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Numerical Methods Formulas
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Numerical Methods
Central Difference Derivative
f\prime(x)\approx\frac{f(x+h)-f(x-h)}{2h}Approximates a derivative with a symmetric second-order difference.
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Central Difference First Derivative
f'(x)\approx\frac{f(x+h)-f(x-h)}{2h}Approximates a first derivative using symmetric samples around x.
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Central Difference Second Derivative
f''(x)\approx\frac{f(x+h)-2f(x)+f(x-h)}{h^2}Approximates a second derivative using three equally spaced samples.
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Classical Runge–Kutta Method
y_{n+1}=y_n+\frac{h}{6}(k_1+2k_2+2k_3+k_4)Combines four slope estimates to produce a fourth-order ODE step.
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Euler Method
y_{n+1}=y_n+h f(t_n,y_n)Advances an ordinary differential equation solution using the current slope.
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Forward Difference Derivative
f\prime(x)\approx\frac{f(x+h)-f(x)}{h}Approximates a derivative with a first-order forward difference.
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Newton Method Iteration
x_{n+1}=x_n-\frac{f(x_n)}{f\prime(x_n)}Iteratively approximates a root of a differentiable function.
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Newton–Raphson Iteration
x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}Iteratively approximates a root using the tangent line at the current estimate.
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Simpson’s Rule
\int_a^b f(x)dx\approx\frac{h}{3}\left[f(x_0)+4\sum_{i\ odd}f(x_i)+2\sum_{i\ even}f(x_i)+f(x_n)\right]Approximates a definite integral with piecewise quadratic interpolation.
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Trapezoidal Rule
\int_a^b f(x)dx\approx\frac{h}{2}\left[f(x_0)+2\sum_{i=1}^{n-1}f(x_i)+f(x_n)\right]Approximates a definite integral with piecewise linear segments.
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