalculation of $M(a,b,x)$.

The answer may be completely incorrect.

(errno $11$)

On entry, $a=⟨value⟩$.

Constraint: $\left|a\right|\le \text{arbnd}=⟨value⟩$.

(errno $31$)

On entry, $b=⟨value⟩$.

Constraint: $\left|b\right|\le \text{arbnd}=⟨value⟩$.

(errno $32$)

On entry, $b=⟨value⟩$.

$M(a,b,x)$ is undefined when $b$ is zero or a negative integer.

(errno $51$)

On entry, $x=⟨value⟩$.

Constraint: $\left|x\right|\le \text{arbnd}=⟨value⟩$.

Warns

NagAlgorithmicWarning

(errno $1$)

Underflow occurred during the evaluation of $M(a,b,x)$.

The returned value may be inaccurate.

(errno $2$)

On completion, overflow occurred in the evaluation of $M(a,b,x)$.

(errno $3$)

All approximations have completed, and the final residual estimate indicates some precision may have been lost.

Relative residual $=⟨value⟩$.

Notes

hyperg_confl_real returns a value for the confluent hypergeometric function ${}_1{F}_1(a;b;x)$ with real parameters $a$ and $b$, and real argument $x$. This function is unbounded or not uniquely defined for $b$ equal to zero or a negative integer.

The associated function hyperg_confl_real_scaled() performs the same operations, but returns $M$ in the scaled form $M=m_f\times 2^{m_s}$ to allow calculations to be performed when $M$ is not representable as a single working precision number. It also accepts the parameters $a$ and $b$ as summations of an integer and a decimal fraction, giving higher accuracy when $a$ or $b$ are close to an integer. In such cases, hyperg_confl_real_scaled() should be used when high accuracy is required.

The confluent hypergeometric function is defined by the confluent series

$${}_1{F}_1(a;b;x)=M(a,b,x)=\sum _0^{\infty }\frac{{\left(a\right)}_s{x}^s}{{\left(b\right)}_{s}s!}=1+\frac{a}{b}x+\frac{a(a+1)}{b(b+1)2!}{x}^2+\cdots$$

where ${\left(a\right)}_s=1\left(a\right)(a+1)(a+2)\dots (a+s-1)$ is the rising factorial of $a$. $M(a,b,x)$ is a solution to the second order ODE (Kummer’s Equation):

$$x\frac{d^{2}M}{d{x}^2}+(b-x)\frac{dM}{dx}-aM=0\text{.}$$

Given the parameters and argument $(a,b,x)$, this function determines a set of safe values $\{({\alpha }_i,{\beta }_i,{\zeta }_i)|i\le 2\}$ and selects an appropriate algorithm to accurately evaluate the functions $M_i({\alpha }_i,{\beta }_i,{\zeta }_i)$. The result is then used to construct the solution to the original problem $M(a,b,x)$ using, where necessary, recurrence relations and/or continuation.

Additionally, an artificial bound, $\text{arbnd}$ is placed on the magnitudes of $a$, $b$ and $x$ to minimize the occurrence of overflow in internal calculations. $\text{arbnd}=0.0001\times I_{max}$, where $I_{max}=\text{machine.integer\_max}$. It should, however, not be assumed that this function will produce an accurate result for all values of $a$, $b$ and $x$ satisfying this criterion.

Please consult the NIST Digital Library of Mathematical Functions for a detailed discussion of the confluent hypergeometric function including special cases, transformations, relations and asymptotic approximations.

References

NIST Digital Library of Mathematical Functions

Pearson, J, 2009, Computation of hypergeometric functions, MSc Dissertation, Mathematical Institute, University of Oxford