diff( f, x ) — numerical derivative of a real or complex function at x
diff( f, x, n ) — nth-order numerical derivative of a real or complex function at x
D( f, x ) — numerical derivative of a real or complex function at x
D( f, x, n ) — nth-order numerical derivative of a real or complex function at x
taylorSeries( f, x0 ) — numerical Taylor series of five terms of a real or complex function around x0 returned as a function
taylorSeries( f, x0, terms ) — numerical Taylor series of an arbitrary number of terms of a real or complex function around x0 returned as a function
gradient( f, point ) — numerical gradient of a real or complex function of multiple variables at the correspondingly dimensioned point
findExtremum( f, point ) — numerical minimum of a real function of multiple variables by gradient descent at the correspondingly dimensioned point
findExtremum( f, point, { findMaximum: true } ) — numerical maximum of a real function of multiple variables by gradient ascent at the correspondingly dimensioned point
integrate( f, [a,b] ) — numerical integral of a real or complex function on the interval [a,b] by an adaptive Simpson algorithm
integrate( f, [a,b], options ) — numerical integral of a real or complex function on the interval [a,b]. Options for this dictionary argument include
method | one of 'euler-maclaurin''romberg', 'adaptive-simpson''tanh-sinh''gaussian', default 'adaptive-simpson' |
tolerance | default 10−10 |
avoidEndpoints | set to true to displace endpoints by tolerance
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discreteIntegral( values, step ) — numerical integral over discrete real values separated by step using Euler-Maclaurin summation
summation( f, [a,b] ) — discrete summation of real or complex function values from a to b inclusive by unit step
summation( f, [a,b,step] ) — discrete summation of real or complex function values from a to b inclusive by arbitrary step
polynomial( x, coefficients ) — value of polynomial with real or complex coefficients at x by Horner’s rule with the coefficient of the highest power first
polynomial( x, coefficients, options ) — value of polynomial with real or complex coefficients at x by Horner’s rule. Options for this dictionary argument include
derivative | return both the value of the polynomial and its derivative as { polynomial: value, derivative: value } |
reverse | set to true to reverse the order of coefficients and make the lowest power first
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partialBell( n, k, arguments ) — partial Bell polynomial with integer indices n and k and an array of length n−k+1 of real arguments
findRoot( f, [a,b] ) — numerical root of a real function on the interval [a,b] by bisection
findRoot( f, a ) — numerical root of a real or complex function starting from a by Newton’s method
findRoot( functions, point ) — simultaneous numerical root of an array of real functions starting from the correspondingly dimensioned point by Newton’s method
All three forms of this function can take a third options argument as a dictionary. The main option is tolerance with default 10−10.
spline( points ) — interpolating cubic spline over the array of two-dimensional points returned as a function
spline( points, value ) — interpolating cubic spline over the array of two-dimensional points with a value of 'function', 'derivative' or 'integral' returned as a function
padeApproximant( coefficients, n, d ) — [n/d] Padé approximant for Taylor coefficients centered at the origin returned as a pair of coefficient arrays { N: N, D: D }
padeApproximant( coefficients, n, d, center ) — [n/d] Padé approximant for Taylor coefficients centered at the center returned as a pair of coefficient arrays { N: N, D: D }
padeApproximant( f, n, d ) — [n/d] Padé approximant using numerical derivatives of a real function centered at the origin returned as a pair of coefficient arrays { N: N, D: D }
padeApproximant( f, n, d, center ) — [n/d] Padé approximant using numerical derivatives of a real function centered at the center returned as a pair of coefficient arrays { N: N, D: D }
ode( f, y0, [x0,x1] ) — numerical solution of the real or complex system dy/dx = f(x,y), y(x0) = y0 on the specified interval.
ode( f, y0, [x0,x1], step, method ) — numerical solution of the real or complex system dy/dx = f(x,y), y(x0) = y0 on the specified interval with specified step size and a method of 'euler' or 'runge-kutta'
For higher-order systems, the function and initial conditions should be vectorized, i.e. the function should return an array of values the same length as the intial conditions array. The solution is returned as an array of arrays of data points, with the independent variable as the first item in each data point array.
fourierSinCoefficient( f, n ) — Fourier sine coefficient of index n of a continuous real function on the interval [0,2π]
fourierSinCoefficient( f, n, period ) — Fourier sine coefficient of index n of a continuous real function on the interval [0,period]
fourierSinCoefficient( points, n ) — Fourier sine coefficient of index n of an array of discrete two-dimensional points
fourierCosCoefficient( f, n ) — Fourier cosine coefficient of index n of a continuous real function on the interval [0,2π]
fourierCosCoefficient( f, n, period ) — Fourier cosine coefficient of index n of a continuous real function on the interval [0,period]
fourierCosCoefficient( points, n ) — Fourier cosine coefficient of index n of an array of discrete two-dimensional points