"If the number is identified as composite, the factor() function then proceeds to the Pollard Rho
factorization algorithm. It is arguably the fastest algorithm for when the second largest prime factor has
less than about 10 digits. There are faster algorithms for when this factor has more digits, but all
known algorithms take expected time that grows exponentially with the number of digits. For example,
the Pollard Rho algorithm would probably take centuries on any current computer to factor the product
of two 50-digit primes. In contrast, the isPrime() function would quickly identify the number as
composite.
"Both the compositivity test and the Pollard Rho algorithm need to square numbers as large as their
inputs, so they quit with a "Warning: ... Simplification might be incomplete." if their inputs exceed about
307 digits."
[6.56] Fourth-order splice joins two functions
A fourth-order splice joins two functions smoothly over an interval. The splice function matches the two
functions at the interval boundaries, and the first derivatives of the splice equal those of the two
functions. The fourth-order splice uses a point in the interval interior which controls the splice behavior.
This splice is useful to join two functions, such as regression models, resulting in a model (with three
equations) which is continuous and smooth, in the first derivative sense, over the interval of interest.
This tip is organized in these sections:
Splice function derivation
The function splice4()
Example: approximate the sin() function
Differentiating the splice
Integrating the splice
Solving for the splice inverse
User interface program for splice4()
Scaling the derivatives
Scaling the splice integral
These programs and functions are developed and described:
splice4() Calculate the splice function coefficients
spli4ui() User interface for splice4()
spli4de() Calculate the splice derivative
spli4in() Calculate a numeric derivative for the splice
spli4x() Calculate x, given the splice value of y. Uses nsolve()
spli4inv() Find an approximate polynomial for the inverse of the splice function
Splice function derivation
Define
x
1
= the left interval bound
x
3
= the right interval bound
x
2
= the interval interior control point, where x
1
< x
2
< x
3
h = x
2
- x
1
= x
3
- x
2
(the splice half-width; the control point is midway between the interval bounds)
f
1
(x) is the left-hand function to be spliced
f
2
(x) is the right-hand function to be spliced
6 - 94
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