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Python Articles
Page 217 of 855
Generate a Vandermonde matrix of the Hermite_e polynomial with float array of points in Python
To generate a Vandermonde matrix of the Hermite_e polynomial with a float array of points, use the numpy.polynomial.hermite_e.hermevander() function. This function returns a pseudo-Vandermonde matrix where each column represents a different degree of the Hermite_e polynomial evaluated at the input points. The Hermite_e polynomials (also called "physicists' Hermite polynomials") are orthogonal polynomials commonly used in quantum mechanics and probability theory. The Vandermonde matrix is useful for polynomial fitting and interpolation. Syntax numpy.polynomial.hermite_e.hermevander(x, deg) Parameters x: Array of points. The dtype is converted to float64 or complex128 depending on whether any elements are complex. ...
Read MoreReturn the Nuclear Norm of the matrix in Linear Algebra in Python
The Nuclear Norm (also known as the trace norm or Schatten 1-norm) is the sum of singular values of a matrix. In Python NumPy, use numpy.linalg.norm() with ord='nuc' parameter to calculate the nuclear norm of a matrix. Syntax numpy.linalg.norm(x, ord='nuc') Parameters x − Input array (must be 2-D for nuclear norm) ord − Order of the norm. Use 'nuc' for nuclear norm What is Nuclear Norm? The nuclear norm of a matrix is the sum of its singular values. It's commonly used in matrix completion and low-rank matrix approximation problems. ...
Read MoreReturn the Frobenius Norm of the matrix in Linear Algebra in Python
The Frobenius norm is a matrix norm that calculates the square root of the sum of squares of all elements in a matrix. In Python, use numpy.linalg.norm() with ord='fro' parameter to compute the Frobenius norm. Syntax numpy.linalg.norm(x, ord='fro') Parameters: x − Input matrix (2-D array) ord − Order of the norm. Use 'fro' for Frobenius norm How Frobenius Norm Works The Frobenius norm is calculated as: ||A|| F = √ Σ Σ |a ij ...
Read MoreIntegrate a Legendre series and multiply the result by a scalar before the integration constant is added in Python
To integrate a Legendre series and multiply the result by a scalar before adding the integration constant, use the polynomial.legendre.legint() method in Python. This method integrates Legendre series coefficients and allows scaling at each iteration before adding integration constants. Syntax numpy.polynomial.legendre.legint(c, m=1, k=[], lbnd=0, scl=1, axis=0) Parameters c − Array of Legendre series coefficients m − Order of integration (default: 1) k − Integration constant(s) (default: []) lbnd − Lower bound of integral (default: 0) scl − Scalar multiplier applied before adding integration constant (default: 1) axis − Axis over which integration is ...
Read MoreIntegrate a Legendre series and set the lower bound of the integral in Python
To integrate a Legendre series, use the polynomial.legendre.legint() method in Python. This method returns the Legendre series coefficients integrated m times from the lower bound (lbnd) along the specified axis. At each iteration, the resulting series is multiplied by a scaling factor and an integration constant is added. Syntax numpy.polynomial.legendre.legint(c, m=1, k=[], lbnd=0, scl=1, axis=0) Parameters Parameter Description Default c Array of Legendre series coefficients Required m Order of integration (must be positive) 1 k Integration constant(s) [] lbnd Lower bound of the integral ...
Read MoreIntegrate a Legendre series and set the integration constant in Python
To integrate a Legendre series in Python, use the polynomial.legendre.legint() method from NumPy. This method integrates Legendre series coefficients and allows you to set integration constants, making it useful for solving differential equations and polynomial manipulations. Syntax numpy.polynomial.legendre.legint(c, m=1, k=[], lbnd=0, scl=1, axis=0) Parameters The legint() method accepts the following parameters: c: Array of Legendre series coefficients m: Order of integration (default: 1) k: Integration constant(s). Use a scalar for single integration or a list for multiple integrations (default: []) lbnd: Lower bound of the integral (default: 0) scl: Scaling factor applied ...
Read MoreDifferentiate a Legendre series and multiply each differentiation by a scalar in Python
To differentiate a Legendre series, use the polynomial.legendre.legder() method in Python. This function returns the Legendre series coefficients differentiated m times along axis, with each differentiation multiplied by a scalar. 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. If multidimensional, 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 scl, resulting in multiplication by scl**m (Default: 1) axis ? Axis over which the ...
Read MoreEvaluate a 3-D Hermite_e series at points (x,y,z) in Python
To evaluate a 3-D Hermite_e series at points (x, y, z), use the numpy.polynomial.hermite_e.hermeval3d() method. This method evaluates a three-dimensional Hermite_e polynomial series and returns values at the specified coordinate points. Syntax numpy.polynomial.hermite_e.hermeval3d(x, y, z, c) Parameters x, y, z − The three-dimensional series is evaluated at points (x, y, z). These arrays must have the same shape. If any parameter is a list or tuple, it is converted to an ndarray. c − Array of coefficients ordered so that the coefficient of the term of multi-degree i, j, k is contained in ...
Read MoreEvaluate a Hermite_e series at multi-dimensional array of points x in Python
To evaluate a Hermite_e series at points x, use the hermite_e.hermeval() method in Python NumPy. This function evaluates Hermite_e polynomials at specified points using coefficient arrays. Syntax numpy.polynomial.hermite_e.hermeval(x, c, tensor=True) Parameters The function accepts three parameters ? x − Points where the series is evaluated. Can be a scalar, list, tuple, or ndarray c − Array of coefficients where c[n] contains coefficients for degree n terms tensor − If True (default), evaluates every column of coefficients for every element of x Example Let's evaluate a Hermite_e series with coefficients ...
Read MoreDifferentiate a Chebyshev series with multidimensional coefficients in Python
To differentiate a Chebyshev series with multidimensional coefficients, use the polynomial.chebder() method in NumPy. The method returns the Chebyshev series coefficients of the derivative, differentiated m times along a specified axis. The argument c is an array of coefficients from low to high degree along each axis. For example, [1, 2, 3] represents the series 1*T_0 + 2*T_1 + 3*T_2 while [[1, 2], [1, 2]] represents 1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y) if axis=0 is x and axis=1 is y. Syntax numpy.polynomial.chebyshev.chebder(c, m=1, scl=1, axis=0) Parameters c − Array of Chebyshev ...
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