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| """ | |
| ==================================================== | |
| Chebyshev Series (:mod:`numpy.polynomial.chebyshev`) | |
| ==================================================== | |
| This module provides a number of objects (mostly functions) useful for | |
| dealing with Chebyshev series, including a `Chebyshev` class that | |
| encapsulates the usual arithmetic operations. (General information | |
| on how this module represents and works with such polynomials is in the | |
| docstring for its "parent" sub-package, `numpy.polynomial`). | |
| Classes | |
| ------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| Chebyshev | |
| Constants | |
| --------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| chebdomain | |
| chebzero | |
| chebone | |
| chebx | |
| Arithmetic | |
| ---------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| chebadd | |
| chebsub | |
| chebmulx | |
| chebmul | |
| chebdiv | |
| chebpow | |
| chebval | |
| chebval2d | |
| chebval3d | |
| chebgrid2d | |
| chebgrid3d | |
| Calculus | |
| -------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| chebder | |
| chebint | |
| Misc Functions | |
| -------------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| chebfromroots | |
| chebroots | |
| chebvander | |
| chebvander2d | |
| chebvander3d | |
| chebgauss | |
| chebweight | |
| chebcompanion | |
| chebfit | |
| chebpts1 | |
| chebpts2 | |
| chebtrim | |
| chebline | |
| cheb2poly | |
| poly2cheb | |
| chebinterpolate | |
| See also | |
| -------- | |
| `numpy.polynomial` | |
| Notes | |
| ----- | |
| The implementations of multiplication, division, integration, and | |
| differentiation use the algebraic identities [1]_: | |
| .. math :: | |
| T_n(x) = \\frac{z^n + z^{-n}}{2} \\\\ | |
| z\\frac{dx}{dz} = \\frac{z - z^{-1}}{2}. | |
| where | |
| .. math :: x = \\frac{z + z^{-1}}{2}. | |
| These identities allow a Chebyshev series to be expressed as a finite, | |
| symmetric Laurent series. In this module, this sort of Laurent series | |
| is referred to as a "z-series." | |
| References | |
| ---------- | |
| .. [1] A. T. Benjamin, et al., "Combinatorial Trigonometry with Chebyshev | |
| Polynomials," *Journal of Statistical Planning and Inference 14*, 2008 | |
| (https://web.archive.org/web/20080221202153/https://www.math.hmc.edu/~benjamin/papers/CombTrig.pdf, pg. 4) | |
| """ | |
| import numpy as np | |
| import numpy.linalg as la | |
| from numpy.core.multiarray import normalize_axis_index | |
| from . import polyutils as pu | |
| from ._polybase import ABCPolyBase | |
| __all__ = [ | |
| 'chebzero', 'chebone', 'chebx', 'chebdomain', 'chebline', 'chebadd', | |
| 'chebsub', 'chebmulx', 'chebmul', 'chebdiv', 'chebpow', 'chebval', | |
| 'chebder', 'chebint', 'cheb2poly', 'poly2cheb', 'chebfromroots', | |
| 'chebvander', 'chebfit', 'chebtrim', 'chebroots', 'chebpts1', | |
| 'chebpts2', 'Chebyshev', 'chebval2d', 'chebval3d', 'chebgrid2d', | |
| 'chebgrid3d', 'chebvander2d', 'chebvander3d', 'chebcompanion', | |
| 'chebgauss', 'chebweight', 'chebinterpolate'] | |
| chebtrim = pu.trimcoef | |
| # | |
| # A collection of functions for manipulating z-series. These are private | |
| # functions and do minimal error checking. | |
| # | |
| def _cseries_to_zseries(c): | |
| """Covert Chebyshev series to z-series. | |
| Covert a Chebyshev series to the equivalent z-series. The result is | |
| never an empty array. The dtype of the return is the same as that of | |
| the input. No checks are run on the arguments as this routine is for | |
| internal use. | |
| Parameters | |
| ---------- | |
| c : 1-D ndarray | |
| Chebyshev coefficients, ordered from low to high | |
| Returns | |
| ------- | |
| zs : 1-D ndarray | |
| Odd length symmetric z-series, ordered from low to high. | |
| """ | |
| n = c.size | |
| zs = np.zeros(2*n-1, dtype=c.dtype) | |
| zs[n-1:] = c/2 | |
| return zs + zs[::-1] | |
| def _zseries_to_cseries(zs): | |
| """Covert z-series to a Chebyshev series. | |
| Covert a z series to the equivalent Chebyshev series. The result is | |
| never an empty array. The dtype of the return is the same as that of | |
| the input. No checks are run on the arguments as this routine is for | |
| internal use. | |
| Parameters | |
| ---------- | |
| zs : 1-D ndarray | |
| Odd length symmetric z-series, ordered from low to high. | |
| Returns | |
| ------- | |
| c : 1-D ndarray | |
| Chebyshev coefficients, ordered from low to high. | |
| """ | |
| n = (zs.size + 1)//2 | |
| c = zs[n-1:].copy() | |
| c[1:n] *= 2 | |
| return c | |
| def _zseries_mul(z1, z2): | |
| """Multiply two z-series. | |
| Multiply two z-series to produce a z-series. | |
| Parameters | |
| ---------- | |
| z1, z2 : 1-D ndarray | |
| The arrays must be 1-D but this is not checked. | |
| Returns | |
| ------- | |
| product : 1-D ndarray | |
| The product z-series. | |
| Notes | |
| ----- | |
| This is simply convolution. If symmetric/anti-symmetric z-series are | |
| denoted by S/A then the following rules apply: | |
| S*S, A*A -> S | |
| S*A, A*S -> A | |
| """ | |
| return np.convolve(z1, z2) | |
| def _zseries_div(z1, z2): | |
| """Divide the first z-series by the second. | |
| Divide `z1` by `z2` and return the quotient and remainder as z-series. | |
| Warning: this implementation only applies when both z1 and z2 have the | |
| same symmetry, which is sufficient for present purposes. | |
| Parameters | |
| ---------- | |
| z1, z2 : 1-D ndarray | |
| The arrays must be 1-D and have the same symmetry, but this is not | |
| checked. | |
| Returns | |
| ------- | |
| (quotient, remainder) : 1-D ndarrays | |
| Quotient and remainder as z-series. | |
| Notes | |
| ----- | |
| This is not the same as polynomial division on account of the desired form | |
| of the remainder. If symmetric/anti-symmetric z-series are denoted by S/A | |
| then the following rules apply: | |
| S/S -> S,S | |
| A/A -> S,A | |
| The restriction to types of the same symmetry could be fixed but seems like | |
| unneeded generality. There is no natural form for the remainder in the case | |
| where there is no symmetry. | |
| """ | |
| z1 = z1.copy() | |
| z2 = z2.copy() | |
| lc1 = len(z1) | |
| lc2 = len(z2) | |
| if lc2 == 1: | |
| z1 /= z2 | |
| return z1, z1[:1]*0 | |
| elif lc1 < lc2: | |
| return z1[:1]*0, z1 | |
| else: | |
| dlen = lc1 - lc2 | |
| scl = z2[0] | |
| z2 /= scl | |
| quo = np.empty(dlen + 1, dtype=z1.dtype) | |
| i = 0 | |
| j = dlen | |
| while i < j: | |
| r = z1[i] | |
| quo[i] = z1[i] | |
| quo[dlen - i] = r | |
| tmp = r*z2 | |
| z1[i:i+lc2] -= tmp | |
| z1[j:j+lc2] -= tmp | |
| i += 1 | |
| j -= 1 | |
| r = z1[i] | |
| quo[i] = r | |
| tmp = r*z2 | |
| z1[i:i+lc2] -= tmp | |
| quo /= scl | |
| rem = z1[i+1:i-1+lc2].copy() | |
| return quo, rem | |
| def _zseries_der(zs): | |
| """Differentiate a z-series. | |
| The derivative is with respect to x, not z. This is achieved using the | |
| chain rule and the value of dx/dz given in the module notes. | |
| Parameters | |
| ---------- | |
| zs : z-series | |
| The z-series to differentiate. | |
| Returns | |
| ------- | |
| derivative : z-series | |
| The derivative | |
| Notes | |
| ----- | |
| The zseries for x (ns) has been multiplied by two in order to avoid | |
| using floats that are incompatible with Decimal and likely other | |
| specialized scalar types. This scaling has been compensated by | |
| multiplying the value of zs by two also so that the two cancels in the | |
| division. | |
| """ | |
| n = len(zs)//2 | |
| ns = np.array([-1, 0, 1], dtype=zs.dtype) | |
| zs *= np.arange(-n, n+1)*2 | |
| d, r = _zseries_div(zs, ns) | |
| return d | |
| def _zseries_int(zs): | |
| """Integrate a z-series. | |
| The integral is with respect to x, not z. This is achieved by a change | |
| of variable using dx/dz given in the module notes. | |
| Parameters | |
| ---------- | |
| zs : z-series | |
| The z-series to integrate | |
| Returns | |
| ------- | |
| integral : z-series | |
| The indefinite integral | |
| Notes | |
| ----- | |
| The zseries for x (ns) has been multiplied by two in order to avoid | |
| using floats that are incompatible with Decimal and likely other | |
| specialized scalar types. This scaling has been compensated by | |
| dividing the resulting zs by two. | |
| """ | |
| n = 1 + len(zs)//2 | |
| ns = np.array([-1, 0, 1], dtype=zs.dtype) | |
| zs = _zseries_mul(zs, ns) | |
| div = np.arange(-n, n+1)*2 | |
| zs[:n] /= div[:n] | |
| zs[n+1:] /= div[n+1:] | |
| zs[n] = 0 | |
| return zs | |
| # | |
| # Chebyshev series functions | |
| # | |
| def poly2cheb(pol): | |
| """ | |
| Convert a polynomial to a Chebyshev 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 Chebyshev series, ordered | |
| from lowest to highest degree. | |
| Parameters | |
| ---------- | |
| pol : array_like | |
| 1-D array containing the polynomial coefficients | |
| Returns | |
| ------- | |
| c : ndarray | |
| 1-D array containing the coefficients of the equivalent Chebyshev | |
| series. | |
| See Also | |
| -------- | |
| cheb2poly | |
| Notes | |
| ----- | |
| The easy way to do conversions between polynomial basis sets | |
| is to use the convert method of a class instance. | |
| Examples | |
| -------- | |
| >>> from numpy import polynomial as P | |
| >>> p = P.Polynomial(range(4)) | |
| >>> p | |
| Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) | |
| >>> c = p.convert(kind=P.Chebyshev) | |
| >>> c | |
| Chebyshev([1. , 3.25, 1. , 0.75], domain=[-1., 1.], window=[-1., 1.]) | |
| >>> P.chebyshev.poly2cheb(range(4)) | |
| array([1. , 3.25, 1. , 0.75]) | |
| """ | |
| [pol] = pu.as_series([pol]) | |
| deg = len(pol) - 1 | |
| res = 0 | |
| for i in range(deg, -1, -1): | |
| res = chebadd(chebmulx(res), pol[i]) | |
| return res | |
| def cheb2poly(c): | |
| """ | |
| Convert a Chebyshev series to a polynomial. | |
| Convert an array representing the coefficients of a Chebyshev 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. | |
| Parameters | |
| ---------- | |
| c : array_like | |
| 1-D array containing the Chebyshev series coefficients, ordered | |
| from lowest order term to highest. | |
| Returns | |
| ------- | |
| pol : ndarray | |
| 1-D array containing the coefficients of the equivalent polynomial | |
| (relative to the "standard" basis) ordered from lowest order term | |
| to highest. | |
| See Also | |
| -------- | |
| poly2cheb | |
| Notes | |
| ----- | |
| The easy way to do conversions between polynomial basis sets | |
| is to use the convert method of a class instance. | |
| Examples | |
| -------- | |
| >>> from numpy import polynomial as P | |
| >>> c = P.Chebyshev(range(4)) | |
| >>> c | |
| Chebyshev([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) | |
| >>> p = c.convert(kind=P.Polynomial) | |
| >>> p | |
| Polynomial([-2., -8., 4., 12.], domain=[-1., 1.], window=[-1., 1.]) | |
| >>> P.chebyshev.cheb2poly(range(4)) | |
| array([-2., -8., 4., 12.]) | |
| """ | |
| from .polynomial import polyadd, polysub, polymulx | |
| [c] = pu.as_series([c]) | |
| n = len(c) | |
| if n < 3: | |
| return c | |
| else: | |
| c0 = c[-2] | |
| c1 = c[-1] | |
| # i is the current degree of c1 | |
| for i in range(n - 1, 1, -1): | |
| tmp = c0 | |
| c0 = polysub(c[i - 2], c1) | |
| c1 = polyadd(tmp, polymulx(c1)*2) | |
| return polyadd(c0, polymulx(c1)) | |
| # | |
| # These are constant arrays are of integer type so as to be compatible | |
| # with the widest range of other types, such as Decimal. | |
| # | |
| # Chebyshev default domain. | |
| chebdomain = np.array([-1, 1]) | |
| # Chebyshev coefficients representing zero. | |
| chebzero = np.array([0]) | |
| # Chebyshev coefficients representing one. | |
| chebone = np.array([1]) | |
| # Chebyshev coefficients representing the identity x. | |
| chebx = np.array([0, 1]) | |
| def chebline(off, scl): | |
| """ | |
| Chebyshev series whose graph is a straight line. | |
| Parameters | |
| ---------- | |
| off, scl : scalars | |
| The specified line is given by ``off + scl*x``. | |
| Returns | |
| ------- | |
| y : ndarray | |
| This module's representation of the Chebyshev series for | |
| ``off + scl*x``. | |
| See Also | |
| -------- | |
| numpy.polynomial.polynomial.polyline | |
| numpy.polynomial.legendre.legline | |
| numpy.polynomial.laguerre.lagline | |
| numpy.polynomial.hermite.hermline | |
| numpy.polynomial.hermite_e.hermeline | |
| Examples | |
| -------- | |
| >>> import numpy.polynomial.chebyshev as C | |
| >>> C.chebline(3,2) | |
| array([3, 2]) | |
| >>> C.chebval(-3, C.chebline(3,2)) # should be -3 | |
| -3.0 | |
| """ | |
| if scl != 0: | |
| return np.array([off, scl]) | |
| else: | |
| return np.array([off]) | |
| def chebfromroots(roots): | |
| """ | |
| Generate a Chebyshev series with given roots. | |
| The function returns the coefficients of the polynomial | |
| .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), | |
| in Chebyshev 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 | |
| .. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x) | |
| The coefficient of the last term is not generally 1 for monic | |
| polynomials in Chebyshev form. | |
| Parameters | |
| ---------- | |
| roots : array_like | |
| Sequence containing the roots. | |
| Returns | |
| ------- | |
| out : ndarray | |
| 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 | |
| -------- | |
| numpy.polynomial.polynomial.polyfromroots | |
| numpy.polynomial.legendre.legfromroots | |
| numpy.polynomial.laguerre.lagfromroots | |
| numpy.polynomial.hermite.hermfromroots | |
| numpy.polynomial.hermite_e.hermefromroots | |
| Examples | |
| -------- | |
| >>> import numpy.polynomial.chebyshev as C | |
| >>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis | |
| array([ 0. , -0.25, 0. , 0.25]) | |
| >>> j = complex(0,1) | |
| >>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis | |
| array([1.5+0.j, 0. +0.j, 0.5+0.j]) | |
| """ | |
| return pu._fromroots(chebline, chebmul, roots) | |
| def chebadd(c1, c2): | |
| """ | |
| Add one Chebyshev series to another. | |
| Returns the sum of two Chebyshev series `c1` + `c2`. The arguments | |
| are sequences of coefficients ordered from lowest order term to | |
| highest, i.e., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``. | |
| Parameters | |
| ---------- | |
| c1, c2 : array_like | |
| 1-D arrays of Chebyshev series coefficients ordered from low to | |
| high. | |
| Returns | |
| ------- | |
| out : ndarray | |
| Array representing the Chebyshev series of their sum. | |
| See Also | |
| -------- | |
| chebsub, chebmulx, chebmul, chebdiv, chebpow | |
| Notes | |
| ----- | |
| Unlike multiplication, division, etc., the sum of two Chebyshev series | |
| is a Chebyshev series (without having to "reproject" the result onto | |
| the basis set) so addition, just like that of "standard" polynomials, | |
| is simply "component-wise." | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import chebyshev as C | |
| >>> c1 = (1,2,3) | |
| >>> c2 = (3,2,1) | |
| >>> C.chebadd(c1,c2) | |
| array([4., 4., 4.]) | |
| """ | |
| return pu._add(c1, c2) | |
| def chebsub(c1, c2): | |
| """ | |
| Subtract one Chebyshev series from another. | |
| Returns the difference of two Chebyshev series `c1` - `c2`. The | |
| sequences of coefficients are from lowest order term to highest, i.e., | |
| [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``. | |
| Parameters | |
| ---------- | |
| c1, c2 : array_like | |
| 1-D arrays of Chebyshev series coefficients ordered from low to | |
| high. | |
| Returns | |
| ------- | |
| out : ndarray | |
| Of Chebyshev series coefficients representing their difference. | |
| See Also | |
| -------- | |
| chebadd, chebmulx, chebmul, chebdiv, chebpow | |
| Notes | |
| ----- | |
| Unlike multiplication, division, etc., the difference of two Chebyshev | |
| series is a Chebyshev series (without having to "reproject" the result | |
| onto the basis set) so subtraction, just like that of "standard" | |
| polynomials, is simply "component-wise." | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import chebyshev as C | |
| >>> c1 = (1,2,3) | |
| >>> c2 = (3,2,1) | |
| >>> C.chebsub(c1,c2) | |
| array([-2., 0., 2.]) | |
| >>> C.chebsub(c2,c1) # -C.chebsub(c1,c2) | |
| array([ 2., 0., -2.]) | |
| """ | |
| return pu._sub(c1, c2) | |
| def chebmulx(c): | |
| """Multiply a Chebyshev series by x. | |
| Multiply the polynomial `c` by x, where x is the independent | |
| variable. | |
| Parameters | |
| ---------- | |
| c : array_like | |
| 1-D array of Chebyshev series coefficients ordered from low to | |
| high. | |
| Returns | |
| ------- | |
| out : ndarray | |
| Array representing the result of the multiplication. | |
| Notes | |
| ----- | |
| .. versionadded:: 1.5.0 | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import chebyshev as C | |
| >>> C.chebmulx([1,2,3]) | |
| array([1. , 2.5, 1. , 1.5]) | |
| """ | |
| # c is a trimmed copy | |
| [c] = pu.as_series([c]) | |
| # The zero series needs special treatment | |
| if len(c) == 1 and c[0] == 0: | |
| return c | |
| prd = np.empty(len(c) + 1, dtype=c.dtype) | |
| prd[0] = c[0]*0 | |
| prd[1] = c[0] | |
| if len(c) > 1: | |
| tmp = c[1:]/2 | |
| prd[2:] = tmp | |
| prd[0:-2] += tmp | |
| return prd | |
| def chebmul(c1, c2): | |
| """ | |
| Multiply one Chebyshev series by another. | |
| Returns the product of two Chebyshev series `c1` * `c2`. The arguments | |
| are sequences of coefficients, from lowest order "term" to highest, | |
| e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``. | |
| Parameters | |
| ---------- | |
| c1, c2 : array_like | |
| 1-D arrays of Chebyshev series coefficients ordered from low to | |
| high. | |
| Returns | |
| ------- | |
| out : ndarray | |
| Of Chebyshev series coefficients representing their product. | |
| See Also | |
| -------- | |
| chebadd, chebsub, chebmulx, chebdiv, chebpow | |
| Notes | |
| ----- | |
| In general, the (polynomial) product of two C-series results in terms | |
| that are not in the Chebyshev polynomial basis set. Thus, to express | |
| the product as a C-series, it is typically necessary to "reproject" | |
| the product onto said basis set, which typically produces | |
| "unintuitive live" (but correct) results; see Examples section below. | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import chebyshev as C | |
| >>> c1 = (1,2,3) | |
| >>> c2 = (3,2,1) | |
| >>> C.chebmul(c1,c2) # multiplication requires "reprojection" | |
| array([ 6.5, 12. , 12. , 4. , 1.5]) | |
| """ | |
| # c1, c2 are trimmed copies | |
| [c1, c2] = pu.as_series([c1, c2]) | |
| z1 = _cseries_to_zseries(c1) | |
| z2 = _cseries_to_zseries(c2) | |
| prd = _zseries_mul(z1, z2) | |
| ret = _zseries_to_cseries(prd) | |
| return pu.trimseq(ret) | |
| def chebdiv(c1, c2): | |
| """ | |
| Divide one Chebyshev series by another. | |
| Returns the quotient-with-remainder of two Chebyshev series | |
| `c1` / `c2`. The arguments are sequences of coefficients from lowest | |
| order "term" to highest, e.g., [1,2,3] represents the series | |
| ``T_0 + 2*T_1 + 3*T_2``. | |
| Parameters | |
| ---------- | |
| c1, c2 : array_like | |
| 1-D arrays of Chebyshev series coefficients ordered from low to | |
| high. | |
| Returns | |
| ------- | |
| [quo, rem] : ndarrays | |
| Of Chebyshev series coefficients representing the quotient and | |
| remainder. | |
| See Also | |
| -------- | |
| chebadd, chebsub, chebmulx, chebmul, chebpow | |
| Notes | |
| ----- | |
| In general, the (polynomial) division of one C-series by another | |
| results in quotient and remainder terms that are not in the Chebyshev | |
| polynomial basis set. Thus, to express these results as C-series, it | |
| is typically necessary to "reproject" the results onto said basis | |
| set, which typically produces "unintuitive" (but correct) results; | |
| see Examples section below. | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import chebyshev as C | |
| >>> c1 = (1,2,3) | |
| >>> c2 = (3,2,1) | |
| >>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not | |
| (array([3.]), array([-8., -4.])) | |
| >>> c2 = (0,1,2,3) | |
| >>> C.chebdiv(c2,c1) # neither "intuitive" | |
| (array([0., 2.]), array([-2., -4.])) | |
| """ | |
| # c1, c2 are trimmed copies | |
| [c1, c2] = pu.as_series([c1, c2]) | |
| if c2[-1] == 0: | |
| raise ZeroDivisionError() | |
| # note: this is more efficient than `pu._div(chebmul, c1, c2)` | |
| lc1 = len(c1) | |
| lc2 = len(c2) | |
| if lc1 < lc2: | |
| return c1[:1]*0, c1 | |
| elif lc2 == 1: | |
| return c1/c2[-1], c1[:1]*0 | |
| else: | |
| z1 = _cseries_to_zseries(c1) | |
| z2 = _cseries_to_zseries(c2) | |
| quo, rem = _zseries_div(z1, z2) | |
| quo = pu.trimseq(_zseries_to_cseries(quo)) | |
| rem = pu.trimseq(_zseries_to_cseries(rem)) | |
| return quo, rem | |
| def chebpow(c, pow, maxpower=16): | |
| """Raise a Chebyshev series to a power. | |
| Returns the Chebyshev 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 ``T_0 + 2*T_1 + 3*T_2.`` | |
| Parameters | |
| ---------- | |
| c : array_like | |
| 1-D array of Chebyshev series coefficients ordered from low to | |
| high. | |
| pow : integer | |
| Power to which the series will be raised | |
| maxpower : integer, optional | |
| Maximum power allowed. This is mainly to limit growth of the series | |
| to unmanageable size. Default is 16 | |
| Returns | |
| ------- | |
| coef : ndarray | |
| Chebyshev series of power. | |
| See Also | |
| -------- | |
| chebadd, chebsub, chebmulx, chebmul, chebdiv | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import chebyshev as C | |
| >>> C.chebpow([1, 2, 3, 4], 2) | |
| array([15.5, 22. , 16. , ..., 12.5, 12. , 8. ]) | |
| """ | |
| # note: this is more efficient than `pu._pow(chebmul, c1, c2)`, as it | |
| # avoids converting between z and c series repeatedly | |
| # c is a trimmed copy | |
| [c] = pu.as_series([c]) | |
| power = int(pow) | |
| if power != pow or power < 0: | |
| raise ValueError("Power must be a non-negative integer.") | |
| elif maxpower is not None and power > maxpower: | |
| raise ValueError("Power is too large") | |
| elif power == 0: | |
| return np.array([1], dtype=c.dtype) | |
| elif power == 1: | |
| return c | |
| else: | |
| # This can be made more efficient by using powers of two | |
| # in the usual way. | |
| zs = _cseries_to_zseries(c) | |
| prd = zs | |
| for i in range(2, power + 1): | |
| prd = np.convolve(prd, zs) | |
| return _zseries_to_cseries(prd) | |
| def chebder(c, m=1, scl=1, axis=0): | |
| """ | |
| Differentiate a Chebyshev series. | |
| Returns the Chebyshev 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*T_0 + 2*T_1 + 3*T_2`` | |
| while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + | |
| 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is | |
| ``y``. | |
| Parameters | |
| ---------- | |
| c : array_like | |
| Array of Chebyshev series coefficients. If c is multidimensional | |
| the different axis correspond to different variables with the | |
| degree in each axis given by the corresponding index. | |
| m : int, optional | |
| Number of derivatives taken, must be non-negative. (Default: 1) | |
| scl : scalar, optional | |
| 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 : int, optional | |
| Axis over which the derivative is taken. (Default: 0). | |
| .. versionadded:: 1.7.0 | |
| Returns | |
| ------- | |
| der : ndarray | |
| Chebyshev series of the derivative. | |
| See Also | |
| -------- | |
| chebint | |
| Notes | |
| ----- | |
| In general, the result of differentiating 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. | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import chebyshev as C | |
| >>> c = (1,2,3,4) | |
| >>> C.chebder(c) | |
| array([14., 12., 24.]) | |
| >>> C.chebder(c,3) | |
| array([96.]) | |
| >>> C.chebder(c,scl=-1) | |
| array([-14., -12., -24.]) | |
| >>> C.chebder(c,2,-1) | |
| array([12., 96.]) | |
| """ | |
| c = np.array(c, ndmin=1, copy=True) | |
| if c.dtype.char in '?bBhHiIlLqQpP': | |
| c = c.astype(np.double) | |
| cnt = pu._deprecate_as_int(m, "the order of derivation") | |
| iaxis = pu._deprecate_as_int(axis, "the axis") | |
| if cnt < 0: | |
| raise ValueError("The order of derivation must be non-negative") | |
| iaxis = normalize_axis_index(iaxis, c.ndim) | |
| if cnt == 0: | |
| return c | |
| c = np.moveaxis(c, iaxis, 0) | |
| n = len(c) | |
| if cnt >= n: | |
| c = c[:1]*0 | |
| else: | |
| for i in range(cnt): | |
| n = n - 1 | |
| c *= scl | |
| der = np.empty((n,) + c.shape[1:], dtype=c.dtype) | |
| for j in range(n, 2, -1): | |
| der[j - 1] = (2*j)*c[j] | |
| c[j - 2] += (j*c[j])/(j - 2) | |
| if n > 1: | |
| der[1] = 4*c[2] | |
| der[0] = c[1] | |
| c = der | |
| c = np.moveaxis(c, 0, iaxis) | |
| return c | |
| def chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0): | |
| """ | |
| Integrate a Chebyshev series. | |
| Returns the Chebyshev 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 ``T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]] | |
| represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + | |
| 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. | |
| Parameters | |
| ---------- | |
| c : array_like | |
| Array of Chebyshev series coefficients. If c is multidimensional | |
| the different axis correspond to different variables with the | |
| degree in each axis given by the corresponding index. | |
| m : int, optional | |
| Order of integration, must be positive. (Default: 1) | |
| k : {[], list, scalar}, optional | |
| Integration constant(s). The value of the first integral at zero | |
| is the first value in the list, the value of the second integral | |
| at zero 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. | |
| lbnd : scalar, optional | |
| The lower bound of the integral. (Default: 0) | |
| scl : scalar, optional | |
| Following each integration the result is *multiplied* by `scl` | |
| before the integration constant is added. (Default: 1) | |
| axis : int, optional | |
| Axis over which the integral is taken. (Default: 0). | |
| .. versionadded:: 1.7.0 | |
| Returns | |
| ------- | |
| S : ndarray | |
| C-series coefficients of the integral. | |
| Raises | |
| ------ | |
| ValueError | |
| If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or | |
| ``np.ndim(scl) != 0``. | |
| See Also | |
| -------- | |
| chebder | |
| Notes | |
| ----- | |
| 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 :math:`u = ax + b` in an integral relative to `x`. Then | |
| :math:`dx = du/a`, so one will need to set `scl` equal to | |
| :math:`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. | |
| Examples | |
| -------- | |
| >>> from numpy.polynomial import chebyshev as C | |
| >>> c = (1,2,3) | |
| >>> C.chebint(c) | |
| array([ 0.5, -0.5, 0.5, 0.5]) | |
| >>> C.chebint(c,3) | |
| array([ 0.03125 , -0.1875 , 0.04166667, -0.05208333, 0.01041667, # may vary | |
| 0.00625 ]) | |
| >>> C.chebint(c, k=3) | |
| array([ 3.5, -0.5, 0.5, 0.5]) | |
| >>> C.chebint(c,lbnd=-2) | |
| array([ 8.5, -0.5, 0.5, 0.5]) | |
| >>> C.chebint(c,scl=-2) | |
| array([-1., 1., -1., -1.]) | |
| """ | |
| c = np.array(c, ndmin=1, copy=True) | |
| if c.dtype.char in '?bBhHiIlLqQpP': | |
| c = c.astype(np.double) | |
| if not np.iterable(k): | |
| k = [k] | |
| cnt = pu._deprecate_as_int(m, "the order of integration") | |
| iaxis = pu._deprecate_as_int(axis, "the axis") | |
| if cnt < 0: | |
| raise ValueError("The order of integration must be non-negative") | |
| if len(k) > cnt: | |
| raise ValueError("Too many integration constants") | |
| if np.ndim(lbnd) != 0: | |
| raise ValueError("lbnd must be a scalar.") | |
| if np.ndim(scl) != 0: | |
| raise ValueError("scl must be a scalar.") | |
| iaxis = normalize_axis_index(iaxis, c.ndim) | |
| if cnt == 0: | |
| return c | |
| c = np.moveaxis(c, iaxis, 0) | |
| k = list(k) + [0]*(cnt - len(k)) | |
| for i in range(cnt): | |
| n = len(c) | |
| c *= scl | |
| if n == 1 and np.all(c[0] == 0): | |
| c[0] += k[i] | |
| else: | |
| tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype) | |
| tmp[0] = c[0]*0 | |
| tmp[1] = c[0] | |
| if n > 1: | |
| tmp[2] = c[1]/4 | |
| for j in range(2, n): | |
| tmp[j + 1] = c[j]/(2*(j + 1)) | |
| tmp[j - 1] -= c[j]/(2*(j - 1)) | |
| tmp[0] += k[i] - chebval(lbnd, tmp) | |
| c = tmp | |
| c = np.moveaxis(c, 0, iaxis) | |
| return c | |
| def chebval(x, c, tensor=True): | |
| """ | |
| Evaluate a Chebyshev series at points x. | |
| If `c` is of length `n + 1`, this function returns the value: | |
| .. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_n(x) | |
| 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. | |
| Parameters | |
| ---------- | |
| x : array_like, compatible object | |
| 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`. | |
| c : array_like | |
| 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`. | |
| tensor : boolean, optional | |
| 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. | |
| .. versionadded:: 1.7.0 | |
| Returns | |
| ------- | |
| values : ndarray, algebra_like | |
| The shape of the return value is described above. | |
| See Also | |
| -------- | |
| chebval2d, chebgrid2d, chebval3d, chebgrid3d | |
| Notes | |
| ----- | |
| The evaluation uses Clenshaw recursion, aka synthetic division. | |
| """ | |
| c = np.array(c, ndmin=1, copy=True) | |
| if c.dtype.char in '?bBhHiIlLqQpP': | |
| c = c.astype(np.double) | |
| if isinstance(x, (tuple, list)): | |
| x = np.asarray(x) | |
| if isinstance(x, np.ndarray) and tensor: | |
| c = c.reshape(c.shape + (1,)*x.ndim) | |
| if len(c) == 1: | |
| c0 = c[0] | |
| c1 = 0 | |
| elif len(c) == 2: | |
| c0 = c[0] | |
| c1 = c[1] | |
| else: | |
| x2 = 2*x | |
| c0 = c[-2] | |
| c1 = c[-1] | |
| for i in range(3, len(c) + 1): | |
| tmp = c0 | |
| c0 = c[-i] - c1 | |
| c1 = tmp + c1*x2 | |
| return c0 + c1*x | |
| def chebval2d(x, y, c): | |
| """ | |
| Evaluate a 2-D Chebyshev series at points (x, y). | |
| This function returns the values: | |
| .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * T_i(x) * T_j(y) | |
| 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. | |
| Parameters | |
| ---------- | |
| x, y : array_like, compatible objects | |
| 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. | |
| c : array_like | |
| 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 2 the remaining indices enumerate multiple | |
| sets of coefficients. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The values of the two dimensional Chebyshev series at points formed | |
| from pairs of corresponding values from `x` and `y`. | |
| See Also | |
| -------- | |
| chebval, chebgrid2d, chebval3d, chebgrid3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._valnd(chebval, c, x, y) | |
| def chebgrid2d(x, y, c): | |
| """ | |
| Evaluate a 2-D Chebyshev series on the Cartesian product of x and y. | |
| This function returns the values: | |
| .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * T_i(a) * T_j(b), | |
| 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. | |
| Parameters | |
| ---------- | |
| x, y : array_like, compatible objects | |
| 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. | |
| c : array_like | |
| 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. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The values of the two dimensional Chebyshev series at points in the | |
| Cartesian product of `x` and `y`. | |
| See Also | |
| -------- | |
| chebval, chebval2d, chebval3d, chebgrid3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._gridnd(chebval, c, x, y) | |
| def chebval3d(x, y, z, c): | |
| """ | |
| Evaluate a 3-D Chebyshev series at points (x, y, z). | |
| This function returns the values: | |
| .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * T_i(x) * T_j(y) * T_k(z) | |
| 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. | |
| Parameters | |
| ---------- | |
| x, y, z : array_like, compatible object | |
| 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. | |
| c : array_like | |
| 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. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The values of the multidimensional polynomial on points formed with | |
| triples of corresponding values from `x`, `y`, and `z`. | |
| See Also | |
| -------- | |
| chebval, chebval2d, chebgrid2d, chebgrid3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._valnd(chebval, c, x, y, z) | |
| def chebgrid3d(x, y, z, c): | |
| """ | |
| Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z. | |
| This function returns the values: | |
| .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * T_i(a) * T_j(b) * T_k(c) | |
| 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. | |
| Parameters | |
| ---------- | |
| x, y, z : array_like, compatible objects | |
| 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. | |
| c : array_like | |
| 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. | |
| Returns | |
| ------- | |
| values : ndarray, compatible object | |
| The values of the two dimensional polynomial at points in the Cartesian | |
| product of `x` and `y`. | |
| See Also | |
| -------- | |
| chebval, chebval2d, chebgrid2d, chebval3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._gridnd(chebval, c, x, y, z) | |
| def chebvander(x, deg): | |
| """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 | |
| .. math:: V[..., i] = T_i(x), | |
| where `0 <= i <= deg`. The leading indices of `V` index the elements of | |
| `x` and the last index is the degree of the Chebyshev polynomial. | |
| If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the | |
| matrix ``V = chebvander(x, n)``, then ``np.dot(V, c)`` and | |
| ``chebval(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 Chebyshev series of the same degree and sample points. | |
| Parameters | |
| ---------- | |
| x : array_like | |
| 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. | |
| deg : int | |
| Degree of the resulting matrix. | |
| Returns | |
| ------- | |
| vander : ndarray | |
| 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 Chebyshev polynomial. The dtype will be the same as | |
| the converted `x`. | |
| """ | |
| ideg = pu._deprecate_as_int(deg, "deg") | |
| if ideg < 0: | |
| raise ValueError("deg must be non-negative") | |
| x = np.array(x, copy=False, ndmin=1) + 0.0 | |
| dims = (ideg + 1,) + x.shape | |
| dtyp = x.dtype | |
| v = np.empty(dims, dtype=dtyp) | |
| # Use forward recursion to generate the entries. | |
| v[0] = x*0 + 1 | |
| if ideg > 0: | |
| x2 = 2*x | |
| v[1] = x | |
| for i in range(2, ideg + 1): | |
| v[i] = v[i-1]*x2 - v[i-2] | |
| return np.moveaxis(v, 0, -1) | |
| def chebvander2d(x, y, deg): | |
| """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 | |
| .. math:: V[..., (deg[1] + 1)*i + j] = T_i(x) * T_j(y), | |
| 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 Chebyshev polynomials. | |
| If ``V = chebvander2d(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 | |
| .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... | |
| and ``np.dot(V, c.flat)`` and ``chebval2d(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 Chebyshev | |
| series of the same degrees and sample points. | |
| Parameters | |
| ---------- | |
| x, y : array_like | |
| 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. | |
| deg : list of ints | |
| List of maximum degrees of the form [x_deg, y_deg]. | |
| Returns | |
| ------- | |
| vander2d : ndarray | |
| The shape of the returned matrix is ``x.shape + (order,)``, where | |
| :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same | |
| as the converted `x` and `y`. | |
| See Also | |
| -------- | |
| chebvander, chebvander3d, chebval2d, chebval3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._vander_nd_flat((chebvander, chebvander), (x, y), deg) | |
| def chebvander3d(x, y, z, deg): | |
| """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 | |
| .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = T_i(x)*T_j(y)*T_k(z), | |
| 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 Chebyshev polynomials. | |
| If ``V = chebvander3d(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 | |
| .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... | |
| and ``np.dot(V, c.flat)`` and ``chebval3d(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 Chebyshev | |
| series of the same degrees and sample points. | |
| Parameters | |
| ---------- | |
| x, y, z : array_like | |
| 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. | |
| deg : list of ints | |
| List of maximum degrees of the form [x_deg, y_deg, z_deg]. | |
| Returns | |
| ------- | |
| vander3d : ndarray | |
| The shape of the returned matrix is ``x.shape + (order,)``, where | |
| :math:`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 | |
| -------- | |
| chebvander, chebvander3d, chebval2d, chebval3d | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| return pu._vander_nd_flat((chebvander, chebvander, chebvander), (x, y, z), deg) | |
| def chebfit(x, y, deg, rcond=None, full=False, w=None): | |
| """ | |
| Least squares fit of Chebyshev series to data. | |
| Return the coefficients of a Chebyshev 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 | |
| .. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x), | |
| where `n` is `deg`. | |
| Parameters | |
| ---------- | |
| x : array_like, shape (M,) | |
| x-coordinates of the M sample points ``(x[i], y[i])``. | |
| y : array_like, shape (M,) or (M, K) | |
| 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. | |
| deg : int or 1-D array_like | |
| 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. | |
| rcond : float, optional | |
| 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. | |
| full : bool, optional | |
| 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. | |
| w : array_like, shape (`M`,), optional | |
| Weights. If not None, the contribution of each point | |
| ``(x[i],y[i])`` to the fit is weighted by ``w[i]``. Ideally the | |
| weights are chosen so that the errors of the products ``w[i]*y[i]`` | |
| all have the same variance. The default value is None. | |
| .. versionadded:: 1.5.0 | |
| Returns | |
| ------- | |
| coef : ndarray, shape (M,) or (M, K) | |
| Chebyshev 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`. | |
| [residuals, rank, singular_values, rcond] : list | |
| These values are only returned if `full` = True | |
| resid -- sum of squared residuals of the least squares fit | |
| rank -- the numerical rank of the scaled Vandermonde matrix | |
| sv -- singular values of the scaled Vandermonde matrix | |
| rcond -- value of `rcond`. | |
| For more details, see `numpy.linalg.lstsq`. | |
| Warns | |
| ----- | |
| RankWarning | |
| 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 | |
| >>> import warnings | |
| >>> warnings.simplefilter('ignore', np.RankWarning) | |
| See Also | |
| -------- | |
| numpy.polynomial.polynomial.polyfit | |
| numpy.polynomial.legendre.legfit | |
| numpy.polynomial.laguerre.lagfit | |
| numpy.polynomial.hermite.hermfit | |
| numpy.polynomial.hermite_e.hermefit | |
| chebval : Evaluates a Chebyshev series. | |
| chebvander : Vandermonde matrix of Chebyshev series. | |
| chebweight : Chebyshev weight function. | |
| numpy.linalg.lstsq : Computes a least-squares fit from the matrix. | |
| scipy.interpolate.UnivariateSpline : Computes spline fits. | |
| Notes | |
| ----- | |
| The solution is the coefficients of the Chebyshev series `p` that | |
| minimizes the sum of the weighted squared errors | |
| .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, | |
| where :math:`w_j` are the weights. This problem is solved by setting up | |
| as the (typically) overdetermined matrix equation | |
| .. math:: V(x) * c = w * y, | |
| 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 Chebyshev 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. | |
| References | |
| ---------- | |
| .. [1] Wikipedia, "Curve fitting", | |
| https://en.wikipedia.org/wiki/Curve_fitting | |
| Examples | |
| -------- | |
| """ | |
| return pu._fit(chebvander, x, y, deg, rcond, full, w) | |
| def chebcompanion(c): | |
| """Return the scaled companion matrix of c. | |
| The basis polynomials are scaled so that the companion matrix is | |
| symmetric when `c` is a Chebyshev 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. | |
| Parameters | |
| ---------- | |
| c : array_like | |
| 1-D array of Chebyshev series coefficients ordered from low to high | |
| degree. | |
| Returns | |
| ------- | |
| mat : ndarray | |
| Scaled companion matrix of dimensions (deg, deg). | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| # c is a trimmed copy | |
| [c] = pu.as_series([c]) | |
| if len(c) < 2: | |
| raise ValueError('Series must have maximum degree of at least 1.') | |
| if len(c) == 2: | |
| return np.array([[-c[0]/c[1]]]) | |
| n = len(c) - 1 | |
| mat = np.zeros((n, n), dtype=c.dtype) | |
| scl = np.array([1.] + [np.sqrt(.5)]*(n-1)) | |
| top = mat.reshape(-1)[1::n+1] | |
| bot = mat.reshape(-1)[n::n+1] | |
| top[0] = np.sqrt(.5) | |
| top[1:] = 1/2 | |
| bot[...] = top | |
| mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*.5 | |
| return mat | |
| def chebroots(c): | |
| """ | |
| Compute the roots of a Chebyshev series. | |
| Return the roots (a.k.a. "zeros") of the polynomial | |
| .. math:: p(x) = \\sum_i c[i] * T_i(x). | |
| Parameters | |
| ---------- | |
| c : 1-D array_like | |
| 1-D array of coefficients. | |
| Returns | |
| ------- | |
| out : ndarray | |
| Array of the roots of the series. If all the roots are real, | |
| then `out` is also real, otherwise it is complex. | |
| See Also | |
| -------- | |
| numpy.polynomial.polynomial.polyroots | |
| numpy.polynomial.legendre.legroots | |
| numpy.polynomial.laguerre.lagroots | |
| numpy.polynomial.hermite.hermroots | |
| numpy.polynomial.hermite_e.hermeroots | |
| Notes | |
| ----- | |
| 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 Chebyshev series basis polynomials aren't powers of `x` so the | |
| results of this function may seem unintuitive. | |
| Examples | |
| -------- | |
| >>> import numpy.polynomial.chebyshev as cheb | |
| >>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots | |
| array([ -5.00000000e-01, 2.60860684e-17, 1.00000000e+00]) # may vary | |
| """ | |
| # c is a trimmed copy | |
| [c] = pu.as_series([c]) | |
| if len(c) < 2: | |
| return np.array([], dtype=c.dtype) | |
| if len(c) == 2: | |
| return np.array([-c[0]/c[1]]) | |
| # rotated companion matrix reduces error | |
| m = chebcompanion(c)[::-1,::-1] | |
| r = la.eigvals(m) | |
| r.sort() | |
| return r | |
| def chebinterpolate(func, deg, args=()): | |
| """Interpolate a function at the Chebyshev points of the first kind. | |
| Returns the Chebyshev series that interpolates `func` at the Chebyshev | |
| points of the first kind in the interval [-1, 1]. The interpolating | |
| series tends to a minmax approximation to `func` with increasing `deg` | |
| if the function is continuous in the interval. | |
| .. versionadded:: 1.14.0 | |
| Parameters | |
| ---------- | |
| func : function | |
| The function to be approximated. It must be a function of a single | |
| variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are | |
| extra arguments passed in the `args` parameter. | |
| deg : int | |
| Degree of the interpolating polynomial | |
| args : tuple, optional | |
| Extra arguments to be used in the function call. Default is no extra | |
| arguments. | |
| Returns | |
| ------- | |
| coef : ndarray, shape (deg + 1,) | |
| Chebyshev coefficients of the interpolating series ordered from low to | |
| high. | |
| Examples | |
| -------- | |
| >>> import numpy.polynomial.chebyshev as C | |
| >>> C.chebfromfunction(lambda x: np.tanh(x) + 0.5, 8) | |
| array([ 5.00000000e-01, 8.11675684e-01, -9.86864911e-17, | |
| -5.42457905e-02, -2.71387850e-16, 4.51658839e-03, | |
| 2.46716228e-17, -3.79694221e-04, -3.26899002e-16]) | |
| Notes | |
| ----- | |
| The Chebyshev polynomials used in the interpolation are orthogonal when | |
| sampled at the Chebyshev points of the first kind. If it is desired to | |
| constrain some of the coefficients they can simply be set to the desired | |
| value after the interpolation, no new interpolation or fit is needed. This | |
| is especially useful if it is known apriori that some of coefficients are | |
| zero. For instance, if the function is even then the coefficients of the | |
| terms of odd degree in the result can be set to zero. | |
| """ | |
| deg = np.asarray(deg) | |
| # check arguments. | |
| if deg.ndim > 0 or deg.dtype.kind not in 'iu' or deg.size == 0: | |
| raise TypeError("deg must be an int") | |
| if deg < 0: | |
| raise ValueError("expected deg >= 0") | |
| order = deg + 1 | |
| xcheb = chebpts1(order) | |
| yfunc = func(xcheb, *args) | |
| m = chebvander(xcheb, deg) | |
| c = np.dot(m.T, yfunc) | |
| c[0] /= order | |
| c[1:] /= 0.5*order | |
| return c | |
| def chebgauss(deg): | |
| """ | |
| Gauss-Chebyshev quadrature. | |
| Computes the sample points and weights for Gauss-Chebyshev quadrature. | |
| These sample points and weights will correctly integrate polynomials of | |
| degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with | |
| the weight function :math:`f(x) = 1/\\sqrt{1 - x^2}`. | |
| Parameters | |
| ---------- | |
| deg : int | |
| Number of sample points and weights. It must be >= 1. | |
| Returns | |
| ------- | |
| x : ndarray | |
| 1-D ndarray containing the sample points. | |
| y : ndarray | |
| 1-D ndarray containing the weights. | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| The results have only been tested up to degree 100, higher degrees may | |
| be problematic. For Gauss-Chebyshev there are closed form solutions for | |
| the sample points and weights. If n = `deg`, then | |
| .. math:: x_i = \\cos(\\pi (2 i - 1) / (2 n)) | |
| .. math:: w_i = \\pi / n | |
| """ | |
| ideg = pu._deprecate_as_int(deg, "deg") | |
| if ideg <= 0: | |
| raise ValueError("deg must be a positive integer") | |
| x = np.cos(np.pi * np.arange(1, 2*ideg, 2) / (2.0*ideg)) | |
| w = np.ones(ideg)*(np.pi/ideg) | |
| return x, w | |
| def chebweight(x): | |
| """ | |
| The weight function of the Chebyshev polynomials. | |
| The weight function is :math:`1/\\sqrt{1 - x^2}` and the interval of | |
| integration is :math:`[-1, 1]`. The Chebyshev polynomials are | |
| orthogonal, but not normalized, with respect to this weight function. | |
| Parameters | |
| ---------- | |
| x : array_like | |
| Values at which the weight function will be computed. | |
| Returns | |
| ------- | |
| w : ndarray | |
| The weight function at `x`. | |
| Notes | |
| ----- | |
| .. versionadded:: 1.7.0 | |
| """ | |
| w = 1./(np.sqrt(1. + x) * np.sqrt(1. - x)) | |
| return w | |
| def chebpts1(npts): | |
| """ | |
| Chebyshev points of the first kind. | |
| The Chebyshev points of the first kind are the points ``cos(x)``, | |
| where ``x = [pi*(k + .5)/npts for k in range(npts)]``. | |
| Parameters | |
| ---------- | |
| npts : int | |
| Number of sample points desired. | |
| Returns | |
| ------- | |
| pts : ndarray | |
| The Chebyshev points of the first kind. | |
| See Also | |
| -------- | |
| chebpts2 | |
| Notes | |
| ----- | |
| .. versionadded:: 1.5.0 | |
| """ | |
| _npts = int(npts) | |
| if _npts != npts: | |
| raise ValueError("npts must be integer") | |
| if _npts < 1: | |
| raise ValueError("npts must be >= 1") | |
| x = 0.5 * np.pi / _npts * np.arange(-_npts+1, _npts+1, 2) | |
| return np.sin(x) | |
| def chebpts2(npts): | |
| """ | |
| Chebyshev points of the second kind. | |
| The Chebyshev points of the second kind are the points ``cos(x)``, | |
| where ``x = [pi*k/(npts - 1) for k in range(npts)]``. | |
| Parameters | |
| ---------- | |
| npts : int | |
| Number of sample points desired. | |
| Returns | |
| ------- | |
| pts : ndarray | |
| The Chebyshev points of the second kind. | |
| Notes | |
| ----- | |
| .. versionadded:: 1.5.0 | |
| """ | |
| _npts = int(npts) | |
| if _npts != npts: | |
| raise ValueError("npts must be integer") | |
| if _npts < 2: | |
| raise ValueError("npts must be >= 2") | |
| x = np.linspace(-np.pi, 0, _npts) | |
| return np.cos(x) | |
| # | |
| # Chebyshev series class | |
| # | |
| class Chebyshev(ABCPolyBase): | |
| """A Chebyshev series class. | |
| The Chebyshev class provides the standard Python numerical methods | |
| '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the | |
| methods listed below. | |
| Parameters | |
| ---------- | |
| coef : array_like | |
| Chebyshev coefficients in order of increasing degree, i.e., | |
| ``(1, 2, 3)`` gives ``1*T_0(x) + 2*T_1(x) + 3*T_2(x)``. | |
| domain : (2,) array_like, optional | |
| Domain to use. The interval ``[domain[0], domain[1]]`` is mapped | |
| to the interval ``[window[0], window[1]]`` by shifting and scaling. | |
| The default value is [-1, 1]. | |
| window : (2,) array_like, optional | |
| Window, see `domain` for its use. The default value is [-1, 1]. | |
| .. versionadded:: 1.6.0 | |
| """ | |
| # Virtual Functions | |
| _add = staticmethod(chebadd) | |
| _sub = staticmethod(chebsub) | |
| _mul = staticmethod(chebmul) | |
| _div = staticmethod(chebdiv) | |
| _pow = staticmethod(chebpow) | |
| _val = staticmethod(chebval) | |
| _int = staticmethod(chebint) | |
| _der = staticmethod(chebder) | |
| _fit = staticmethod(chebfit) | |
| _line = staticmethod(chebline) | |
| _roots = staticmethod(chebroots) | |
| _fromroots = staticmethod(chebfromroots) | |
| def interpolate(cls, func, deg, domain=None, args=()): | |
| """Interpolate a function at the Chebyshev points of the first kind. | |
| Returns the series that interpolates `func` at the Chebyshev points of | |
| the first kind scaled and shifted to the `domain`. The resulting series | |
| tends to a minmax approximation of `func` when the function is | |
| continuous in the domain. | |
| .. versionadded:: 1.14.0 | |
| Parameters | |
| ---------- | |
| func : function | |
| The function to be interpolated. It must be a function of a single | |
| variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are | |
| extra arguments passed in the `args` parameter. | |
| deg : int | |
| Degree of the interpolating polynomial. | |
| domain : {None, [beg, end]}, optional | |
| Domain over which `func` is interpolated. The default is None, in | |
| which case the domain is [-1, 1]. | |
| args : tuple, optional | |
| Extra arguments to be used in the function call. Default is no | |
| extra arguments. | |
| Returns | |
| ------- | |
| polynomial : Chebyshev instance | |
| Interpolating Chebyshev instance. | |
| Notes | |
| ----- | |
| See `numpy.polynomial.chebfromfunction` for more details. | |
| """ | |
| if domain is None: | |
| domain = cls.domain | |
| xfunc = lambda x: func(pu.mapdomain(x, cls.window, domain), *args) | |
| coef = chebinterpolate(xfunc, deg) | |
| return cls(coef, domain=domain) | |
| # Virtual properties | |
| domain = np.array(chebdomain) | |
| window = np.array(chebdomain) | |
| basis_name = 'T' | |