bessel

Synopsis

B = bessel(alpha,x)

Description

The differential equation

is called Bessel's equation, and its solutions are known as Bessel functions.

B = bessel(alpha x) computes Bessel functions of the first kind, or modified Bessel functions of the first kind, for real, nonnegative alpha and nonnegative x:

For real elements of x (no imaginary component), bessel uses the besselj function to compute the Bessel function of the first kind.

For imaginary elements of x (no real component), bessel uses the besseli function to compute the modified Bessel function of the first kind.

For complex elements of x (both real and imaginary component), bessel uses the bessela function to compute the Bessel function of the first kind.
If alpha is a scalar and x is a vector, B is a vector the same length as x. If x is a vector of length m and alpha is a vector of length n, then B is an m-by-n matrix and B(i,k) is bessel(alpha(k), x(i)). The elements of x can be any nonnegative values, in any order. For alpha, the increment between elements must be 1, and all elements must be between 0 and 1000, inclusive.

Examples

Plot Bessel functions of order 0 and 1:

x = 0:.25:10;
plot(x,[bessel(0,x);bessel(1,x)])

Algorithm

For real x, bessel calls the function besselj. besselj evaluates the Bessel function using three-term recurrence for small-valued x, or an asymptotic series for large x. For purely complex x, bessel calls besseli, which uses three-term backward recurrence for most x, and an asymptotic expansion for large x. For complex x, bessel calls bessela, which uses either a power series or an asymptotic expansion depending on roundoff error considerations. See reference [1].

For some values of arguments, bessela's results may be severely contaminated by roundoff error. [J,digits] = bessela(alpha,x) returns an estimate of the number of correct significant digits in the computed result. digits is the log10 of the estimated relative error, so a value of 14 or 15 corresponds to nearly full accuracy in IEEE or VAX arithmetic, while 1 or 2 indicates nearly useless results. Any negative value of digits is replaced by zero, the corresponding J set to NaN and a division by zero warning message is generated. If either alpha or x is less than 50, digits are at least 8. In the (alpha,x) plane, the region of least accuracy is near the line alpha = x, so small values of alpha and large values of x, or vice versa, give the most accurate results. For example:

----------------
x   alpha  digits  
----------------
25  25     11.8    
25  75     14.4    
75  25     14.5    
75  75     1.9     
----------------

References

[1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series #55, Dover Publications, 1965, sections 9.1.1, 9.1.89 and 9.12, formulas 9.1.10 and 9.2.5.