bessI
bessI (n,x [, algorithm [, accuracy ]])
Obsolete -- use Besseli.
The bessI function returns the modified Bessel function of the first kind, In (x ), of ordern and argument x.
For real x, the optional parameter algorithm selects between a faster, less accurate calculation method and slower, more accurate methods. In addition, when algorithm is zero or absent, the order n is truncated to an integer.
When algorithm is included and is 1, accuracy can be used to specify the desired fractional accuracy. See below for details about algorithms.
if x is complex, a complex result is returned. In this case, algorithm and accuracy are ignored. The order n can be fractional, and must be real.
Details
The algorithm parameter has three options, each selecting a different calculation method:
| Algorithm | What You Get |
|---|---|
| 0 (default) | Uses a calculation method that has fractional accuracy better than 10-6 everywhere and is generally better than 10-8. This method does not handle fractional order n; the order is truncated to an integer before the calculation is performed. |
| Algorithm 0 is fastest by a large margin. | |
| 1 | Fractional order is allowed. The calculation is performed using methods described in Numerical Recipes in C, edition 2, section 6.7 "Bessel Functions of Fractional Order", pp. 240-245. |
| Using algorithm 1, accuracy specifies the fractional accuracy that you desire. That is, if you set accuracy to 1e-7 (that is, 10-7), that means that you wish that the absolute value of (factual-freturned)/factual be better than 10-7. Asking for less accuracy gives some increase in speed. | |
| You pay a heavy price for higher accuracy or fractional order. When algorithm is nonzero, calculation time is increased by an order of magnitude for small x ; at larger x the penalty is even greater. | |
| If accuracy is greater than 10-8 and n is an integer, algorithm 0 is used. | |
| The algorithm used calculates bessI and bessK simultaneously. Igor stores both values, and if a call to bessI is followed by a call to bessK (or bessK is followed by bessI) with the same n, x, and accuracy the previously-stored value is returned, making the second call very fast. | |
| 2 | Fractional order is allowed. The calculation is performed using code from the SLATEC library. The accuracy achievable is often better than algorithm 1, but not always. Algorithm 2 is 1.5 to 3 times faster than algorithm 1, but still slower than algorithm 0. The accuracy parameter is ignored. |
The achievable accuracy of algorithms 1 and 2 is a complicated function n and x. To see a summary of accuracy achievable, pull down the File menu and select Example Experiments→Testing and Misc→Bessel Accuracy.