Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y). The pseudo-Vandermonde matrix is defined by
where 0 <= i <= deg[0] and 0 <= j <= deg[1]. The leading indices of V index the points (x, y) and the last index encodes the degrees of the Hermite polynomials.
If V = hermvander2d(x, y, [xdeg, ydeg]), then the columns of V correspond to the elements of a 2-D coefficient array c of shape (xdeg + 1, ydeg + 1) in the order
and np.dot(V, c.flat) and hermval2d(x, y, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Hermite series of the same degrees and sample points.
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.
List of maximum degrees of the form [x_deg, y_deg].
The shape of the returned matrix is x.shape + (order,), where \(order = (deg[0]+1)*(deg[1]+1)\). The dtype will be the same as the converted x and y.
See also
New in version 1.7.0.


Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y, z). If l, m, n are the given degrees in x, y, z, then The pseudo-Vandermonde matrix is defined by
where 0 <= i <= l, 0 <= j <= m, and 0 <= j <= n. The leading indices of V index the points (x, y, z) and the last index encodes the degrees of the Hermite polynomials.
If V = hermvander3d(x, y, z, [xdeg, ydeg, zdeg]), then the columns of V correspond to the elements of a 3-D coefficient array c of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
and np.dot(V, c.flat) and hermval3d(x, y, z, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Hermite series of the same degrees and sample points.
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.
List of maximum degrees of the form [x_deg, y_deg, z_deg].
The shape of the returned matrix is x.shape + (order,), where \(order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)\). The dtype will be the same as the converted x, y, and z.
See also
New in version 1.7.0.


Weight function of the Hermite polynomials.
The weight function is \(\exp(-x^2)\) and the interval of integration is \([-\inf, \inf]\). the Hermite polynomials are orthogonal, but not normalized, with respect to this weight function.
Values at which the weight function will be computed.
The weight function at x.
New in version 1.7.0.


An array object represents a multidimensional, homogeneous array of fixed-size items. An associated data-type object describes the format of each element in the array (its byte-order, how many bytes it occupies in memory, whether it is an integer, a floating point number, or something else, etc.)
Arrays should be constructed using array, zeros or empty (refer to the See Also section below). The parameters given here refer to a low-level method (ndarray(…)) for instantiating an array.
For more information, refer to the numpy module and examine the methods and attributes of an array.
Shape of created array.
Any object that can be interpreted as a numpy data type.
Used to fill the array with data.
Offset of array data in buffer.
Strides of data in memory.
Row-major (C-style) or column-major (Fortran-style) order.
See also
Construct an array.
Create an array, each element of which is zero.
Create an array, but leave its allocated memory unchanged (i.e., it contains “garbage”).
Create a data-type.
An ndarray alias generic w.r.t. its dtype.type.
There are two modes of creating an array using __new__:
No __init__ method is needed because the array is fully initialized after the __new__ method.
These examples illustrate the low-level ndarray constructor. Refer to the See Also section above for easier ways of constructing an ndarray.
First mode, buffer is None:
Second mode:
Transpose of the array.
The array’s elements, in memory.
Describes the format of the elements in the array.
Dictionary containing information related to memory use, e.g., ‘C_CONTIGUOUS’, ‘OWNDATA’, ‘WRITEABLE’, etc.
Flattened version of the array as an iterator. The iterator allows assignments, e.g., x.flat = 3 (See ndarray.flat for assignment examples; TODO).
Imaginary part of the array.
Real part of the array.
Number of elements in the array.
The memory use of each array element in bytes.
The total number of bytes required to store the array data, i.e., itemsize * size.
The array’s number of dimensions.
Shape of the array.
The step-size required to move from one element to the next in memory. For example, a contiguous (3, 4) array of type int16 in C-order has strides (8, 2). This implies that to move from element to element in memory requires jumps of 2 bytes. To move from row-to-row, one needs to jump 8 bytes at a time (2 * 4).
Class containing properties of the array needed for interaction with ctypes.
If the array is a view into another array, that array is its base (unless that array is also a view). The base array is where the array data is actually stored.


Convert a polynomial to a Hermite series.
Convert an array representing the coefficients of a polynomial (relative to the “standard” basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Hermite series, ordered from lowest to highest degree.
1-D array containing the polynomial coefficients
1-D array containing the coefficients of the equivalent Hermite series.
See also
The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance.


Convert a Hermite series to a polynomial.
Convert an array representing the coefficients of a Hermite series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the “standard” basis) ordered from lowest to highest degree.
1-D array containing the Hermite series coefficients, ordered from lowest order term to highest.
1-D array containing the coefficients of the equivalent polynomial (relative to the “standard” basis) ordered from lowest order term to highest.
See also
The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance.


Add one Hermite series to another.
Returns the sum of two Hermite series c1 + c2. 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.
1-D arrays of Hermite series coefficients ordered from low to high.
Array representing the Hermite series of their sum.
See also
Unlike multiplication, division, etc., the sum of two Hermite series is a Hermite series (without having to “reproject” the result onto the basis set) so addition, just like that of “standard” polynomials, is simply “component-wise.”


Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is symmetric when c is an HermiteE basis polynomial. This provides better eigenvalue estimates than the unscaled case and for basis polynomials the eigenvalues are guaranteed to be real if numpy.linalg.eigvalsh is used to obtain them.
1-D array of HermiteE series coefficients ordered from low to high degree.
Scaled companion matrix of dimensions (deg, deg).
New in version 1.7.0.


Differentiate a Hermite_e series.
Returns the series coefficients c differentiated m times along axis. At each iteration the result is multiplied by scl (the scaling factor is for use in a linear change of variable). The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series 1*He_0 + 2*He_1 + 3*He_2 while [[1,2],[1,2]] represents 1*He_0(x)*He_0(y) + 1*He_1(x)*He_0(y)
+ 2*He_0(x)*He_1(y) + 2*He_1(x)*He_1(y) if axis=0 is x and axis=1 is y.
Array of Hermite_e series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.
Number of derivatives taken, must be non-negative. (Default: 1)
Each differentiation is multiplied by scl. The end result is multiplication by scl**m. This is for use in a linear change of variable. (Default: 1)
Axis over which the derivative is taken. (Default: 0).
New in version 1.7.0.
Hermite series of the derivative.
See also
In general, the result of differentiating a Hermite series does not resemble the same operation on a power series. Thus the result of this function may be “unintuitive,” albeit correct; see Examples section below.


Divide one Hermite series by another.
Returns the quotient-with-remainder of two Hermite series c1 / c2. The arguments are sequences of coefficients from lowest order “term” to highest, e.g., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2.
1-D arrays of Hermite series coefficients ordered from low to high.
Of Hermite series coefficients representing the quotient and remainder.
See also
In general, the (polynomial) division of one Hermite series by another results in quotient and remainder terms that are not in the Hermite polynomial basis set. Thus, to express these results as a Hermite series, it is necessary to “reproject” the results onto the Hermite basis set, which may produce “unintuitive” (but correct) results; see Examples section below.


Least squares fit of Hermite series to data.
Return the coefficients of a HermiteE series of degree deg that is the least squares fit to the data values y given at points x. If y is 1-D the returned coefficients will also be 1-D. If y is 2-D multiple fits are done, one for each column of y, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form
where n is deg.
x-coordinates of the M sample points (x[i], y[i]).
y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column.
Degree(s) of the fitting polynomials. If deg is a single integer all terms up to and including the deg’th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead.
Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases.
Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned.
Weights. If not None, the weight w[i] applies to the unsquared residual y[i] - y_hat[i] at x[i]. Ideally the weights are chosen so that the errors of the products w[i]*y[i] all have the same variance. When using inverse-variance weighting, use w[i] = 1/sigma(y[i]). The default value is None.
Hermite coefficients ordered from low to high. If y was 2-D, the coefficients for the data in column k of y are in column k.
These values are only returned if full == True
For more details, see numpy.linalg.lstsq.
The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if full = False. The warnings can be turned off by
See also
Evaluates a Hermite series.
pseudo Vandermonde matrix of Hermite series.
HermiteE weight function.
Computes a least-squares fit from the matrix.
Computes spline fits.
The solution is the coefficients of the HermiteE series p that minimizes the sum of the weighted squared errors
where the \(w_j\) are the weights. This problem is solved by setting up the (typically) overdetermined matrix equation
where V is the pseudo Vandermonde matrix of x, the elements of c are the coefficients to be solved for, and the elements of y are the observed values. This equation is then solved using the singular value decomposition of V.
If some of the singular values of V are so small that they are neglected, then a RankWarning will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The rcond parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error.
Fits using HermiteE series are probably most useful when the data can be approximated by sqrt(w(x)) * p(x), where w(x) is the HermiteE weight. In that case the weight sqrt(w(x[i])) should be used together with data values y[i]/sqrt(w(x[i])). The weight function is available as hermeweight.
Wikipedia, “Curve fitting”, https://en.wikipedia.org/wiki/Curve_fitting


Generate a HermiteE series with given roots.
The function returns the coefficients of the polynomial
in HermiteE form, where the r_n are the roots specified in roots. If a zero has multiplicity n, then it must appear in roots n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then roots looks something like [2, 2, 2, 3, 3]. The roots can appear in any order.
If the returned coefficients are c, then
The coefficient of the last term is not generally 1 for monic polynomials in HermiteE form.
Sequence containing the roots.
1-D array of coefficients. If all roots are real then out is a real array, if some of the roots are complex, then out is complex even if all the coefficients in the result are real (see Examples below).
See also


Gauss-HermiteE quadrature.
Computes the sample points and weights for Gauss-HermiteE quadrature. These sample points and weights will correctly integrate polynomials of degree \(2*deg - 1\) or less over the interval \([-\inf, \inf]\) with the weight function \(f(x) = \exp(-x^2/2)\).
Number of sample points and weights. It must be >= 1.
1-D ndarray containing the sample points.
1-D ndarray containing the weights.
New in version 1.7.0.
The results have only been tested up to degree 100, higher degrees may be problematic. The weights are determined by using the fact that
where \(c\) is a constant independent of \(k\) and \(x_k\) is the k’th root of \(He_n\), and then scaling the results to get the right value when integrating 1.


Evaluate a 2-D HermiteE series on the Cartesian product of x and y.
This function returns the values:
where the points (a, b) consist of all pairs formed by taking a from x and b from y. The resulting points form a grid with x in the first dimension and y in the second.
The parameters x and y are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either x and y or their elements must support multiplication and addition both with themselves and with the elements of c.
If c has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape.
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, if it isn’t an ndarray, it is treated as a scalar.
Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate multiple sets of coefficients.
The values of the two dimensional polynomial at points in the Cartesian product of x and y.
See also
New in version 1.7.0.


Evaluate a 3-D HermiteE series on the Cartesian product of x, y, and z.
This function returns the values:
where the points (a, b, c) consist of all triples formed by taking a from x, b from y, and c from z. The resulting points form a grid with x in the first dimension, y in the second, and z in the third.
The parameters x, y, and z are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either x, y, and z or their elements must support multiplication and addition both with themselves and with the elements of c.
If c has fewer than three 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 + y.shape + z.shape.
The three dimensional series is evaluated at the points in the Cartesian product of x, y, and z. If 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 ndarray, it is treated as a scalar.
Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate multiple sets of coefficients.
The values of the two dimensional polynomial at points in the Cartesian product of x and y.
See also
New in version 1.7.0.


Integrate a Hermite_e series.
Returns the Hermite_e series coefficients c integrated m times from lbnd along axis. At each iteration the resulting series is multiplied by scl and an integration constant, k, is added. The scaling factor is for use in a linear change of variable. (“Buyer beware”: note that, depending on what one is doing, one may want scl to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series H_0 + 2*H_1 + 3*H_2 while [[1,2],[1,2]] represents 1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) +
2*H_1(x)*H_1(y) if axis=0 is x and axis=1 is y.
Array of Hermite_e series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.
Order of integration, must be positive. (Default: 1)
Integration constant(s). The value of the first integral at lbnd is the first value in the list, the value of the second integral at lbnd is the second value, etc. If k == [] (the default), all constants are set to zero. If m == 1, a single scalar can be given instead of a list.
The lower bound of the integral. (Default: 0)
Following each integration the result is multiplied by scl before the integration constant is added. (Default: 1)
Axis over which the integral is taken. (Default: 0).
New in version 1.7.0.
Hermite_e series coefficients of the integral.
If m < 0, len(k) > m, np.ndim(lbnd) != 0, or np.ndim(scl) != 0.
See also
Note that the result of each integration is multiplied by scl. Why is this important to note? Say one is making a linear change of variable \(u = ax + b\) in an integral relative to x. Then \(dx = du/a\), so one will need to set scl equal to \(1/a\) - perhaps not what one would have first thought.
Also note that, in general, the result of integrating a C-series needs to be “reprojected” onto the C-series basis set. Thus, typically, the result of this function is “unintuitive,” albeit correct; see Examples section below.


Hermite series whose graph is a straight line.
The specified line is given by off + scl*x.
This module’s representation of the Hermite series for off + scl*x.
See also


Multiply one Hermite series by another.
Returns the product of two Hermite series c1 * c2. The arguments are sequences of coefficients, from lowest order “term” to highest, e.g., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2.
1-D arrays of Hermite series coefficients ordered from low to high.
Of Hermite series coefficients representing their product.
See also
In general, the (polynomial) product of two C-series results in terms that are not in the Hermite polynomial basis set. Thus, to express the product as a Hermite series, it is necessary to “reproject” the product onto said basis set, which may produce “unintuitive” (but correct) results; see Examples section below.


Multiply a Hermite series by x.
Multiply the Hermite series c by x, where x is the independent variable.
1-D array of Hermite series coefficients ordered from low to high.
Array representing the result of the multiplication.
The multiplication uses the recursion relationship for Hermite polynomials in the form


Raise a Hermite series to a power.
Returns the Hermite series c raised to the power pow. The argument c is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series P_0 + 2*P_1 + 3*P_2.
1-D array of Hermite series coefficients ordered from low to high.
Power to which the series will be raised
Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16
Hermite series of power.
See also


Compute the roots of a HermiteE series.
Return the roots (a.k.a. “zeros”) of the polynomial
1-D array of coefficients.
Array of the roots of the series. If all the roots are real, then out is also real, otherwise it is complex.
See also
The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton’s method.
The HermiteE series basis polynomials aren’t powers of x so the results of this function may seem unintuitive.


Subtract one Hermite series from another.
Returns the difference of two Hermite series c1 - c2. The sequences of coefficients are from lowest order term to highest, i.e., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2.
1-D arrays of Hermite series coefficients ordered from low to high.
Of Hermite series coefficients representing their difference.
See also
Unlike multiplication, division, etc., the difference of two Hermite series is a Hermite series (without having to “reproject” the result onto the basis set) so subtraction, just like that of “standard” polynomials, is simply “component-wise.”


Remove “small” “trailing” coefficients from a polynomial.
“Small” means “small in absolute value” and is controlled by the parameter tol; “trailing” means highest order coefficient(s), e.g., in [0, 1, 1, 0, 0] (which represents 0 + x + x**2 + 0*x**3 + 0*x**4) both the 3-rd and 4-th order coefficients would be “trimmed.”
1-d array of coefficients, ordered from lowest order to highest.
Trailing (i.e., highest order) elements with absolute value less than or equal to tol (default value is zero) are removed.
1-d array with trailing zeros removed. If the resulting series would be empty, a series containing a single zero is returned.
If tol < 0
See also


Evaluate an HermiteE series at points x.
If c is of length n + 1, this function returns the value:
The parameter x is converted to an array only if it is a tuple or a list, otherwise it is treated as a scalar. In either case, either x or its elements must support multiplication and addition both with themselves and with the elements of c.
If c is a 1-D array, then p(x) will have the same shape as x. If c is multidimensional, then the shape of the result depends on the value of tensor. If tensor is true the shape will be c.shape[1:] + x.shape. If tensor is false the shape will be c.shape[1:]. Note that scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so they should be avoided if efficiency is a concern.
If x is a list or tuple, it is converted to an ndarray, otherwise it is left unchanged and treated as a scalar. In either case, x or its elements must support addition and multiplication with with themselves and with the elements of c.
Array of coefficients ordered so that the coefficients for terms of degree n are contained in c[n]. If c is multidimensional the remaining indices enumerate multiple polynomials. In the two dimensional case the coefficients may be thought of as stored in the columns of c.
If True, the shape of the coefficient array is extended with ones on the right, one for each dimension of x. Scalars have dimension 0 for this action. The result is that every column of coefficients in c is evaluated for every element of x. If False, x is broadcast over the columns of c for the evaluation. This keyword is useful when c is multidimensional. The default value is True.
New in version 1.7.0.
The shape of the return value is described above.
See also
The evaluation uses Clenshaw recursion, aka synthetic division.


Evaluate a 2-D HermiteE series at points (x, y).
This function returns the values:
The parameters x and y are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either x and y or their elements must support multiplication and addition both with themselves and with the elements of c.
If c is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape.
The two dimensional series is evaluated at the points (x, y), where x and y must have the same shape. If x or y is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn’t an ndarray it is treated as a scalar.
Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate multiple sets of coefficients.
The values of the two dimensional polynomial at points formed with pairs of corresponding values from x and y.
See also
New in version 1.7.0.


Evaluate a 3-D Hermite_e series at points (x, y, z).
This function returns the values:
The parameters x, y, and z are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either x, y, and z or their elements must support multiplication and addition both with themselves and with the elements of c.
If 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 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 ndarray it is treated as a scalar.
Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in c[i,j,k]. If c has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients.
The values of the multidimensional polynomial on points formed with triples of corresponding values from x, y, and z.
See also
New in version 1.7.0.


Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree deg and sample points x. The pseudo-Vandermonde matrix is defined by
where 0 <= i <= deg. The leading indices of V index the elements of x and the last index is the degree of the HermiteE polynomial.
If c is a 1-D array of coefficients of length n + 1 and V is the array V = hermevander(x, n), then np.dot(V, c) and hermeval(x, c) are the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of HermiteE series of the same degree and sample points.
Array of points. The dtype is converted to float64 or complex128 depending on whether any of the elements are complex. If x is scalar it is converted to a 1-D array.
Degree of the resulting matrix.
The pseudo-Vandermonde matrix. The shape of the returned matrix is x.shape + (deg + 1,), where The last index is the degree of the corresponding HermiteE polynomial. The dtype will be the same as the converted x.


Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y). The pseudo-Vandermonde matrix is defined by
where 0 <= i <= deg[0] and 0 <= j <= deg[1]. The leading indices of V index the points (x, y) and the last index encodes the degrees of the HermiteE polynomials.
If V = hermevander2d(x, y, [xdeg, ydeg]), then the columns of V correspond to the elements of a 2-D coefficient array c of shape (xdeg + 1, ydeg + 1) in the order
and np.dot(V, c.flat) and hermeval2d(x, y, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D HermiteE series of the same degrees and sample points.
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.
List of maximum degrees of the form [x_deg, y_deg].
The shape of the returned matrix is x.shape + (order,), where \(order = (deg[0]+1)*(deg[1]+1)\). The dtype will be the same as the converted x and y.
See also
New in version 1.7.0.


Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y, z). If l, m, n are the given degrees in x, y, z, then Hehe pseudo-Vandermonde matrix is defined by
where 0 <= i <= l, 0 <= j <= m, and 0 <= j <= n. The leading indices of V index the points (x, y, z) and the last index encodes the degrees of the HermiteE polynomials.
If V = hermevander3d(x, y, z, [xdeg, ydeg, zdeg]), then the columns of V correspond to the elements of a 3-D coefficient array c of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
and np.dot(V, c.flat) and hermeval3d(x, y, z, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D HermiteE series of the same degrees and sample points.
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.
List of maximum degrees of the form [x_deg, y_deg, z_deg].
The shape of the returned matrix is x.shape + (order,), where \(order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)\). The dtype will be the same as the converted x, y, and z.
See also
New in version 1.7.0.


Weight function of the Hermite_e polynomials.
The weight function is \(\exp(-x^2/2)\) and the interval of integration is \([-\inf, \inf]\). the HermiteE polynomials are orthogonal, but not normalized, with respect to this weight function.
Values at which the weight function will be computed.
The weight function at x.
New in version 1.7.0.


An array object represents a multidimensional, homogeneous array of fixed-size items. An associated data-type object describes the format of each element in the array (its byte-order, how many bytes it occupies in memory, whether it is an integer, a floating point number, or something else, etc.)
Arrays should be constructed using array, zeros or empty (refer to the See Also section below). The parameters given here refer to a low-level method (ndarray(…)) for instantiating an array.
For more information, refer to the numpy module and examine the methods and attributes of an array.
Shape of created array.
Any object that can be interpreted as a numpy data type.
Used to fill the array with data.
Offset of array data in buffer.
Strides of data in memory.
Row-major (C-style) or column-major (Fortran-style) order.
See also
Construct an array.
Create an array, each element of which is zero.
Create an array, but leave its allocated memory unchanged (i.e., it contains “garbage”).
Create a data-type.
An ndarray alias generic w.r.t. its dtype.type.
There are two modes of creating an array using __new__:
No __init__ method is needed because the array is fully initialized after the __new__ method.
These examples illustrate the low-level ndarray constructor. Refer to the See Also section above for easier ways of constructing an ndarray.
First mode, buffer is None:
Second mode:
Transpose of the array.
The array’s elements, in memory.
Describes the format of the elements in the array.
Dictionary containing information related to memory use, e.g., ‘C_CONTIGUOUS’, ‘OWNDATA’, ‘WRITEABLE’, etc.
Flattened version of the array as an iterator. The iterator allows assignments, e.g., x.flat = 3 (See ndarray.flat for assignment examples; TODO).
Imaginary part of the array.
Real part of the array.
Number of elements in the array.
The memory use of each array element in bytes.
The total number of bytes required to store the array data, i.e., itemsize * size.
The array’s number of dimensions.
Shape of the array.
The step-size required to move from one element to the next in memory. For example, a contiguous (3, 4) array of type int16 in C-order has strides (8, 2). This implies that to move from element to element in memory requires jumps of 2 bytes. To move from row-to-row, one needs to jump 8 bytes at a time (2 * 4).
Class containing properties of the array needed for interaction with ctypes.
If the array is a view into another array, that array is its base (unless that array is also a view). The base array is where the array data is actually stored.


method
Series basis polynomial of degree deg.
Returns the series representing the basis polynomial of degree deg.
New in version 1.7.0.
Degree of the basis polynomial for the series. Must be >= 0.
If given, the array must be of the form [beg, end], where beg and end are the endpoints of the domain. If None is given then the class domain is used. The default is None.
If given, the resulting array must be if the form [beg, end], where beg and end are the endpoints of the window. If None is given then the class window is used. The default is None.
A series with the coefficient of the deg term set to one and all others zero.


method
Convert series to series of this class.
The series is expected to be an instance of some polynomial series of one of the types supported by by the numpy.polynomial module, but could be some other class that supports the convert method.
New in version 1.7.0.
The series instance to be converted.
If given, the array must be of the form [beg, end], where beg and end are the endpoints of the domain. If None is given then the class domain is used. The default is None.
If given, the resulting array must be if the form [beg, end], where beg and end are the endpoints of the window. If None is given then the class window is used. The default is None.
A series of the same kind as the calling class and equal to series when evaluated.
See also
similar instance method


method
Convert series to a different kind and/or domain and/or window.
The domain of the converted series. If the value is None, the default domain of kind is used.
The polynomial series type class to which the current instance should be converted. If kind is None, then the class of the current instance is used.
The window of the converted series. If the value is None, the default window of kind is used.
The returned class can be of different type than the current instance and/or have a different domain and/or different window.
Conversion between domains and class types can result in numerically ill defined series.


method
Truncate series to the given degree.
Reduce the degree of the series to deg by discarding the high order terms. If deg is greater than the current degree a copy of the current series is returned. This can be useful in least squares where the coefficients of the high degree terms may be very small.
New in version 1.5.0.
The series is reduced to degree deg by discarding the high order terms. The value of deg must be a non-negative integer.
New instance of series with reduced degree.


method
The degree of the series.
New in version 1.5.0.
Degree of the series, one less than the number of coefficients.


method
Differentiate.
Return a series instance of that is the derivative of the current series.
Find the derivative of order m.
A new series representing the derivative. The domain is the same as the domain of the differentiated series.


method
Least squares fit to data.
Return a series instance that is the least squares fit to the data y sampled at x. The domain of the returned instance can be specified and this will often result in a superior fit with less chance of ill conditioning.
x-coordinates of the M sample points (x[i], y[i]).
y-coordinates of the M sample points (x[i], y[i]).
Degree(s) of the fitting polynomials. If deg is a single integer all terms up to and including the deg’th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead.
Domain to use for the returned series. If None, then a minimal domain that covers the points x is chosen. If [] the class domain is used. The default value was the class domain in NumPy 1.4 and None in later versions. The [] option was added in numpy 1.5.0.
Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases.
Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned.
Weights. If not None, the weight w[i] applies to the unsquared residual y[i] - y_hat[i] at x[i]. Ideally the weights are chosen so that the errors of the products w[i]*y[i] all have the same variance. When using inverse-variance weighting, use w[i] = 1/sigma(y[i]). The default value is None.
New in version 1.5.0.
Window to use for the returned series. The default value is the default class domain
New in version 1.6.0.
A series that represents the least squares fit to the data and has the domain and window specified in the call. If the coefficients for the unscaled and unshifted basis polynomials are of interest, do new_series.convert().coef.
These values are only returned if full == True
For more details, see linalg.lstsq.


method
Return series instance that has the specified roots.
Returns a series representing the product (x - r[0])*(x - r[1])*...*(x - r[n-1]), where r is a list of roots.
List of roots.
Domain for the resulting series. If None the domain is the interval from the smallest root to the largest. If [] the domain is the class domain. The default is [].
Window for the returned series. If None the class window is used. The default is None.
Series with the specified roots.


method
Check if coefficients match.
New in version 1.6.0.
The other class must have the coef attribute.
True if the coefficients are the same, False otherwise.


method
Check if domains match.
New in version 1.6.0.
The other class must have the domain attribute.
True if the domains are the same, False otherwise.


method
Check if types match.
New in version 1.7.0.
Class instance.
True if other is same class as self


method
Check if windows match.
New in version 1.6.0.
The other class must have the window attribute.
True if the windows are the same, False otherwise.


method
Identity function.
If p is the returned series, then p(x) == x for all values of x.
If given, the array must be of the form [beg, end], where beg and end are the endpoints of the domain. If None is given then the class domain is used. The default is None.
If given, the resulting array must be if the form [beg, end], where beg and end are the endpoints of the window. If None is given then the class window is used. The default is None.
Series of representing the identity.


method
Integrate.
Return a series instance that is the definite integral of the current series.
The number of integrations to perform.
Integration constants. The first constant is applied to the first integration, the second to the second, and so on. The list of values must less than or equal to m in length and any missing values are set to zero.
The lower bound of the definite integral.
A new series representing the integral. The domain is the same as the domain of the integrated series.


method
Return x, y values at equally spaced points in domain.
Returns the x, y values at n linearly spaced points across the domain. Here y is the value of the polynomial at the points x. By default the domain is the same as that of the series instance. This method is intended mostly as a plotting aid.
New in version 1.5.0.
Number of point pairs to return. The default value is 100.
If not None, the specified domain is used instead of that of the calling instance. It should be of the form [beg,end]. The default is None which case the class domain is used.
x is equal to linspace(self.domain[0], self.domain[1], n) and y is the series evaluated at element of x.


method
Return the mapping parameters.
The returned values define a linear map off + scl*x that is applied to the input arguments before the series is evaluated. The map depends on the domain and window; if the current domain is equal to the window the resulting map is the identity. If the coefficients of the series instance are to be used by themselves outside this class, then the linear function must be substituted for the x in the standard representation of the base polynomials.
The mapping function is defined by off + scl*x.
If the current domain is the interval [l1, r1] and the window is [l2, r2], then the linear mapping function L is defined by the equations:


method
Return the roots of the series polynomial.
Compute the roots for the series. Note that the accuracy of the roots decrease the further outside the domain they lie.
Array containing the roots of the series.


method
Remove trailing coefficients
Remove trailing coefficients until a coefficient is reached whose absolute value greater than tol or the beginning of the series is reached. If all the coefficients would be removed the series is set to [0]. A new series instance is returned with the new coefficients. The current instance remains unchanged.
All trailing coefficients less than tol will be removed.
New instance of series with trimmed coefficients.


method
Truncate series to length size.
Reduce the series to length size by discarding the high degree terms. The value of size must be a positive integer. This can be useful in least squares where the coefficients of the high degree terms may be very small.
The series is reduced to length size by discarding the high degree terms. The value of size must be a positive integer.
New instance of series with truncated coefficients.


Convert a polynomial to a Hermite series.
Convert an array representing the coefficients of a polynomial (relative to the “standard” basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Hermite series, ordered from lowest to highest degree.
1-D array containing the polynomial coefficients
1-D array containing the coefficients of the equivalent Hermite series.
See also
The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance.


Convert a Laguerre series to a polynomial.
Convert an array representing the coefficients of a Laguerre series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the “standard” basis) ordered from lowest to highest degree.
1-D array containing the Laguerre series coefficients, ordered from lowest order term to highest.
1-D array containing the coefficients of the equivalent polynomial (relative to the “standard” basis) ordered from lowest order term to highest.
See also
The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance.


Add one Laguerre series to another.
Returns the sum of two Laguerre series c1 + c2. 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.
1-D arrays of Laguerre series coefficients ordered from low to high.
Array representing the Laguerre series of their sum.
See also
Unlike multiplication, division, etc., the sum of two Laguerre series is a Laguerre series (without having to “reproject” the result onto the basis set) so addition, just like that of “standard” polynomials, is simply “component-wise.”


Return the companion matrix of c.
The usual companion matrix of the Laguerre polynomials is already symmetric when c is a basis Laguerre polynomial, so no scaling is applied.
1-D array of Laguerre series coefficients ordered from low to high degree.
Companion matrix of dimensions (deg, deg).
New in version 1.7.0.


Differentiate a Laguerre series.
Returns the Laguerre series coefficients c differentiated m times along axis. At each iteration the result is multiplied by scl (the scaling factor is for use in a linear change of variable). The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series 1*L_0 + 2*L_1 + 3*L_2 while [[1,2],[1,2]] represents 1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y) if axis=0 is x and axis=1 is y.
Array of Laguerre series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.
Number of derivatives taken, must be non-negative. (Default: 1)
Each differentiation is multiplied by scl. The end result is multiplication by scl**m. This is for use in a linear change of variable. (Default: 1)
Axis over which the derivative is taken. (Default: 0).
New in version 1.7.0.
Laguerre series of the derivative.
See also
In general, the result of differentiating a Laguerre series does not resemble the same operation on a power series. Thus the result of this function may be “unintuitive,” albeit correct; see Examples section below.


Divide one Laguerre series by another.
Returns the quotient-with-remainder of two Laguerre series c1 / c2. The arguments are sequences of coefficients from lowest order “term” to highest, e.g., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2.
1-D arrays of Laguerre series coefficients ordered from low to high.
Of Laguerre series coefficients representing the quotient and remainder.
See also
In general, the (polynomial) division of one Laguerre series by another results in quotient and remainder terms that are not in the Laguerre polynomial basis set. Thus, to express these results as a Laguerre series, it is necessary to “reproject” the results onto the Laguerre basis set, which may produce “unintuitive” (but correct) results; see Examples section below.


An array object represents a multidimensional, homogeneous array of fixed-size items. An associated data-type object describes the format of each element in the array (its byte-order, how many bytes it occupies in memory, whether it is an integer, a floating point number, or something else, etc.)
Arrays should be constructed using array, zeros or empty (refer to the See Also section below). The parameters given here refer to a low-level method (ndarray(…)) for instantiating an array.
For more information, refer to the numpy module and examine the methods and attributes of an array.
Shape of created array.
Any object that can be interpreted as a numpy data type.
Used to fill the array with data.
Offset of array data in buffer.
Strides of data in memory.
Row-major (C-style) or column-major (Fortran-style) order.
See also
Construct an array.
Create an array, each element of which is zero.
Create an array, but leave its allocated memory unchanged (i.e., it contains “garbage”).
Create a data-type.
An ndarray alias generic w.r.t. its dtype.type.
There are two modes of creating an array using __new__:
No __init__ method is needed because the array is fully initialized after the __new__ method.
These examples illustrate the low-level ndarray constructor. Refer to the See Also section above for easier ways of constructing an ndarray.
First mode, buffer is None:
Second mode:
Transpose of the array.
The array’s elements, in memory.
Describes the format of the elements in the array.
Dictionary containing information related to memory use, e.g., ‘C_CONTIGUOUS’, ‘OWNDATA’, ‘WRITEABLE’, etc.
Flattened version of the array as an iterator. The iterator allows assignments, e.g., x.flat = 3 (See ndarray.flat for assignment examples; TODO).
Imaginary part of the array.
Real part of the array.
Number of elements in the array.
The memory use of each array element in bytes.
The total number of bytes required to store the array data, i.e., itemsize * size.
The array’s number of dimensions.
Shape of the array.
The step-size required to move from one element to the next in memory. For example, a contiguous (3, 4) array of type int16 in C-order has strides (8, 2). This implies that to move from element to element in memory requires jumps of 2 bytes. To move from row-to-row, one needs to jump 8 bytes at a time (2 * 4).
Class containing properties of the array needed for interaction with ctypes.
If the array is a view into another array, that array is its base (unless that array is also a view). The base array is where the array data is actually stored.


Least squares fit of Laguerre series to data.
Return the coefficients of a Laguerre series of degree deg that is the least squares fit to the data values y given at points x. If y is 1-D the returned coefficients will also be 1-D. If y is 2-D multiple fits are done, one for each column of y, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form
where n is deg.
x-coordinates of the M sample points (x[i], y[i]).
y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column.
Degree(s) of the fitting polynomials. If deg is a single integer all terms up to and including the deg’th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead.
Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases.
Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned.
Weights. If not None, the weight w[i] applies to the unsquared residual y[i] - y_hat[i] at x[i]. Ideally the weights are chosen so that the errors of the products w[i]*y[i] all have the same variance. When using inverse-variance weighting, use w[i] = 1/sigma(y[i]). The default value is None.
Laguerre coefficients ordered from low to high. If y was 2-D, the coefficients for the data in column k of y are in column k.
These values are only returned if full == True
For more details, see numpy.linalg.lstsq.
The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if full == False. The warnings can be turned off by
See also
Evaluates a Laguerre series.
pseudo Vandermonde matrix of Laguerre series.
Laguerre weight function.
Computes a least-squares fit from the matrix.
Computes spline fits.
The solution is the coefficients of the Laguerre series p that minimizes the sum of the weighted squared errors
where the \(w_j\) are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation
where V is the weighted pseudo Vandermonde matrix of x, c are the coefficients to be solved for, w are the weights, and y are the observed values. This equation is then solved using the singular value decomposition of V.
If some of the singular values of V are so small that they are neglected, then a RankWarning will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The rcond parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error.
Fits using Laguerre series are probably most useful when the data can be approximated by sqrt(w(x)) * p(x), where w(x) is the Laguerre weight. In that case the weight sqrt(w(x[i])) should be used together with data values y[i]/sqrt(w(x[i])). The weight function is available as lagweight.
Wikipedia, “Curve fitting”, https://en.wikipedia.org/wiki/Curve_fitting


Generate a Laguerre series with given roots.
The function returns the coefficients of the polynomial
in Laguerre form, where the r_n are the roots specified in roots. If a zero has multiplicity n, then it must appear in roots n times. For instance, if 2 is a root of multiplicity three and 3 is a root of multiplicity 2, then roots looks something like [2, 2, 2, 3, 3]. The roots can appear in any order.
If the returned coefficients are c, then
The coefficient of the last term is not generally 1 for monic polynomials in Laguerre form.
Sequence containing the roots.
1-D array of coefficients. If all roots are real then out is a real array, if some of the roots are complex, then out is complex even if all the coefficients in the result are real (see Examples below).
See also


Gauss-Laguerre quadrature.
Computes the sample points and weights for Gauss-Laguerre quadrature. These sample points and weights will correctly integrate polynomials of degree \(2*deg - 1\) or less over the interval \([0, \inf]\) with the weight function \(f(x) = \exp(-x)\).
Number of sample points and weights. It must be >= 1.
1-D ndarray containing the sample points.
1-D ndarray containing the weights.
New in version 1.7.0.
The results have only been tested up to degree 100 higher degrees may be problematic. The weights are determined by using the fact that
where \(c\) is a constant independent of \(k\) and \(x_k\) is the k’th root of \(L_n\), and then scaling the results to get the right value when integrating 1.


Evaluate a 2-D Laguerre series on the Cartesian product of x and y.
This function returns the values:
where the points (a, b) consist of all pairs formed by taking a from x and b from y. The resulting points form a grid with x in the first dimension and y in the second.
The parameters x and y are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either x and y or their elements must support multiplication and addition both with themselves and with the elements of c.
If c has fewer than two dimensions, ones are implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape + y.shape.
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, if it isn’t an ndarray, it is treated as a scalar.
Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate multiple sets of coefficients.
The values of the two dimensional Chebyshev series at points in the Cartesian product of x and y.
See also
New in version 1.7.0.


Evaluate a 3-D Laguerre series on the Cartesian product of x, y, and z.
This function returns the values:
where the points (a, b, c) consist of all triples formed by taking a from x, b from y, and c from z. The resulting points form a grid with x in the first dimension, y in the second, and z in the third.
The parameters x, y, and z are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars. In either case, either x, y, and z or their elements must support multiplication and addition both with themselves and with the elements of c.
If c has fewer than three 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 + y.shape + z.shape.
The three dimensional series is evaluated at the points in the Cartesian product of x, y, and z. If 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 ndarray, it is treated as a scalar.
Array of coefficients ordered so that the coefficients for terms of degree i,j are contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate multiple sets of coefficients.
The values of the two dimensional polynomial at points in the Cartesian product of x and y.
See also
New in version 1.7.0.


Integrate a Laguerre series.
Returns the Laguerre series coefficients c integrated m times from lbnd along axis. At each iteration the resulting series is multiplied by scl and an integration constant, k, is added. The scaling factor is for use in a linear change of variable. (“Buyer beware”: note that, depending on what one is doing, one may want scl to be the reciprocal of what one might expect; for more information, see the Notes section below.) The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series L_0 + 2*L_1 + 3*L_2 while [[1,2],[1,2]] represents 1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) +
2*L_1(x)*L_1(y) if axis=0 is x and axis=1 is y.
Array of Laguerre series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.
Order of integration, must be positive. (Default: 1)
Integration constant(s). The value of the first integral at lbnd is the first value in the list, the value of the second integral at lbnd is the second value, etc. If k == [] (the default), all constants are set to zero. If m == 1, a single scalar can be given instead of a list.
The lower bound of the integral. (Default: 0)
Following each integration the result is multiplied by scl before the integration constant is added. (Default: 1)
Axis over which the integral is taken. (Default: 0).
New in version 1.7.0.
Laguerre series coefficients of the integral.
If m < 0, len(k) > m, np.ndim(lbnd) != 0, or np.ndim(scl) != 0.
See also
Note that the result of each integration is multiplied by scl. Why is this important to note? Say one is making a linear change of variable \(u = ax + b\) in an integral relative to x. Then \(dx = du/a\), so one will need to set scl equal to \(1/a\) - perhaps not what one would have first thought.
Also note that, in general, the result of integrating a C-series needs to be “reprojected” onto the C-series basis set. Thus, typically, the result of this function is “unintuitive,” albeit correct; see Examples section below.


Laguerre series whose graph is a straight line.
The specified line is given by off + scl*x.
This module’s representation of the Laguerre series for off + scl*x.
See also


Multiply one Laguerre series by another.
Returns the product of two Laguerre series c1 * c2. The arguments are sequences of coefficients, from lowest order “term” to highest, e.g., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2.
1-D arrays of Laguerre series coefficients ordered from low to high.
Of Laguerre series coefficients representing their product.
See also
In general, the (polynomial) product of two C-series results in terms that are not in the Laguerre polynomial basis set. Thus, to express the product as a Laguerre series, it is necessary to “reproject” the product onto said basis set, which may produce “unintuitive” (but correct) results; see Examples section below.


Multiply a Laguerre series by x.
Multiply the Laguerre series c by x, where x is the independent variable.
1-D array of Laguerre series coefficients ordered from low to high.
Array representing the result of the multiplication.
See also
The multiplication uses the recursion relationship for Laguerre polynomials in the form


Raise a Laguerre series to a power.
Returns the Laguerre series c raised to the power pow. The argument c is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series P_0 + 2*P_1 + 3*P_2.
1-D array of Laguerre series coefficients ordered from low to high.
Power to which the series will be raised
Maximum power allowed. This is mainly to limit growth of the series to unmanageable size. Default is 16
Laguerre series of power.
See also


Compute the roots of a Laguerre series.
Return the roots (a.k.a. “zeros”) of the polynomial
1-D array of coefficients.
Array of the roots of the series. If all the roots are real, then out is also real, otherwise it is complex.
See also
The root estimates are obtained as the eigenvalues of the companion matrix, Roots far from the origin of the complex plane may have large errors due to the numerical instability of the series for such values. Roots with multiplicity greater than 1 will also show larger errors as the value of the series near such points is relatively insensitive to errors in the roots. Isolated roots near the origin can be improved by a few iterations of Newton’s method.
The Laguerre series basis polynomials aren’t powers of x so the results of this function may seem unintuitive.


Subtract one Laguerre series from another.
Returns the difference of two Laguerre series c1 - c2. The sequences of coefficients are from lowest order term to highest, i.e., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2.
1-D arrays of Laguerre series coefficients ordered from low to high.
Of Laguerre series coefficients representing their difference.
See also
Unlike multiplication, division, etc., the difference of two Laguerre series is a Laguerre series (without having to “reproject” the result onto the basis set) so subtraction, just like that of “standard” polynomials, is simply “component-wise.”


Remove “small” “trailing” coefficients from a polynomial.
“Small” means “small in absolute value” and is controlled by the parameter tol; “trailing” means highest order coefficient(s), e.g., in [0, 1, 1, 0, 0] (which represents 0 + x + x**2 + 0*x**3 + 0*x**4) both the 3-rd and 4-th order coefficients would be “trimmed.”
1-d array of coefficients, ordered from lowest order to highest.
Trailing (i.e., highest order) elements with absolute value less than or equal to tol (default value is zero) are removed.
1-d array with trailing zeros removed. If the resulting series would be empty, a series containing a single zero is returned.
If tol < 0
See also


method
Series basis polynomial of degree deg.
Returns the series representing the basis polynomial of degree deg.
New in version 1.7.0.
Degree of the basis polynomial for the series. Must be >= 0.
If given, the array must be of the form [beg, end], where beg and end are the endpoints of the domain. If None is given then the class domain is used. The default is None.
If given, the resulting array must be if the form [beg, end], where beg and end are the endpoints of the window. If None is given then the class window is used. The default is None.
A series with the coefficient of the deg term set to one and all others zero.


method
Convert series to series of this class.
The series is expected to be an instance of some polynomial series of one of the types supported by by the numpy.polynomial module, but could be some other class that supports the convert method.
New in version 1.7.0.
The series instance to be converted.
If given, the array must be of the form [beg, end], where beg and end are the endpoints of the domain. If None is given then the class domain is used. The default is None.
If given, the resulting array must be if the form [beg, end], where beg and end are the endpoints of the window. If None is given then the class window is used. The default is None.
A series of the same kind as the calling class and equal to series when evaluated.
See also
similar instance method


method
Convert series to a different kind and/or domain and/or window.
The domain of the converted series. If the value is None, the default domain of kind is used.
The polynomial series type class to which the current instance should be converted. If kind is None, then the class of the current instance is used.
The window of the converted series. If the value is None, the default window of kind is used.
The returned class can be of different type than the current instance and/or have a different domain and/or different window.
Conversion between domains and class types can result in numerically ill defined series.


method
Truncate series to the given degree.
Reduce the degree of the series to deg by discarding the high order terms. If deg is greater than the current degree a copy of the current series is returned. This can be useful in least squares where the coefficients of the high degree terms may be very small.
New in version 1.5.0.
The series is reduced to degree deg by discarding the high order terms. The value of deg must be a non-negative integer.
New instance of series with reduced degree.


method
The degree of the series.
New in version 1.5.0.
Degree of the series, one less than the number of coefficients.


method
Differentiate.
Return a series instance of that is the derivative of the current series.
Find the derivative of order m.
A new series representing the derivative. The domain is the same as the domain of the differentiated series.


method
Least squares fit to data.
Return a series instance that is the least squares fit to the data y sampled at x. The domain of the returned instance can be specified and this will often result in a superior fit with less chance of ill conditioning.
x-coordinates of the M sample points (x[i], y[i]).
y-coordinates of the M sample points (x[i], y[i]).
Degree(s) of the fitting polynomials. If deg is a single integer all terms up to and including the deg’th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead.
Domain to use for the returned series. If None, then a minimal domain that covers the points x is chosen. If [] the class domain is used. The default value was the class domain in NumPy 1.4 and None in later versions. The [] option was added in numpy 1.5.0.
Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases.
Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned.
Weights. If not None, the weight w[i] applies to the unsquared residual y[i] - y_hat[i] at x[i]. Ideally the weights are chosen so that the errors of the products w[i]*y[i] all have the same variance. When using inverse-variance weighting, use w[i] = 1/sigma(y[i]). The default value is None.
New in version 1.5.0.
Window to use for the returned series. The default value is the default class domain
New in version 1.6.0.
A series that represents the least squares fit to the data and has the domain and window specified in the call. If the coefficients for the unscaled and unshifted basis polynomials are of interest, do new_series.convert().coef.
These values are only returned if full == True
For more details, see linalg.lstsq.


method
Return series instance that has the specified roots.
Returns a series representing the product (x - r[0])*(x - r[1])*...*(x - r[n-1]), where r is a list of roots.
List of roots.
Domain for the resulting series. If None the domain is the interval from the smallest root to the largest. If [] the domain is the class domain. The default is [].
Window for the returned series. If None the class window is used. The default is None.
Series with the specified roots.


method
Check if coefficients match.
New in version 1.6.0.
The other class must have the coef attribute.
True if the coefficients are the same, False otherwise.


method
Check if domains match.
New in version 1.6.0.
The other class must have the domain attribute.
True if the domains are the same, False otherwise.


method
Check if types match.
New in version 1.7.0.
Class instance.
True if other is same class as self


method
Check if windows match.
New in version 1.6.0.
The other class must have the window attribute.
True if the windows are the same, False otherwise.


method
Identity function.
If p is the returned series, then p(x) == x for all values of x.
If given, the array must be of the form [beg, end], where beg and end are the endpoints of the domain. If None is given then the class domain is used. The default is None.
If given, the resulting array must be if the form [beg, end], where beg and end are the endpoints of the window. If None is given then the class window is used. The default is None.
Series of representing the identity.


method
Integrate.
Return a series instance that is the definite integral of the current series.
The number of integrations to perform.
Integration constants. The first constant is applied to the first integration, the second to the second, and so on. The list of values must less than or equal to m in length and any missing values are set to zero.
The lower bound of the definite integral.
A new series representing the integral. The domain is the same as the domain of the integrated series.


method
Return x, y values at equally spaced points in domain.
Returns the x, y values at n linearly spaced points across the domain. Here y is the value of the polynomial at the points x. By default the domain is the same as that of the series instance. This method is intended mostly as a plotting aid.
New in version 1.5.0.
Number of point pairs to return. The default value is 100.
If not None, the specified domain is used instead of that of the calling instance. It should be of the form [beg,end]. The default is None which case the class domain is used.
x is equal to linspace(self.domain[0], self.domain[1], n) and y is the series evaluated at element of x.


method
Return the mapping parameters.
The returned values define a linear map off + scl*x that is applied to the input arguments before the series is evaluated. The map depends on the domain and window; if the current domain is equal to the window the resulting map is the identity. If the coefficients of the series instance are to be used by themselves outside this class, then the linear function must be substituted for the x in the standard representation of the base polynomials.
The mapping function is defined by off + scl*x.
If the current domain is the interval [l1, r1] and the window is [l2, r2], then the linear mapping function L is defined by the equations:


method
Return the roots of the series polynomial.
Compute the roots for the series. Note that the accuracy of the roots decrease the further outside the domain they lie.
Array containing the roots of the series.


method
Remove trailing coefficients
Remove trailing coefficients until a coefficient is reached whose absolute value greater than tol or the beginning of the series is reached. If all the coefficients would be removed the series is set to [0]. A new series instance is returned with the new coefficients. The current instance remains unchanged.
All trailing coefficients less than tol will be removed.
New instance of series with trimmed coefficients.


method
Truncate series to length size.
Reduce the series to length size by discarding the high degree terms. The value of size must be a positive integer. This can be useful in least squares where the coefficients of the high degree terms may be very small.
The series is reduced to length size by discarding the high degree terms. The value of size must be a positive integer.
New instance of series with truncated coefficients.


Evaluate a Laguerre series at points x.
If c is of length n + 1, this function returns the value:
The parameter x is converted to an array only if it is a tuple or a list, otherwise it is treated as a scalar. In either case, either x or its elements must support multiplication and addition both with themselves and with the elements of c.
If c is a 1-D array, then p(x) will have the same shape as x. If c is multidimensional, then the shape of the result depends on the value of tensor. If tensor is true the shape will be c.shape[1:] + x.shape. If tensor is false the shape will be c.shape[1:]. Note that scalars have shape (,).
Trailing zeros in the coefficients will be used in the evaluation, so they should be avoided if efficiency is a concern.
If x is a list or tuple, it is converted to an ndarray, otherwise it is left unchanged and treated as a scalar. In either case, x or its elements must support addition and multiplication with with themselves and with the elements of c.
Array of coefficients ordered so that the coefficients for terms of degree n are contained in c[n]. If c is multidimensional the remaining indices enumerate multiple polynomials. In the two dimensional case the coefficients may be thought of as stored in the columns of c.
If True, the shape of the coefficient array is extended with ones on the right, one for each dimension of x. Scalars have dimension 0 for this action. The result is that every column of coefficients in c is evaluated for every element of x. If False, x is broadcast over the columns of c for the evaluation. This keyword is useful when c is multidimensional. The default value is True.
New in version 1.7.0.
The shape of the return value is described above.
See also
The evaluation uses Clenshaw recursion, aka synthetic division.


Evaluate a 2-D Laguerre series at points (x, y).
This function returns the values:
The parameters x and y are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either x and y or their elements must support multiplication and addition both with themselves and with the elements of c.
If c is a 1-D array a one is implicitly appended to its shape to make it 2-D. The shape of the result will be c.shape[2:] + x.shape.
The two dimensional series is evaluated at the points (x, y), where x and y must have the same shape. If x or y is a list or tuple, it is first converted to an ndarray, otherwise it is left unchanged and if it isn’t an ndarray it is treated as a scalar.
Array of coefficients ordered so that the coefficient of the term of multi-degree i,j is contained in c[i,j]. If c has dimension greater than two the remaining indices enumerate multiple sets of coefficients.
The values of the two dimensional polynomial at points formed with pairs of corresponding values from x and y.
See also
New in version 1.7.0.


Evaluate a 3-D Laguerre series at points (x, y, z).
This function returns the values:
The parameters x, y, and z are converted to arrays only if they are tuples or a lists, otherwise they are treated as a scalars and they must have the same shape after conversion. In either case, either x, y, and z or their elements must support multiplication and addition both with themselves and with the elements of c.
If 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 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 ndarray it is treated as a scalar.
Array of coefficients ordered so that the coefficient of the term of multi-degree i,j,k is contained in c[i,j,k]. If c has dimension greater than 3 the remaining indices enumerate multiple sets of coefficients.
The values of the multidimensional polynomial on points formed with triples of corresponding values from x, y, and z.
See also
New in version 1.7.0.


Pseudo-Vandermonde matrix of given degree.
Returns the pseudo-Vandermonde matrix of degree deg and sample points x. The pseudo-Vandermonde matrix is defined by
where 0 <= i <= deg. The leading indices of V index the elements of x and the last index is the degree of the Laguerre polynomial.
If c is a 1-D array of coefficients of length n + 1 and V is the array V = lagvander(x, n), then np.dot(V, c) and lagval(x, c) are the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of Laguerre series of the same degree and sample points.
Array of points. The dtype is converted to float64 or complex128 depending on whether any of the elements are complex. If x is scalar it is converted to a 1-D array.
Degree of the resulting matrix.
The pseudo-Vandermonde matrix. The shape of the returned matrix is x.shape + (deg + 1,), where The last index is the degree of the corresponding Laguerre polynomial. The dtype will be the same as the converted x.


Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y). The pseudo-Vandermonde matrix is defined by
where 0 <= i <= deg[0] and 0 <= j <= deg[1]. The leading indices of V index the points (x, y) and the last index encodes the degrees of the Laguerre polynomials.
If V = lagvander2d(x, y, [xdeg, ydeg]), then the columns of V correspond to the elements of a 2-D coefficient array c of shape (xdeg + 1, ydeg + 1) in the order
and np.dot(V, c.flat) and lagval2d(x, y, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 2-D Laguerre series of the same degrees and sample points.
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.
List of maximum degrees of the form [x_deg, y_deg].
The shape of the returned matrix is x.shape + (order,), where \(order = (deg[0]+1)*(deg[1]+1)\). The dtype will be the same as the converted x and y.
See also
New in version 1.7.0.


Pseudo-Vandermonde matrix of given degrees.
Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y, z). If l, m, n are the given degrees in x, y, z, then The pseudo-Vandermonde matrix is defined by
where 0 <= i <= l, 0 <= j <= m, and 0 <= j <= n. The leading indices of V index the points (x, y, z) and the last index encodes the degrees of the Laguerre polynomials.
If V = lagvander3d(x, y, z, [xdeg, ydeg, zdeg]), then the columns of V correspond to the elements of a 3-D coefficient array c of shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
and np.dot(V, c.flat) and lagval3d(x, y, z, c) will be the same up to roundoff. This equivalence is useful both for least squares fitting and for the evaluation of a large number of 3-D Laguerre series of the same degrees and sample points.
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.
List of maximum degrees of the form [x_deg, y_deg, z_deg].
The shape of the returned matrix is x.shape + (order,), where \(order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)\). The dtype will be the same as the converted x, y, and z.
See also
New in version 1.7.0.


Weight function of the Laguerre polynomials.
The weight function is \(exp(-x)\) and the interval of integration is \([0, \inf]\). The Laguerre polynomials are orthogonal, but not normalized, with respect to this weight function.
Values at which the weight function will be computed.
The weight function at x.
New in version 1.7.0.


An array object represents a multidimensional, homogeneous array of fixed-size items. An associated data-type object describes the format of each element in the array (its byte-order, how many bytes it occupies in memory, whether it is an integer, a floating point number, or something else, etc.)
Arrays should be constructed using array, zeros or empty (refer to the See Also section below). The parameters given here refer to a low-level method (ndarray(…)) for instantiating an array.
For more information, refer to the numpy module and examine the methods and attributes of an array.
Shape of created array.
Any object that can be interpreted as a numpy data type.
Used to fill the array with data.
Offset of array data in buffer.
Strides of data in memory.
Row-major (C-style) or column-major (Fortran-style) order.
See also
Construct an array.
Create an array, each element of which is zero.
Create an array, but leave its allocated memory unchanged (i.e., it contains “garbage”).
Create a data-type.
An ndarray alias generic w.r.t. its dtype.type.
There are two modes of creating an array using __new__:
No __init__ method is needed because the array is fully initialized after the __new__ method.
These examples illustrate the low-level ndarray constructor. Refer to the See Also section above for easier ways of constructing an ndarray.
First mode, buffer is None:
Second mode:
Transpose of the array.
The array’s elements, in memory.
Describes the format of the elements in the array.
Dictionary containing information related to memory use, e.g., ‘C_CONTIGUOUS’, ‘OWNDATA’, ‘WRITEABLE’, etc.
Flattened version of the array as an iterator. The iterator allows assignments, e.g., x.flat = 3 (See ndarray.flat for assignment examples; TODO).
Imaginary part of the array.
Real part of the array.
Number of elements in the array.
The memory use of each array element in bytes.
The total number of bytes required to store the array data, i.e., itemsize * size.
The array’s number of dimensions.
Shape of the array.
The step-size required to move from one element to the next in memory. For example, a contiguous (3, 4) array of type int16 in C-order has strides (8, 2). This implies that to move from element to element in memory requires jumps of 2 bytes. To move from row-to-row, one needs to jump 8 bytes at a time (2 * 4).
Class containing properties of the array needed for interaction with ctypes.
If the array is a view into another array, that array is its base (unless that array is also a view). The base array is where the array data is actually stored.


Convert a polynomial to a Laguerre series.
Convert an array representing the coefficients of a polynomial (relative to the “standard” basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Laguerre series, ordered from lowest to highest degree.
1-D array containing the polynomial coefficients
1-D array containing the coefficients of the equivalent Laguerre series.
See also
The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance.


Convert a Legendre series to a polynomial.
Convert an array representing the coefficients of a Legendre series, ordered from lowest degree to highest, to an array of the coefficients of the equivalent polynomial (relative to the “standard” basis) ordered from lowest to highest degree.
1-D array containing the Legendre series coefficients, ordered from lowest order term to highest.
1-D array containing the coefficients of the equivalent polynomial (relative to the “standard” basis) ordered from lowest order term to highest.
See also
The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance.


Add one Legendre series to another.
Returns the sum of two Legendre series c1 + c2. 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.
1-D arrays of Legendre series coefficients ordered from low to high.
Array representing the Legendre series of their sum.
See also
Unlike multiplication, division, etc., the sum of two Legendre series is a Legendre series (without having to “reproject” the result onto the basis set) so addition, just like that of “standard” polynomials, is simply “component-wise.”


Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is symmetric when c is an Legendre basis polynomial. This provides better eigenvalue estimates than the unscaled case and for basis polynomials the eigenvalues are guaranteed to be real if numpy.linalg.eigvalsh is used to obtain them.
1-D array of Legendre series coefficients ordered from low to high degree.
Scaled companion matrix of dimensions (deg, deg).
New in version 1.7.0.


Differentiate a Legendre series.
Returns the Legendre series coefficients c differentiated m times along axis. At each iteration the result is multiplied by scl (the scaling factor is for use in a linear change of variable). The argument c is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series 1*L_0 + 2*L_1 + 3*L_2 while [[1,2],[1,2]] represents 1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y) if axis=0 is x and axis=1 is y.
Array of Legendre series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index.
Number of derivatives taken, must be non-negative. (Default: 1)
Each differentiation is multiplied by scl. The end result is multiplication by scl**m. This is for use in a linear change of variable. (Default: 1)
Axis over which the derivative is taken. (Default: 0).
New in version 1.7.0.
Legendre series of the derivative.
See also
In general, the result of differentiating a Legendre series does not resemble the same operation on a power series. Thus the result of this function may be “unintuitive,” albeit correct; see Examples section below.


Divide one Legendre series by another.
Returns the quotient-with-remainder of two Legendre series c1 / c2. The arguments are sequences of coefficients from lowest order “term” to highest, e.g., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2.
1-D arrays of Legendre series coefficients ordered from low to high.
Of Legendre series coefficients representing the quotient and remainder.
See also
In general, the (polynomial) division of one Legendre series by another results in quotient and remainder terms that are not in the Legendre polynomial basis set. Thus, to express these results as a Legendre series, it is necessary to “reproject” the results onto the Legendre basis set, which may produce “unintuitive” (but correct) results; see Examples section below.


method
Series basis polynomial of degree deg.
Returns the series representing the basis polynomial of degree deg.
New in version 1.7.0.
Degree of the basis polynomial for the series. Must be >= 0.
If given, the array must be of the form [beg, end], where beg and end are the endpoints of the domain. If None is given then the class domain is used. The default is None.
If given, the resulting array must be if the form [beg, end], where beg and end are the endpoints of the window. If None is given then the class window is used. The default is None.
A series with the coefficient of the deg term set to one and all others zero.


method
Convert series to series of this class.
The series is expected to be an instance of some polynomial series of one of the types supported by by the numpy.polynomial module, but could be some other class that supports the convert method.
New in version 1.7.0.
The series instance to be converted.
If given, the array must be of the form [beg, end], where beg and end are the endpoints of the domain. If None is given then the class domain is used. The default is None.
If given, the resulting array must be if the form [beg, end], where beg and end are the endpoints of the window. If None is given then the class window is used. The default is None.
A series of the same kind as the calling class and equal to series when evaluated.
See also
similar instance method


method
Convert series to a different kind and/or domain and/or window.
The domain of the converted series. If the value is None, the default domain of kind is used.
The polynomial series type class to which the current instance should be converted. If kind is None, then the class of the current instance is used.
The window of the converted series. If the value is None, the default window of kind is used.
The returned class can be of different type than the current instance and/or have a different domain and/or different window.
Conversion between domains and class types can result in numerically ill defined series.


method
Truncate series to the given degree.
Reduce the degree of the series to deg by discarding the high order terms. If deg is greater than the current degree a copy of the current series is returned. This can be useful in least squares where the coefficients of the high degree terms may be very small.
New in version 1.5.0.
The series is reduced to degree deg by discarding the high order terms. The value of deg must be a non-negative integer.
New instance of series with reduced degree.


method
The degree of the series.
New in version 1.5.0.
Degree of the series, one less than the number of coefficients.


method
Differentiate.
Return a series instance of that is the derivative of the current series.
Find the derivative of order m.
A new series representing the derivative. The domain is the same as the domain of the differentiated series.


method
Least squares fit to data.
Return a series instance that is the least squares fit to the data y sampled at x. The domain of the returned instance can be specified and this will often result in a superior fit with less chance of ill conditioning.
x-coordinates of the M sample points (x[i], y[i]).
y-coordinates of the M sample points (x[i], y[i]).
Degree(s) of the fitting polynomials. If deg is a single integer all terms up to and including the deg’th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead.
Domain to use for the returned series. If None, then a minimal domain that covers the points x is chosen. If [] the class domain is used. The default value was the class domain in NumPy 1.4 and None in later versions. The [] option was added in numpy 1.5.0.
Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases.
Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned.
Weights. If not None, the weight w[i] applies to the unsquared residual y[i] - y_hat[i] at x[i]. Ideally the weights are chosen so that the errors of the products w[i]*y[i] all have the same variance. When using inverse-variance weighting, use w[i] = 1/sigma(y[i]). The default value is None.
New in version 1.5.0.
Window to use for the returned series. The default value is the default class domain
New in version 1.6.0.
A series that represents the least squares fit to the data and has the domain and window specified in the call. If the coefficients for the unscaled and unshifted basis polynomials are of interest, do new_series.convert().coef.
These values are only returned if full == True
For more details, see linalg.lstsq.


method
Return series instance that has the specified roots.
Returns a series representing the product (x - r[0])*(x - r[1])*...*(x - r[n-1]), where r is a list of roots.
List of roots.
Domain for the resulting series. If None the domain is the interval from the smallest root to the largest. If [] the domain is the class domain. The default is [].
Window for the returned series. If None the class window is used. The default is None.
Series with the specified roots.


method
Check if coefficients match.
New in version 1.6.0.
The other class must have the coef attribute.
True if the coefficients are the same, False otherwise.


method
Check if domains match.
New in version 1.6.0.
The other class must have the domain attribute.
True if the domains are the same, False otherwise.


method
Check if types match.
New in version 1.7.0.
Class instance.
True if other is same class as self


method
Check if windows match.
New in version 1.6.0.
The other class must have the window attribute.
True if the windows are the same, False otherwise.


method
Identity function.
If p is the returned series, then p(x) == x for all values of x.
If given, the array must be of the form [beg, end], where beg and end are the endpoints of the domain. If None is given then the class domain is used. The default is None.
If given, the resulting array must be if the form [beg, end], where beg and end are the endpoints of the window. If None is given then the class window is used. The default is None.
Series of representing the identity.


method
Integrate.
Return a series instance that is the definite integral of the current series.
The number of integrations to perform.
Integration constants. The first constant is applied to the first integration, the second to the second, and so on. The list of values must less than or equal to m in length and any missing values are set to zero.
The lower bound of the definite integral.
A new series representing the integral. The domain is the same as the domain of the integrated series.


method
Return x, y values at equally spaced points in domain.
Returns the x, y values at n linearly spaced points across the domain. Here y is the value of the polynomial at the points x. By default the domain is the same as that of the series instance. This method is intended mostly as a plotting aid.
New in version 1.5.0.
Number of point pairs to return. The default value is 100.
If not None, the specified domain is used instead of that of the calling instance. It should be of the form [beg,end]. The default is None which case the class domain is used.
x is equal to linspace(self.domain[0], self.domain[1], n) and y is the series evaluated at element of x.


method
Return the mapping parameters.
The returned values define a linear map off + scl*x that is applied to the input arguments before the series is evaluated. The map depends on the domain and window; if the current domain is equal to the window the resulting map is the identity. If the coefficients of the series instance are to be used by themselves outside this class, then the linear function must be substituted for the x in the standard representation of the base polynomials.
The mapping function is defined by off + scl*x.
If the current domain is the interval [l1, r1] and the window is [l2, r2], then the linear mapping function L is defined by the equations:


method
Return the roots of the series polynomial.
Compute the roots for the series. Note that the accuracy of the roots decrease the further outside the domain they lie.
Array containing the roots of the series.


method
Remove trailing coefficients
Remove trailing coefficients until a coefficient is reached whose absolute value greater than tol or the beginning of the series is reached. If all the coefficients would be removed the series is set to [0]. A new series instance is returned with the new coefficients. The current instance remains unchanged.
All trailing coefficients less than tol will be removed.
New instance of series with trimmed coefficients.


method
Truncate series to length size.
Reduce the series to length size by discarding the high degree terms. The value of size must be a positive integer. This can be useful in least squares where the coefficients of the high degree terms may be very small.
The series is reduced to length size by discarding the high degree terms. The value of size must be a positive integer.
New instance of series with truncated coefficients.


Least squares fit of Legendre series to data.
Return the coefficients of a Legendre series of degree deg that is the least squares fit to the data values y given at points x. If y is 1-D the returned coefficients will also be 1-D. If y is 2-D multiple fits are done, one for each column of y, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form
where n is deg.
x-coordinates of the M sample points (x[i], y[i]).
y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column.
Degree(s) of the fitting polynomials. If deg is a single integer all terms up to and including the deg’th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead.
Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases.
Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned.
Weights. If not None, the weight w[i] applies to the unsquared residual y[i] - y_hat[i] at x[i]. Ideally the weights are chosen so that the errors of the products w[i]*y[i] all have the same variance. When using inverse-variance weighting, use w[i] = 1/sigma(y[i]). The default value is None.
New in version 1.5.0.
Legendre coefficients ordered from low to high. If y was 2-D, the coefficients for the data in column k of y are in column k. If deg is specified as a list, coefficients for terms not included in the fit are set equal to zero in the returned coef.
These values are only returned if full == True
For more details, see numpy.linalg.lstsq.
The rank of the coefficient matrix in the least-squares fit is deficient. The warning is only raised if full == False. The warnings can be turned off by
See also
Evaluates a Legendre series.
Vandermonde matrix of Legendre series.
Legendre weight function (= 1).
Computes a least-squares fit from the matrix.
Computes spline fits.
The solution is the coefficients of the Legendre series p that minimizes the sum of the weighted squared errors
where \(w_j\) are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation
where V is the weighted pseudo Vandermonde matrix of x, c are the coefficients to be solved for, w are the weights, and y are the observed values. This equation is then solved using the singular value decomposition of V.
If some of the singular values of V are so small that they are neglected, then a RankWarning will be issued. This means that the coefficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The rcond parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error.
Fits using Legendre series are usually better conditioned than fits using power series, but much can depend on the distribution of the sample points and the smoothness of the data. If the quality of the fit is inadequate splines may be a good alternative.
Wikipedia, “Curve fitting”, https://en.wikipedia.org/wiki/Curve_fitting