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1.16.1
Provides utility methods for computing derivatives and integrals of functions. More...
Static Public Member Functions | |
| static double | derivative (MathFunction func, double x) |
| Returns the first derivative of the function func evaluated at x. | |
| static double | derivative (MathFunction func, double x, int n) |
| Returns the \(n\)th derivative of function func evaluated at x. | |
| static double | finiteDifferenceDerivative (MathFunction func, double x, int n, double h) |
| Computes and returns an estimate of the \(n\)th derivative of the function \(f(x)\). | |
| 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. | |
| 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. | |
| 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. | |
| static double | integral (MathFunction func, double a, double b) |
| Returns the integral of the function func over \([a, b]\). | |
| 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. | |
| static double | gaussLobatto (MathFunction func, double a, double b, double tol) |
| Computes and returns a numerical approximation of the integral of. | |
| 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. | |
Static Public Attributes | |
| static double | H = 1e-6 |
| Step length in \(x\) to compute derivatives. | |
| static int | NUMINTERVALS = 1024 |
| Default number of intervals for Simpson’s integral. | |
Provides utility methods for computing derivatives and integrals of functions.
Definition at line 35 of file MathFunctionUtil.java.
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Returns the first derivative of the function func evaluated at x.
If the given function implements
MathFunctionWithFirstDerivative, this method calls derivative(double). Otherwise, if the function implements MathFunctionWithDerivative, this method calls derivative(double, int). If the function does not implement any of these two interfaces, the method uses finiteCenteredDifferenceDerivative(MathFunction, double, double) to obtain an estimate of the derivative.
| func | the function to derivate. |
| x | the evaluation point. |
Definition at line 92 of file MathFunctionUtil.java.
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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.
| func | the function to derivate. |
| x | the evaluation point. |
| n | the order of the derivative. |
Definition at line 120 of file MathFunctionUtil.java.
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Returns \((f(x + h) - f(x - h))/(2h)\), an estimate of the first derivative of \(f(x)\) using centered differences.
| func | the function to derivate. |
| x | the evaluation point. |
| h | the error. |
Definition at line 174 of file MathFunctionUtil.java.
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Computes and returns an estimate of the \(n\)th derivative of the function \(f(x)\) using finite centered differences.
If \(n\) is even, this method returns finiteDifferenceDerivative(func, x - *n/2, n, h), with
\(h=\epsilon^n\).
| func | the function to derivate. |
| x | the evaluation point. |
| n | the order of the derivative. |
| h | the error. |
Definition at line 193 of file MathFunctionUtil.java.
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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 \(\Delta f_i = f_{i+1} - f_i\).
| func | the function to derivate. |
| x | the evaluation point. |
| n | the order of the derivative. |
| h | the error. |
Definition at line 149 of file MathFunctionUtil.java.
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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.
| func | the function being integrated. |
| a | the left bound |
| b | the right bound. |
| tol | error. |
Definition at line 311 of file MathFunctionUtil.java.
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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.
| 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) |
Definition at line 359 of file MathFunctionUtil.java.
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Returns the integral of the function func over \([a, b]\).
If the given function implements MathFunctionWithIntegral, this returns integral(double, double). Otherwise, this calls simpsonIntegral(MathFunction, double, double, int) with NUMINTERVALS intervals.
| func | the function to integrate. |
| a | the lower bound. |
| b | the upper bound. |
Definition at line 248 of file MathFunctionUtil.java.
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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.
| x | the \(X\) coordinates. |
| y | the \(Y\) coordinates. |
Definition at line 215 of file MathFunctionUtil.java.
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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.
| func | the function being integrated. |
| a | the left bound |
| b | the right bound. |
| numIntervals | the number of intervals. |
Definition at line 271 of file MathFunctionUtil.java.
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Step length in \(x\) to compute derivatives.
Default: \(10^{-6}\).
Definition at line 40 of file MathFunctionUtil.java.
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Default number of intervals for Simpson’s integral.
Definition at line 73 of file MathFunctionUtil.java.
1.16.1