Estimate the derivative of the scalar-to-scalar function f at x using a 2-sided method (central difference).
Estimate the derivative of the scalar-to-scalar function f at x using a 2-sided method (central difference). Approximate the tangent line at (x, f(x)) with the secant line through points (x-h, f(x-h)) and (x+h, f(x+h)). Tends to be MORE ACCURATE than the 1-sided method.
the function whose derivative is sought
the point (scalar) at which to estimate the derivative
http://www.math.montana.edu/frankw/ccp/modeling/continuous/heatflow2/firstder.htm
Estimate the derivative of the scalar-to-scalar function f at x using a 1-sided method (forward difference).
Estimate the derivative of the scalar-to-scalar function f at x using a 1-sided method (forward difference). Approximate the tangent line at (x, f(x)) with the secant line through points (x, f(x)) and (x+h, f(x+h)).
the function whose derivative is sought
the point (scalar) at which to estimate the derivative
Estimate the second derivative of the scalar-to-scalar function f at x using the central difference formula for second derivatives.
Estimate the second derivative of the scalar-to-scalar function f at x using the central difference formula for second derivatives.
the function whose second derivative is sought
the point (scalar) at which to estimate the derivative
Estimate the gradient of the vector-to-scalar function f at point x returning a value for the partial derivative for each dimension of x.
Estimate the gradient of the vector-to-scalar function f at point x returning a value for the partial derivative for each dimension of x.
the function whose gradient is sought
the point (vector) at which to estimate the gradient
Compute the gradient of the vector-to-scalar function f using partial derivative functions evaluated at point x.
Compute the gradient of the vector-to-scalar function f using partial derivative functions evaluated at point x. Return a value for the partial derivative for each dimension of the vector x.
the array of partial derivative functions
the point (vector) at which to compute the gradient
Estimate the Hessian of the vector-to-scalar function f at point x returning a matrix of second partial derivative.
Estimate the Hessian of the vector-to-scalar function f at point x returning a matrix of second partial derivative.
the function whose Hessian is sought
the point (vector) at which to estimate the Hessian
Compute the Jacobian matrix for a vector-valued function represented as an array of scalar-valued functions.
Compute the Jacobian matrix for a vector-valued function represented as an array of scalar-valued functions. The i-th row in the matrix is the gradient of the i-th function.
the array of functions whose Jacobian is sought
the point (vector) at which to estimate the Jacobian
Estimate the Laplacian of the vector-to-scalar function f at point x returning the sum of the pure second partial derivatives.
Estimate the Laplacian of the vector-to-scalar function f at point x returning the sum of the pure second partial derivatives.
the function whose Hessian is sought
the point (vector) at which to estimate the Hessian
Estimate the ith partial derivative of the vector-to-scalar function f at point x returning the value for the partial derivative for dimension i.
Estimate the ith partial derivative of the vector-to-scalar function f at point x returning the value for the partial derivative for dimension i.
the function whose partial derivative is sought
the point (vector) at which to estimate the partial derivative
the dimension to compute the partial derivative on
Estimate the (i,j)th second partial derivative of the vector-to-scalar function f at point x returning the value for the second partial derivative for dimensions (i, j).
Estimate the (i,j)th second partial derivative of the vector-to-scalar function f at point x returning the value for the second partial derivative for dimensions (i, j). If i = j, the second partial derivative is called "pure", otherwise it is a "cross" second partial derivative.
the function whose second partial derivative is sought
the point (vector) at which to estimate the second partial derivative
the first dimension to compute the second partial derivative on
the second dimension to compute the second partial derivative on
Reset the step size from its default step size to one more suitable for your function.
Reset the step size from its default step size to one more suitable for your function. A heuristic for the central difference method is to let h = max (|x|,1) * (machine-epsilon)^(1/3) For double precision, the machine-epsilon is about 1E-16.
the new step size to reset h to
http://www.karenkopecky.net/Teaching/eco613614/Notes_NumericalDifferentiation.pdf
Compute the slope of the vector-to-scalar function f defined on mixed real/integer vectors.
Compute the slope of the vector-to-scalar function f defined on mixed real/integer vectors.
the function whose slope is sought
the point (vector) at which to estimate the slope
the number of dimensions that are real-valued (rest are integers)
The
Calculusobject contains function for computing derivatives, gradients and Jacobians.