Simpson's method of approximating the integral of a function, f, on the interval [a,b].
| Parameters |
f : Function The function to be evaluated.
a : Number The left endpoint of the interval.
b : Number The right endpoint of the interval.
eps : Number An error bound.
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|---|---|
| Returns |
num : Number The approximation of the integral of f on [a,b].
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| Errors |
This function does not raise any errors. |
Lanczos' approximation to the gamma function of a number, n.
| Parameters |
n : Number A number.
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|---|---|
| Returns |
num : Number Gamma of n.
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| Errors |
This function does not raise any errors. |
Calculates the limit of a function, f, at a point x. The point can be approached from the left, right, or middle (a combination of the left and right).
| Parameters |
f : Function The function to be evaluated.
x : Number The point for which the limit will be calculated.
approach : String A desired approach. left, right and middle are the possible approaches.
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|---|---|
| Returns |
num : Number The limit of f at x.
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| Errors |
An error is thrown if:
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The Monte-Carlo method for approximating the integral of a singlevariate or multivariate function, f, over a given interval(s). The number of intervals must match the number of variables of the function. The nth element of I is the interval for the nth variable of the function.
| Parameters |
f : Function The function to be evaluated.
N : Number The number of function evaluations.
I : Array An array of arrays, where each inner array is an interval containing
the endpoints.
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|---|---|
| Returns |
num : Number An approximation to the integral of f with N function evaluations.
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| Errors |
An error is thrown if:
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Calculates the point differential of a function f at a point x. Currently only supports one-dimensional functions.
| Parameters |
f : Function The function to be evaluated.
x : Number The point for which the point differential will be calculated.
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|---|---|
| Returns |
num : Number The point differential of f at x.
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| Errors |
This function does not raise any errors. |
Calculates the Riemann sum for a one-variable function f on the interval [a,b] with n equally-spaced divisons. If sampler is given, that function will be used to calculate which value to sample on each subinterval; otherwise, the left endpoint will be used.
| Parameters |
f : Function The function to be evaluated.
a : Number The left endpoint of the interval.
b : Number The right endpoint of the interval.
n : Number The number of subintervals.
sampler : Function, optional A function that determines what value is to be used for sampling
on a subinterval.
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|---|---|
| Returns |
num : Number The Riemann sum of f on [a,b] with n divisions.
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| Errors |
This function does not raise any errors. |
Simpson's method of approximating the integral of a function, f, on the interval [a,b].
| Parameters |
f : Function The function to be evaluated.
a : Number The left endpoint of the interval.
b : Number The right endpoint of the interval.
|
|---|---|
| Returns |
num : Number The approximation of the integral of f on [a,b].
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| Errors |
This function does not raise any errors. |
The helper function used for adaptive Simpson's method of approximating the integral of a function, f, on the interval [a,b].
| Parameters |
f : Function The function to be evaluated.
a : Number The left endpoint of the interval.
b : Number The right endpoint of the interval.
whole : Number The value of the integral of f on [a,b]
(Simpson's approximation, in the caes of adaptiveSimpson).
eps : Number An error bound.
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|---|---|
| Returns |
num : Number A recursive evaluation of the left and right sides.
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| Errors |
This function does not raise any errors. |
Striling's approximation to the gamma function of a number, n.
| Parameters |
n : Number A number.
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|---|---|
| Returns |
num : Number Gamma of n.
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| Errors |
This function does not raise any errors. |