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Technical articles with clear explanations and examples
Evaluate a 3D Legendre series at points (x, y, z) in Python
To evaluate a 3D Legendre series at points (x, y, z), use the polynomial.legendre.legval3d() method in Python NumPy. The method returns the values of the multidimensional polynomial on points formed with triples of corresponding values from x, y, and z. If the coefficient array c has fewer than 3 dimensions, ones are implicitly appended to its shape to make it 3-D. The shape of the result will be c.shape[3:] + x.shape. The coordinates x, y, and z must have the same shape, and if any are lists or tuples, they are first converted to ndarrays. Syntax ...
Read MoreEvaluate a Legendre series at array of points x in Python
To evaluate a Legendre series at an array of points x, use the polynomial.legendre.legval() method in Python NumPy. This method takes coefficients of a Legendre polynomial and evaluates it at specified points. 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. The elements must support addition and multiplication operations. c: Array of coefficients ordered so that coefficients for terms of degree n are in c[n]. For multidimensional arrays, remaining indices enumerate multiple polynomials. ...
Read MoreEvaluate a Legendre series at points x broadcast over the columns of the coefficient in Python
To evaluate a Legendre series at points x, use the polynomial.legendre.legval() method in Python NumPy. This function broadcasts x over the columns of the coefficient array, making it useful for evaluating multiple polynomials simultaneously. 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, otherwise treated as a scalar. c: Array of coefficients ordered so that coefficients for terms of degree n are in c[n]. If multidimensional, remaining indices enumerate multiple polynomials stored in columns. ...
Read MoreEvaluate a Legendre series at points x in Python
To evaluate a Legendre series at specific points x in Python, use the polynomial.legendre.legval() method from NumPy. This function computes the value of a Legendre polynomial series at given points using the provided coefficients. Syntax numpy.polynomial.legendre.legval(x, c, tensor=True) Parameters x: Points at which to evaluate the Legendre series. Can be a scalar, list, tuple, or ndarray. c: Array of coefficients ordered so that coefficients for terms of degree n are in c[n]. For multidimensional arrays, remaining indices enumerate multiple polynomials. tensor: If True (default), the coefficient array shape is extended for broadcasting. ...
Read MoreRaise a Legendre series to a power in Python
To raise a Legendre series to a power, use the polynomial.legendre.legpow() method in Python NumPy. The method returns the Legendre series c raised to the power pow. The argument c is a sequence of coefficients ordered from low to high. For example, [1, 2, 3] represents the series P₀ + 2*P₁ + 3*P₂. Syntax numpy.polynomial.legendre.legpow(c, pow, maxpower=16) Parameters The function accepts the following parameters ? c ? 1-D array of Legendre series coefficients ordered from low to high pow ? Power to which the series will be raised maxpower ? Maximum power ...
Read MoreIntegrate a Hermite series in Python
To integrate a Hermite series in Python, use the hermite.hermint() method from NumPy. This function performs polynomial integration on Hermite series coefficients and returns the integrated coefficients. Syntax numpy.polynomial.hermite.hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0) Parameters c − Array of Hermite series coefficients. For multidimensional arrays, different axes correspond to different variables m − Order of integration (must be positive, default: 1) k − Integration constant(s). If empty list (default), all constants are zero lbnd − Lower bound of the integral (default: 0) scl − Scalar multiplier applied after each integration (default: 1) axis ...
Read MoreGenerate a Pseudo Vandermonde matrix of the Laguerre polynomial and x, y, z floating array of points in Python
To generate a pseudo Vandermonde matrix of the Laguerre polynomial with x, y, z sample points, use the laguerre.lagvander3d() in Python NumPy. The parameters x, y, z are arrays of points with the same shape. The parameter deg is a list of maximum degrees of the form [x_deg, y_deg, z_deg]. Syntax numpy.polynomial.laguerre.lagvander3d(x, y, z, deg) Parameters The function accepts the following parameters ? x, y, z − Arrays of point coordinates, all of the same shape deg − List of maximum degrees [x_deg, y_deg, z_deg] Example Let's create arrays ...
Read MoreDivide one Legendre series by another in Python
To divide one Legendre series by another, use the polynomial.legendre.legdiv() method in Python NumPy. The method returns a tuple containing the quotient and remainder of Legendre series coefficients. The function performs polynomial division where the arguments are sequences of coefficients from lowest order "term" to highest. For example, [1, 2, 3] represents the series P_0 + 2*P_1 + 3*P_2. The parameters c1 and c2 are 1-D arrays of Legendre series coefficients ordered from low to high. Syntax numpy.polynomial.legendre.legdiv(c1, c2) Parameters: c1 − 1-D array of Legendre series coefficients (dividend) c2 − 1-D array ...
Read MoreMultiply one Legendre series to another in Python
To multiply one Legendre series with another, use the polynomial.legendre.legmul() method in NumPy. The method returns an array representing the Legendre series of their product. The arguments are sequences of coefficients ordered from lowest order term to highest, i.e., [1, 2, 3] represents the series P_0 + 2*P_1 + 3*P_2. Syntax numpy.polynomial.legendre.legmul(c1, c2) Parameters The parameters c1 and c2 are 1-D arrays of Legendre series coefficients ordered from low to high degree. Example Let's multiply two Legendre series using coefficient arrays ? import numpy as np from numpy.polynomial import legendre ...
Read MoreMultiply a Legendre series by an independent variable in Python
To multiply a Legendre series by an independent variable x, use the polynomial.legendre.legmulx() method in NumPy. This method takes a 1-D array of Legendre series coefficients and returns the result of multiplying the series by x. Syntax numpy.polynomial.legendre.legmulx(c) Parameters The parameter c is a 1-D array of Legendre series coefficients ordered from low to high degree. For example, [1, 2, 3] represents the series P₀ + 2×P₁ + 3×P₂, where Pₙ are Legendre polynomials. Example Let's create a Legendre series and multiply it by x ? import numpy as np ...
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