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Technical articles with clear explanations and examples
Add one Hermite_e series to another in Python
To add one Hermite_e series to another, use the polynomial.hermite_e.hermeadd() method in NumPy. The method returns an array representing the Hermite_e series of their sum. 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.hermite_e.hermeadd(c1, c2) Parameters: c1, c2 − 1-D arrays of Hermite_e series coefficients ordered from low to high Returns: Array representing the sum of the two Hermite_e series Example Let's create two Hermite_e series and add them together − ...
Read MoreConvert a polynomial to Legendre series in Python
To convert a polynomial to a Legendre series, use the legendre.poly2leg() method in Python NumPy. This function converts an array representing the coefficients of a polynomial (ordered from lowest degree to highest) to an array of the coefficients of the equivalent Legendre series. Syntax numpy.polynomial.legendre.poly2leg(pol) Parameters: pol − 1-D array containing the polynomial coefficients ordered from lowest to highest degree Returns: 1-D array containing the coefficients of the equivalent Legendre series. Example Let's convert a polynomial with coefficients [1, 2, 3, 4, 5] to its Legendre series representation ? ...
Read MoreWhat is the difference between freedom of information and information privacy?
Freedom of Information and Information Privacy are two fundamental concepts that often exist in tension with each other. While Freedom of Information promotes transparency and public access to data, Information Privacy protects individual rights to control personal information. Freedom of Information (FOI) Freedom of Information is a principle that defines the right of individuals and the public to access information relevant to their interests. The United Nations recognizes FOI as a basic human right, arguing that it enables government accountability through institutional transparency and is essential for maintaining the Rule of Law. Key Characteristics of FOI ...
Read MoreGenerate a Pseudo Vandermonde matrix of the Legendre polynomial and x, y complex array of points in Python
To generate a pseudo Vandermonde matrix of the Legendre polynomial, use the legendre.legvander2d() method in Python NumPy. This method returns the pseudo-Vandermonde matrix for two-dimensional Legendre polynomial evaluation. The pseudo Vandermonde matrix is used for polynomial fitting and interpolation. Each row corresponds to a point coordinate, and each column represents a basis function formed by the product of Legendre polynomials of different degrees. Syntax numpy.polynomial.legendre.legvander2d(x, y, deg) Parameters x, y − Arrays of point coordinates with the same shape. Complex numbers are supported. deg − List of maximum degrees in the form [x_deg, ...
Read MoreEvaluate a 2D Legendre series on the Cartesian product of x and y in Python
To evaluate a 2D Legendre series on the Cartesian product of x and y, use the polynomial.legendre.leggrid2d() method in NumPy. This method evaluates a two-dimensional Legendre series at points formed by the Cartesian product of x and y coordinates. The method returns values of the two-dimensional Legendre series at specified grid points. If the coefficient array c has fewer than two dimensions, ones are implicitly appended to make it 2D. The result shape will be c.shape[2:] + x.shape + y.shape. Syntax numpy.polynomial.legendre.leggrid2d(x, y, c) Parameters x, y: The two-dimensional series is evaluated at ...
Read MoreEvaluate a 3D Legendre series at points (x,y,z) with 2D array of coefficient 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 3D. The shape of the result will be c.shape[3:] + x.shape. Syntax numpy.polynomial.legendre.legval3d(x, y, z, c) Parameters x, y, z: The three dimensional series is evaluated at the points (x, y, z), where ...
Read MoreGenerate a Pseudo Vandermonde matrix of the Hermite_e polynomial and x, y, z floating array of points in Python
To generate a pseudo Vandermonde matrix of the Hermite_e polynomial and x, y, z sample points, use the hermite_e.hermevander3d() in Python NumPy. The method returns the pseudo-Vandermonde matrix. The parameter 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 the list of maximum degrees of the form [x_deg, y_deg, z_deg]. Syntax numpy.polynomial.hermite_e.hermevander3d(x, y, z, deg) Parameters x, y, z: Arrays of ...
Read MoreGenerate a Pseudo Vandermonde matrix of the Legendre polynomial and x, y floating array of points in Python
To generate a pseudo Vandermonde matrix of the Legendre polynomial, use the legendre.legvander2d() method in Python NumPy. This method creates a 2D pseudo-Vandermonde matrix where each row corresponds to a point coordinate and columns represent different polynomial degrees. The pseudo-Vandermonde matrix evaluates Legendre polynomials at given points for all degree combinations up to specified maximum degrees. The returned matrix shape is x.shape + (deg[0] + 1) * (deg[1] + 1), where the last dimension contains polynomial evaluations. Syntax numpy.polynomial.legendre.legvander2d(x, y, deg) Parameters The function accepts the following parameters: x, y − ...
Read MoreGenerate a Pseudo Vandermonde matrix of the Legendre polynomial and x, y array of points in Python
To generate a pseudo Vandermonde matrix of the Legendre polynomial, use the legendre.legvander2d() method in NumPy. This method creates a 2D pseudo-Vandermonde matrix where each column represents evaluations of Legendre polynomial basis functions at given coordinate points. The method returns a matrix with shape x.shape + (deg + 1, ), where the last index corresponds to the degree of the Legendre polynomial. The parameters x and y are arrays of point coordinates with the same shape, and deg is a list specifying maximum degrees as [x_deg, y_deg]. Basic Example Let's create coordinate arrays and generate the pseudo ...
Read MoreGenerate a Vandermonde matrix of the Legendre polynomial with complex array of points in Python
To generate a pseudo Vandermonde matrix of the Legendre polynomial with complex points, use the polynomial.legvander() method in NumPy. This method returns a matrix where each row represents the evaluation of Legendre polynomials at a given point, with columns corresponding to different polynomial degrees. The returned matrix has shape x.shape + (deg + 1, ), where the last index represents the degree of the corresponding Legendre polynomial. The data type matches the input array after conversion to float64 or complex128. Syntax numpy.polynomial.legendre.legvander(x, deg) Parameters x − Array of points. Scalar values are ...
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