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To integrate a Hermite_e series and set a custom lower bound, use the hermite_e.hermeint() method in NumPy. This function performs polynomial integration on Hermite_e series coefficients with flexible integration parameters. Syntax numpy.polynomial.hermite_e.hermeint(c, m=1, k=[], lbnd=0, scl=1, axis=0) Parameters The function accepts the following parameters ? c ? Array of Hermite_e series coefficients m ? Order of integration (must be positive, default: 1) k ? Integration constants (default: []) lbnd ? Lower bound of the integral (default: 0) scl ? Scalar multiplier applied after each integration (default: 1) axis ? Axis over which ... Read More
To evaluate a Legendre series at tuple of points x, use the polynomial.legendre.legval() method in Python NumPy. This function evaluates a Legendre polynomial series at given points using coefficients. Syntax numpy.polynomial.legendre.legval(x, c, tensor=True) Parameters x: Array of points at which to evaluate the series. If x is a list or tuple, it is converted to an ndarray. c: Array of coefficients ordered so that coefficients for terms of degree n are contained in c[n]. For multidimensional arrays, remaining indices enumerate multiple polynomials. tensor: If True (default), the coefficient array shape is extended ... Read More
To differentiate a Hermite_e series and multiply each differentiation by a scalar, use the hermite_e.hermeder() method in Python. This function computes derivatives of Hermite_e series with optional scalar multiplication. Syntax numpy.polynomial.hermite_e.hermeder(c, m=1, scl=1, axis=-1) Parameters The function takes the following parameters: c: Array of Hermite_e series coefficients. For multidimensional arrays, different axes correspond to different variables m: Number of derivatives taken, must be non-negative (Default: 1) scl: Scalar multiplier. Each differentiation is multiplied by this value (Default: 1) axis: Axis over which the derivative is taken (Default: -1) Example ... Read More
To differentiate a Legendre series with multidimensional coefficients, use the polynomial.legendre.legder() method in Python. This method returns the Legendre series coefficients differentiated m times along the specified axis. Syntax numpy.polynomial.legendre.legder(c, m=1, scl=1, axis=0) Parameters The function accepts the following parameters: c: Array of Legendre series coefficients. For multidimensional arrays, different axes correspond to different variables m: Number of derivatives taken, must be non-negative (Default: 1) scl: Scalar multiplier for each differentiation. Final result is multiplied by scl**m (Default: 1) axis: Axis over which the derivative is taken (Default: 0) Example: ... Read More
To differentiate a Legendre series with multidimensional coefficients, use the numpy.polynomial.legendre.legder() method. This function returns the Legendre series coefficients differentiated m times along a specified axis. Syntax numpy.polynomial.legendre.legder(c, m=1, scl=1, axis=0) Parameters The function accepts the following parameters − c − Array of Legendre series coefficients. For multidimensional arrays, different axes correspond to different variables m − Number of derivatives taken (default: 1). Must be non-negative scl − Scalar multiplier for each differentiation (default: 1). Final result is multiplied by scl**m axis − Axis over which the derivative is taken (default: 0) ... Read More
To evaluate a 2-D Hermite_e series on the Cartesian product of x and y, use the hermite_e.hermegrid2d() method in Python. This method returns the values of the two-dimensional polynomial at points in the Cartesian product of x and y coordinates. Syntax numpy.polynomial.hermite_e.hermegrid2d(x, y, c) Parameters The parameters are: x, y: The two dimensional series is evaluated at points in the Cartesian product of x and y. If x or y is a list or tuple, it is first converted to an ndarray c: Array of coefficients ordered so that coefficients for terms ... Read More
To evaluate a 2-D Hermite_e series on the Cartesian product of x and y, use the hermite_e.hermegrid2d() method in Python. This method returns the values of the two-dimensional polynomial at points in the Cartesian product of x and y. Syntax numpy.polynomial.hermite_e.hermegrid2d(x, y, c) Parameters The method accepts the following parameters ? x, y ? The two-dimensional series is evaluated at the points in the Cartesian product of x and y. If x or y is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and, ... Read More
To generate a pseudo Vandermonde matrix of the Legendre polynomial with x, y, z sample points, use the legendre.legvander3d() method in NumPy. This function returns a 3D pseudo-Vandermonde matrix of degrees deg and sample points (x, y, z). The parameters x, y, z are arrays of point coordinates, all of the same shape. The dtypes will be converted to either float64 or complex128 depending on whether any of the elements are complex. Scalars are converted to 1-D arrays. The parameter deg is a list of maximum degrees of the form [x_deg, y_deg, z_deg]. Syntax legendre.legvander3d(x, y, ... Read More
The pseudo-Vandermonde matrix of Legendre polynomials is a mathematical construct used in polynomial interpolation and approximation. In NumPy, the legendre.legvander3d() function generates this matrix for three-dimensional sample points (x, y, z). Syntax numpy.polynomial.legendre.legvander3d(x, y, z, deg) Parameters The function accepts the following parameters ? x, y, z ? Arrays of point coordinates with the same shape deg ? List of maximum degrees [x_deg, y_deg, z_deg] Example Here's how to generate a pseudo-Vandermonde matrix using three-dimensional sample points ? import numpy as np from numpy.polynomial import legendre as ... Read More
To evaluate a 3D Hermite_e series at points (x, y, z), use the hermite_e.hermeval3d() method in NumPy. The method returns the values of the multidimensional polynomial on points formed with triples of corresponding values from x, y, and z. The first parameter consists of x, y, z coordinates. The three dimensional series is evaluated at the points (x, y, z), where x, y, and z must have the same shape. If any of x, y, or z is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn't an ... Read More