Provides utility methods for computing derivatives and integrals of functions.
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| static double | derivative (MathFunction func, double x) |
| | Returns the first derivative of the function func evaluated at x. More...
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| static double | derivative (MathFunction func, double x, int n) |
| | Returns the \(n\)th derivative of function func evaluated at x. More...
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| static double | finiteDifferenceDerivative (MathFunction func, double x, int n, double h) |
| | Computes and returns an estimate of the \(n\)th derivative of the function \(f(x)\). More...
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| static double | finiteCenteredDifferenceDerivative (MathFunction func, double x, double h) |
| | Returns \((f(x + h) - f(x - h))/(2h)\), an estimate of the first derivative of \(f(x)\) using centered differences. More...
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| static double | finiteCenteredDifferenceDerivative (MathFunction func, double x, int n, double h) |
| | Computes and returns an estimate of the \(n\)th derivative of the function \(f(x)\) using finite centered differences. More...
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| static double [][] | removeNaNs (double[] x, double[] y) |
| | Removes any point (NaN, y) or (x, NaN) from x and y, and returns a 2D array containing the filtered points. More...
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| static double | integral (MathFunction func, double a, double b) |
| | Returns the integral of the function func over \([a, b]\). More...
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| static double | simpsonIntegral (MathFunction func, double a, double b, int numIntervals) |
| | Computes and returns an approximation of the integral of func over \([a, b]\), using the Simpson’s \(1/3\) method with numIntervals intervals. More...
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| static double | gaussLobatto (MathFunction func, double a, double b, double tol) |
| | Computes and returns a numerical approximation of the integral of \(f(x)\) over \([a, b]\), using Gauss-Lobatto adaptive quadrature with 5 nodes, with tolerance tol. More...
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| static double | gaussLobatto (MathFunction func, double a, double b, double tol, double[][] T) |
| | Similar to method gaussLobatto(MathFunction, double, double, double), but also returns in T[0] the subintervals of integration, and in T[1], the partial values of the integral over the corresponding subintervals. More...
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Provides utility methods for computing derivatives and integrals of functions.
| static double derivative |
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MathFunction |
func, |
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double |
x, |
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int |
n |
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Returns the \(n\)th derivative of function func evaluated at x.
If \(n=0\), this returns \(f(x)\). If \(n=1\), this calls derivative(MathFunction, double) and returns the resulting first derivative. Otherwise, if the function implements MathFunctionWithDerivative, this method calls derivative(double, int). If the function does not implement this interface, the method uses finiteCenteredDifferenceDerivative(MathFunction, double, int, double) if \(n\) is even, or finiteDifferenceDerivative(MathFunction, double, int, double) if \(n\) is odd, to obtain a numerical approximation of the derivative.
- Parameters
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| func | the function to derivate. |
| x | the evaluation point. |
| n | the order of the derivative. |
- Returns
- the \(n\)th derivative.
| static double finiteDifferenceDerivative |
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MathFunction |
func, |
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double |
x, |
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int |
n, |
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double |
h |
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Computes and returns an estimate of the \(n\)th derivative of the function \(f(x)\).
This method estimates
\[ \frac{d^nf(x)}{dx^n}, \]
the \(n\)th derivative of \(f(x)\) evaluated at \(x\). This method first computes \(f_i=f(x+i\epsilon)\), for \(i=0,…,n\), with \(\epsilon=h^{1/n}\). The estimate is then given by \(\Delta^nf_0/h\), where \(\Delta^nf_i=\Delta^{n-1}f_{i+1} - \Delta^{n-1}f_i\), and \(\Delta f_i = f_{i+1} - f_i\).
- Parameters
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| func | the function to derivate. |
| x | the evaluation point. |
| n | the order of the derivative. |
| h | the error. |
- Returns
- the estimate of the derivative.
| static double gaussLobatto |
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MathFunction |
func, |
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double |
a, |
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double |
b, |
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double |
tol |
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Computes and returns a numerical approximation of the integral of \(f(x)\) over \([a, b]\), using Gauss-Lobatto adaptive quadrature with 5 nodes, with tolerance tol.
This method estimates
\[ \int_a^b f(x)dx, \]
where \(f(x)\) is the function defined by func. Whenever the estimated error is larger than tol, the interval \([a, b]\) will be halved in two smaller intervals, and the method will recursively call itself on the two smaller intervals until the estimated error is smaller than tol.
- Parameters
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| func | the function being integrated. |
| a | the left bound |
| b | the right bound. |
| tol | error. |
- Returns
- the approximate value of the integral.
| static double gaussLobatto |
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MathFunction |
func, |
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double |
a, |
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double |
b, |
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double |
tol, |
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double |
T[][] |
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Similar to method gaussLobatto(MathFunction, double, double, double), but also returns in T[0] the subintervals of integration, and in T[1], the partial values of the integral over the corresponding subintervals.
Thus T[0][0] \(= x_0 = a\) and T[0][n] \(=x_n =b\); T[1][i] contains the value of the integral over the subinterval \([x_{i-1}, x_i]\); we also have T[1][0] \(=0\). The sum over all T[1][i], for \(i=1, …, n\) gives the value of the integral over \([a,b]\), which is the value returned by this method. WARNING: The user must reserve the 2 elements of the first dimension (T[0] and T[1]) before calling this method.
- Parameters
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| func | function being integrated |
| a | left bound of interval |
| b | right bound of interval |
| tol | error |
| T | \((x,y)\) = (values of partial intervals,partial values of integral) |
- Returns
- value of the integral
| static double [][] removeNaNs |
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double [] |
x, |
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double [] |
y |
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Removes any point (NaN, y) or (x, NaN) from x and y, and returns a 2D array containing the filtered points.
This method filters each pair (x[i], y[i]) containing at least one NaN element. It constructs a 2D array containing the two filtered arrays, whose size is smaller than or equal to x.length.
- Parameters
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| x | the \(X\) coordinates. |
| y | the \(Y\) coordinates. |
- Returns
- the filtered \(X\) and \(Y\) arrays.
| static double simpsonIntegral |
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MathFunction |
func, |
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double |
a, |
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double |
b, |
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int |
numIntervals |
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Computes and returns an approximation of the integral of func over \([a, b]\), using the Simpson’s \(1/3\) method with numIntervals intervals.
This method estimates
\[ \int_a^b f(x)dx, \]
where \(f(x)\) is the function defined by func evaluated at \(x\), by dividing \([a, b]\) in \(n=\) numIntervals intervals of length \(h=(b - a)/n\). The integral is estimated by
\[ \frac{h}{3}(f(a)+4f(a+h)+2f(a+2h)+4f(a+3h)+\cdots+f(b)) \]
This method assumes that \(a\le b<\infty\), and \(n\) is even.
- Parameters
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| func | the function being integrated. |
| a | the left bound |
| b | the right bound. |
| numIntervals | the number of intervals. |
- Returns
- the approximate value of the integral.
