File size: 36,366 Bytes
7885a28 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 |
import warnings
import numpy as np
from scipy.special import factorial
from scipy._lib._util import (_asarray_validated, float_factorial, check_random_state,
_transition_to_rng)
__all__ = ["KroghInterpolator", "krogh_interpolate",
"BarycentricInterpolator", "barycentric_interpolate",
"approximate_taylor_polynomial"]
def _isscalar(x):
"""Check whether x is if a scalar type, or 0-dim"""
return np.isscalar(x) or hasattr(x, 'shape') and x.shape == ()
class _Interpolator1D:
"""
Common features in univariate interpolation
Deal with input data type and interpolation axis rolling. The
actual interpolator can assume the y-data is of shape (n, r) where
`n` is the number of x-points, and `r` the number of variables,
and use self.dtype as the y-data type.
Attributes
----------
_y_axis
Axis along which the interpolation goes in the original array
_y_extra_shape
Additional trailing shape of the input arrays, excluding
the interpolation axis.
dtype
Dtype of the y-data arrays. Can be set via _set_dtype, which
forces it to be float or complex.
Methods
-------
__call__
_prepare_x
_finish_y
_reshape_yi
_set_yi
_set_dtype
_evaluate
"""
__slots__ = ('_y_axis', '_y_extra_shape', 'dtype')
def __init__(self, xi=None, yi=None, axis=None):
self._y_axis = axis
self._y_extra_shape = None
self.dtype = None
if yi is not None:
self._set_yi(yi, xi=xi, axis=axis)
def __call__(self, x):
"""
Evaluate the interpolant
Parameters
----------
x : array_like
Point or points at which to evaluate the interpolant.
Returns
-------
y : array_like
Interpolated values. Shape is determined by replacing
the interpolation axis in the original array with the shape of `x`.
Notes
-----
Input values `x` must be convertible to `float` values like `int`
or `float`.
"""
x, x_shape = self._prepare_x(x)
y = self._evaluate(x)
return self._finish_y(y, x_shape)
def _evaluate(self, x):
"""
Actually evaluate the value of the interpolator.
"""
raise NotImplementedError()
def _prepare_x(self, x):
"""Reshape input x array to 1-D"""
x = _asarray_validated(x, check_finite=False, as_inexact=True)
x_shape = x.shape
return x.ravel(), x_shape
def _finish_y(self, y, x_shape):
"""Reshape interpolated y back to an N-D array similar to initial y"""
y = y.reshape(x_shape + self._y_extra_shape)
if self._y_axis != 0 and x_shape != ():
nx = len(x_shape)
ny = len(self._y_extra_shape)
s = (list(range(nx, nx + self._y_axis))
+ list(range(nx)) + list(range(nx+self._y_axis, nx+ny)))
y = y.transpose(s)
return y
def _reshape_yi(self, yi, check=False):
yi = np.moveaxis(np.asarray(yi), self._y_axis, 0)
if check and yi.shape[1:] != self._y_extra_shape:
ok_shape = (f"{self._y_extra_shape[-self._y_axis:]!r} + (N,) + "
f"{self._y_extra_shape[:-self._y_axis]!r}")
raise ValueError(f"Data must be of shape {ok_shape}")
return yi.reshape((yi.shape[0], -1))
def _set_yi(self, yi, xi=None, axis=None):
if axis is None:
axis = self._y_axis
if axis is None:
raise ValueError("no interpolation axis specified")
yi = np.asarray(yi)
shape = yi.shape
if shape == ():
shape = (1,)
if xi is not None and shape[axis] != len(xi):
raise ValueError("x and y arrays must be equal in length along "
"interpolation axis.")
self._y_axis = (axis % yi.ndim)
self._y_extra_shape = yi.shape[:self._y_axis] + yi.shape[self._y_axis+1:]
self.dtype = None
self._set_dtype(yi.dtype)
def _set_dtype(self, dtype, union=False):
if np.issubdtype(dtype, np.complexfloating) \
or np.issubdtype(self.dtype, np.complexfloating):
self.dtype = np.complex128
else:
if not union or self.dtype != np.complex128:
self.dtype = np.float64
class _Interpolator1DWithDerivatives(_Interpolator1D):
def derivatives(self, x, der=None):
"""
Evaluate several derivatives of the polynomial at the point `x`
Produce an array of derivatives evaluated at the point `x`.
Parameters
----------
x : array_like
Point or points at which to evaluate the derivatives
der : int or list or None, optional
How many derivatives to evaluate, or None for all potentially
nonzero derivatives (that is, a number equal to the number
of points), or a list of derivatives to evaluate. This number
includes the function value as the '0th' derivative.
Returns
-------
d : ndarray
Array with derivatives; ``d[j]`` contains the jth derivative.
Shape of ``d[j]`` is determined by replacing the interpolation
axis in the original array with the shape of `x`.
Examples
--------
>>> from scipy.interpolate import KroghInterpolator
>>> KroghInterpolator([0,0,0],[1,2,3]).derivatives(0)
array([1.0,2.0,3.0])
>>> KroghInterpolator([0,0,0],[1,2,3]).derivatives([0,0])
array([[1.0,1.0],
[2.0,2.0],
[3.0,3.0]])
"""
x, x_shape = self._prepare_x(x)
y = self._evaluate_derivatives(x, der)
y = y.reshape((y.shape[0],) + x_shape + self._y_extra_shape)
if self._y_axis != 0 and x_shape != ():
nx = len(x_shape)
ny = len(self._y_extra_shape)
s = ([0] + list(range(nx+1, nx + self._y_axis+1))
+ list(range(1, nx+1)) +
list(range(nx+1+self._y_axis, nx+ny+1)))
y = y.transpose(s)
return y
def derivative(self, x, der=1):
"""
Evaluate a single derivative of the polynomial at the point `x`.
Parameters
----------
x : array_like
Point or points at which to evaluate the derivatives
der : integer, optional
Which derivative to evaluate (default: first derivative).
This number includes the function value as 0th derivative.
Returns
-------
d : ndarray
Derivative interpolated at the x-points. Shape of `d` is
determined by replacing the interpolation axis in the
original array with the shape of `x`.
Notes
-----
This may be computed by evaluating all derivatives up to the desired
one (using self.derivatives()) and then discarding the rest.
"""
x, x_shape = self._prepare_x(x)
y = self._evaluate_derivatives(x, der+1)
return self._finish_y(y[der], x_shape)
def _evaluate_derivatives(self, x, der=None):
"""
Actually evaluate the derivatives.
Parameters
----------
x : array_like
1D array of points at which to evaluate the derivatives
der : integer, optional
The number of derivatives to evaluate, from 'order 0' (der=1)
to order der-1. If omitted, return all possibly-non-zero
derivatives, ie 0 to order n-1.
Returns
-------
d : ndarray
Array of shape ``(der, x.size, self.yi.shape[1])`` containing
the derivatives from 0 to der-1
"""
raise NotImplementedError()
class KroghInterpolator(_Interpolator1DWithDerivatives):
"""
Interpolating polynomial for a set of points.
The polynomial passes through all the pairs ``(xi, yi)``. One may
additionally specify a number of derivatives at each point `xi`;
this is done by repeating the value `xi` and specifying the
derivatives as successive `yi` values.
Allows evaluation of the polynomial and all its derivatives.
For reasons of numerical stability, this function does not compute
the coefficients of the polynomial, although they can be obtained
by evaluating all the derivatives.
Parameters
----------
xi : array_like, shape (npoints, )
Known x-coordinates. Must be sorted in increasing order.
yi : array_like, shape (..., npoints, ...)
Known y-coordinates. When an xi occurs two or more times in
a row, the corresponding yi's represent derivative values. The length of `yi`
along the interpolation axis must be equal to the length of `xi`. Use the
`axis` parameter to select the correct axis.
axis : int, optional
Axis in the `yi` array corresponding to the x-coordinate values. Defaults to
``axis=0``.
Notes
-----
Be aware that the algorithms implemented here are not necessarily
the most numerically stable known. Moreover, even in a world of
exact computation, unless the x coordinates are chosen very
carefully - Chebyshev zeros (e.g., cos(i*pi/n)) are a good choice -
polynomial interpolation itself is a very ill-conditioned process
due to the Runge phenomenon. In general, even with well-chosen
x values, degrees higher than about thirty cause problems with
numerical instability in this code.
Based on [1]_.
References
----------
.. [1] Krogh, "Efficient Algorithms for Polynomial Interpolation
and Numerical Differentiation", 1970.
Examples
--------
To produce a polynomial that is zero at 0 and 1 and has
derivative 2 at 0, call
>>> from scipy.interpolate import KroghInterpolator
>>> KroghInterpolator([0,0,1],[0,2,0])
This constructs the quadratic :math:`2x^2-2x`. The derivative condition
is indicated by the repeated zero in the `xi` array; the corresponding
yi values are 0, the function value, and 2, the derivative value.
For another example, given `xi`, `yi`, and a derivative `ypi` for each
point, appropriate arrays can be constructed as:
>>> import numpy as np
>>> rng = np.random.default_rng()
>>> xi = np.linspace(0, 1, 5)
>>> yi, ypi = rng.random((2, 5))
>>> xi_k, yi_k = np.repeat(xi, 2), np.ravel(np.dstack((yi,ypi)))
>>> KroghInterpolator(xi_k, yi_k)
To produce a vector-valued polynomial, supply a higher-dimensional
array for `yi`:
>>> KroghInterpolator([0,1],[[2,3],[4,5]])
This constructs a linear polynomial giving (2,3) at 0 and (4,5) at 1.
"""
def __init__(self, xi, yi, axis=0):
super().__init__(xi, yi, axis)
self.xi = np.asarray(xi)
self.yi = self._reshape_yi(yi)
self.n, self.r = self.yi.shape
if (deg := self.xi.size) > 30:
warnings.warn(f"{deg} degrees provided, degrees higher than about"
" thirty cause problems with numerical instability "
"with 'KroghInterpolator'", stacklevel=2)
c = np.zeros((self.n+1, self.r), dtype=self.dtype)
c[0] = self.yi[0]
Vk = np.zeros((self.n, self.r), dtype=self.dtype)
for k in range(1, self.n):
s = 0
while s <= k and xi[k-s] == xi[k]:
s += 1
s -= 1
Vk[0] = self.yi[k]/float_factorial(s)
for i in range(k-s):
if xi[i] == xi[k]:
raise ValueError("Elements of `xi` can't be equal.")
if s == 0:
Vk[i+1] = (c[i]-Vk[i])/(xi[i]-xi[k])
else:
Vk[i+1] = (Vk[i+1]-Vk[i])/(xi[i]-xi[k])
c[k] = Vk[k-s]
self.c = c
def _evaluate(self, x):
pi = 1
p = np.zeros((len(x), self.r), dtype=self.dtype)
p += self.c[0,np.newaxis,:]
for k in range(1, self.n):
w = x - self.xi[k-1]
pi = w*pi
p += pi[:,np.newaxis] * self.c[k]
return p
def _evaluate_derivatives(self, x, der=None):
n = self.n
r = self.r
if der is None:
der = self.n
pi = np.zeros((n, len(x)))
w = np.zeros((n, len(x)))
pi[0] = 1
p = np.zeros((len(x), self.r), dtype=self.dtype)
p += self.c[0, np.newaxis, :]
for k in range(1, n):
w[k-1] = x - self.xi[k-1]
pi[k] = w[k-1] * pi[k-1]
p += pi[k, :, np.newaxis] * self.c[k]
cn = np.zeros((max(der, n+1), len(x), r), dtype=self.dtype)
cn[:n+1, :, :] += self.c[:n+1, np.newaxis, :]
cn[0] = p
for k in range(1, n):
for i in range(1, n-k+1):
pi[i] = w[k+i-1]*pi[i-1] + pi[i]
cn[k] = cn[k] + pi[i, :, np.newaxis]*cn[k+i]
cn[k] *= float_factorial(k)
cn[n, :, :] = 0
return cn[:der]
def krogh_interpolate(xi, yi, x, der=0, axis=0):
"""
Convenience function for polynomial interpolation.
See `KroghInterpolator` for more details.
Parameters
----------
xi : array_like
Interpolation points (known x-coordinates).
yi : array_like
Known y-coordinates, of shape ``(xi.size, R)``. Interpreted as
vectors of length R, or scalars if R=1.
x : array_like
Point or points at which to evaluate the derivatives.
der : int or list or None, optional
How many derivatives to evaluate, or None for all potentially
nonzero derivatives (that is, a number equal to the number
of points), or a list of derivatives to evaluate. This number
includes the function value as the '0th' derivative.
axis : int, optional
Axis in the `yi` array corresponding to the x-coordinate values.
Returns
-------
d : ndarray
If the interpolator's values are R-D then the
returned array will be the number of derivatives by N by R.
If `x` is a scalar, the middle dimension will be dropped; if
the `yi` are scalars then the last dimension will be dropped.
See Also
--------
KroghInterpolator : Krogh interpolator
Notes
-----
Construction of the interpolating polynomial is a relatively expensive
process. If you want to evaluate it repeatedly consider using the class
KroghInterpolator (which is what this function uses).
Examples
--------
We can interpolate 2D observed data using Krogh interpolation:
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.interpolate import krogh_interpolate
>>> x_observed = np.linspace(0.0, 10.0, 11)
>>> y_observed = np.sin(x_observed)
>>> x = np.linspace(min(x_observed), max(x_observed), num=100)
>>> y = krogh_interpolate(x_observed, y_observed, x)
>>> plt.plot(x_observed, y_observed, "o", label="observation")
>>> plt.plot(x, y, label="krogh interpolation")
>>> plt.legend()
>>> plt.show()
"""
P = KroghInterpolator(xi, yi, axis=axis)
if der == 0:
return P(x)
elif _isscalar(der):
return P.derivative(x, der=der)
else:
return P.derivatives(x, der=np.amax(der)+1)[der]
def approximate_taylor_polynomial(f,x,degree,scale,order=None):
"""
Estimate the Taylor polynomial of f at x by polynomial fitting.
Parameters
----------
f : callable
The function whose Taylor polynomial is sought. Should accept
a vector of `x` values.
x : scalar
The point at which the polynomial is to be evaluated.
degree : int
The degree of the Taylor polynomial
scale : scalar
The width of the interval to use to evaluate the Taylor polynomial.
Function values spread over a range this wide are used to fit the
polynomial. Must be chosen carefully.
order : int or None, optional
The order of the polynomial to be used in the fitting; `f` will be
evaluated ``order+1`` times. If None, use `degree`.
Returns
-------
p : poly1d instance
The Taylor polynomial (translated to the origin, so that
for example p(0)=f(x)).
Notes
-----
The appropriate choice of "scale" is a trade-off; too large and the
function differs from its Taylor polynomial too much to get a good
answer, too small and round-off errors overwhelm the higher-order terms.
The algorithm used becomes numerically unstable around order 30 even
under ideal circumstances.
Choosing order somewhat larger than degree may improve the higher-order
terms.
Examples
--------
We can calculate Taylor approximation polynomials of sin function with
various degrees:
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.interpolate import approximate_taylor_polynomial
>>> x = np.linspace(-10.0, 10.0, num=100)
>>> plt.plot(x, np.sin(x), label="sin curve")
>>> for degree in np.arange(1, 15, step=2):
... sin_taylor = approximate_taylor_polynomial(np.sin, 0, degree, 1,
... order=degree + 2)
... plt.plot(x, sin_taylor(x), label=f"degree={degree}")
>>> plt.legend(bbox_to_anchor=(1.05, 1), loc='upper left',
... borderaxespad=0.0, shadow=True)
>>> plt.tight_layout()
>>> plt.axis([-10, 10, -10, 10])
>>> plt.show()
"""
if order is None:
order = degree
n = order+1
# Choose n points that cluster near the endpoints of the interval in
# a way that avoids the Runge phenomenon. Ensure, by including the
# endpoint or not as appropriate, that one point always falls at x
# exactly.
xs = scale*np.cos(np.linspace(0,np.pi,n,endpoint=n % 1)) + x
P = KroghInterpolator(xs, f(xs))
d = P.derivatives(x,der=degree+1)
return np.poly1d((d/factorial(np.arange(degree+1)))[::-1])
class BarycentricInterpolator(_Interpolator1DWithDerivatives):
r"""Interpolating polynomial for a set of points.
Constructs a polynomial that passes through a given set of points.
Allows evaluation of the polynomial and all its derivatives,
efficient changing of the y-values to be interpolated,
and updating by adding more x- and y-values.
For reasons of numerical stability, this function does not compute
the coefficients of the polynomial.
The values `yi` need to be provided before the function is
evaluated, but none of the preprocessing depends on them, so rapid
updates are possible.
Parameters
----------
xi : array_like, shape (npoints, )
1-D array of x coordinates of the points the polynomial
should pass through
yi : array_like, shape (..., npoints, ...), optional
N-D array of y coordinates of the points the polynomial should pass through.
If None, the y values will be supplied later via the `set_y` method.
The length of `yi` along the interpolation axis must be equal to the length
of `xi`. Use the ``axis`` parameter to select correct axis.
axis : int, optional
Axis in the yi array corresponding to the x-coordinate values. Defaults
to ``axis=0``.
wi : array_like, optional
The barycentric weights for the chosen interpolation points `xi`.
If absent or None, the weights will be computed from `xi` (default).
This allows for the reuse of the weights `wi` if several interpolants
are being calculated using the same nodes `xi`, without re-computation.
rng : {None, int, `numpy.random.Generator`}, optional
If `rng` is passed by keyword, types other than `numpy.random.Generator` are
passed to `numpy.random.default_rng` to instantiate a ``Generator``.
If `rng` is already a ``Generator`` instance, then the provided instance is
used. Specify `rng` for repeatable interpolation.
If this argument `random_state` is passed by keyword,
legacy behavior for the argument `random_state` applies:
- If `random_state` is None (or `numpy.random`), the `numpy.random.RandomState`
singleton is used.
- If `random_state` is an int, a new ``RandomState`` instance is used,
seeded with `random_state`.
- If `random_state` is already a ``Generator`` or ``RandomState`` instance then
that instance is used.
.. versionchanged:: 1.15.0
As part of the `SPEC-007 <https://scientific-python.org/specs/spec-0007/>`_
transition from use of `numpy.random.RandomState` to
`numpy.random.Generator` this keyword was changed from `random_state` to `rng`.
For an interim period, both keywords will continue to work (only specify
one of them). After the interim period using the `random_state` keyword will emit
warnings. The behavior of the `random_state` and `rng` keywords is outlined above.
Notes
-----
This class uses a "barycentric interpolation" method that treats
the problem as a special case of rational function interpolation.
This algorithm is quite stable, numerically, but even in a world of
exact computation, unless the x coordinates are chosen very
carefully - Chebyshev zeros (e.g., cos(i*pi/n)) are a good choice -
polynomial interpolation itself is a very ill-conditioned process
due to the Runge phenomenon.
Based on Berrut and Trefethen 2004, "Barycentric Lagrange Interpolation".
Examples
--------
To produce a quintic barycentric interpolant approximating the function
:math:`\sin x`, and its first four derivatives, using six randomly-spaced
nodes in :math:`(0, \frac{\pi}{2})`:
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.interpolate import BarycentricInterpolator
>>> rng = np.random.default_rng()
>>> xi = rng.random(6) * np.pi/2
>>> f, f_d1, f_d2, f_d3, f_d4 = np.sin, np.cos, lambda x: -np.sin(x), lambda x: -np.cos(x), np.sin
>>> P = BarycentricInterpolator(xi, f(xi), random_state=rng)
>>> fig, axs = plt.subplots(5, 1, sharex=True, layout='constrained', figsize=(7,10))
>>> x = np.linspace(0, np.pi, 100)
>>> axs[0].plot(x, P(x), 'r:', x, f(x), 'k--', xi, f(xi), 'xk')
>>> axs[1].plot(x, P.derivative(x), 'r:', x, f_d1(x), 'k--', xi, f_d1(xi), 'xk')
>>> axs[2].plot(x, P.derivative(x, 2), 'r:', x, f_d2(x), 'k--', xi, f_d2(xi), 'xk')
>>> axs[3].plot(x, P.derivative(x, 3), 'r:', x, f_d3(x), 'k--', xi, f_d3(xi), 'xk')
>>> axs[4].plot(x, P.derivative(x, 4), 'r:', x, f_d4(x), 'k--', xi, f_d4(xi), 'xk')
>>> axs[0].set_xlim(0, np.pi)
>>> axs[4].set_xlabel(r"$x$")
>>> axs[4].set_xticks([i * np.pi / 4 for i in range(5)],
... ["0", r"$\frac{\pi}{4}$", r"$\frac{\pi}{2}$", r"$\frac{3\pi}{4}$", r"$\pi$"])
>>> axs[0].set_ylabel("$f(x)$")
>>> axs[1].set_ylabel("$f'(x)$")
>>> axs[2].set_ylabel("$f''(x)$")
>>> axs[3].set_ylabel("$f^{(3)}(x)$")
>>> axs[4].set_ylabel("$f^{(4)}(x)$")
>>> labels = ['Interpolation nodes', 'True function $f$', 'Barycentric interpolation']
>>> axs[0].legend(axs[0].get_lines()[::-1], labels, bbox_to_anchor=(0., 1.02, 1., .102),
... loc='lower left', ncols=3, mode="expand", borderaxespad=0., frameon=False)
>>> plt.show()
""" # numpy/numpydoc#87 # noqa: E501
@_transition_to_rng("random_state", replace_doc=False)
def __init__(self, xi, yi=None, axis=0, *, wi=None, rng=None):
super().__init__(xi, yi, axis)
rng = check_random_state(rng)
self.xi = np.asarray(xi, dtype=np.float64)
self.set_yi(yi)
self.n = len(self.xi)
# cache derivative object to avoid re-computing the weights with every call.
self._diff_cij = None
if wi is not None:
self.wi = wi
else:
# See page 510 of Berrut and Trefethen 2004 for an explanation of the
# capacity scaling and the suggestion of using a random permutation of
# the input factors.
# At the moment, the permutation is not performed for xi that are
# appended later through the add_xi interface. It's not clear to me how
# to implement that and it seems that most situations that require
# these numerical stability improvements will be able to provide all
# the points to the constructor.
self._inv_capacity = 4.0 / (np.max(self.xi) - np.min(self.xi))
permute = rng.permutation(self.n, )
inv_permute = np.zeros(self.n, dtype=np.int32)
inv_permute[permute] = np.arange(self.n)
self.wi = np.zeros(self.n)
for i in range(self.n):
dist = self._inv_capacity * (self.xi[i] - self.xi[permute])
dist[inv_permute[i]] = 1.0
prod = np.prod(dist)
if prod == 0.0:
raise ValueError("Interpolation points xi must be"
" distinct.")
self.wi[i] = 1.0 / prod
def set_yi(self, yi, axis=None):
"""
Update the y values to be interpolated
The barycentric interpolation algorithm requires the calculation
of weights, but these depend only on the `xi`. The `yi` can be changed
at any time.
Parameters
----------
yi : array_like
The y-coordinates of the points the polynomial will pass through.
If None, the y values must be supplied later.
axis : int, optional
Axis in the `yi` array corresponding to the x-coordinate values.
"""
if yi is None:
self.yi = None
return
self._set_yi(yi, xi=self.xi, axis=axis)
self.yi = self._reshape_yi(yi)
self.n, self.r = self.yi.shape
self._diff_baryint = None
def add_xi(self, xi, yi=None):
"""
Add more x values to the set to be interpolated
The barycentric interpolation algorithm allows easy updating by
adding more points for the polynomial to pass through.
Parameters
----------
xi : array_like
The x coordinates of the points that the polynomial should pass
through.
yi : array_like, optional
The y coordinates of the points the polynomial should pass through.
Should have shape ``(xi.size, R)``; if R > 1 then the polynomial is
vector-valued.
If `yi` is not given, the y values will be supplied later. `yi`
should be given if and only if the interpolator has y values
specified.
Notes
-----
The new points added by `add_xi` are not randomly permuted
so there is potential for numerical instability,
especially for a large number of points. If this
happens, please reconstruct interpolation from scratch instead.
"""
if yi is not None:
if self.yi is None:
raise ValueError("No previous yi value to update!")
yi = self._reshape_yi(yi, check=True)
self.yi = np.vstack((self.yi,yi))
else:
if self.yi is not None:
raise ValueError("No update to yi provided!")
old_n = self.n
self.xi = np.concatenate((self.xi,xi))
self.n = len(self.xi)
self.wi **= -1
old_wi = self.wi
self.wi = np.zeros(self.n)
self.wi[:old_n] = old_wi
for j in range(old_n, self.n):
self.wi[:j] *= self._inv_capacity * (self.xi[j]-self.xi[:j])
self.wi[j] = np.multiply.reduce(
self._inv_capacity * (self.xi[:j]-self.xi[j])
)
self.wi **= -1
self._diff_cij = None
self._diff_baryint = None
def __call__(self, x):
"""Evaluate the interpolating polynomial at the points x
Parameters
----------
x : array_like
Point or points at which to evaluate the interpolant.
Returns
-------
y : array_like
Interpolated values. Shape is determined by replacing
the interpolation axis in the original array with the shape of `x`.
Notes
-----
Currently the code computes an outer product between `x` and the
weights, that is, it constructs an intermediate array of size
``(N, len(x))``, where N is the degree of the polynomial.
"""
return _Interpolator1D.__call__(self, x)
def _evaluate(self, x):
if x.size == 0:
p = np.zeros((0, self.r), dtype=self.dtype)
else:
c = x[..., np.newaxis] - self.xi
z = c == 0
c[z] = 1
c = self.wi / c
with np.errstate(divide='ignore'):
p = np.dot(c, self.yi) / np.sum(c, axis=-1)[..., np.newaxis]
# Now fix where x==some xi
r = np.nonzero(z)
if len(r) == 1: # evaluation at a scalar
if len(r[0]) > 0: # equals one of the points
p = self.yi[r[0][0]]
else:
p[r[:-1]] = self.yi[r[-1]]
return p
def derivative(self, x, der=1):
"""
Evaluate a single derivative of the polynomial at the point x.
Parameters
----------
x : array_like
Point or points at which to evaluate the derivatives
der : integer, optional
Which derivative to evaluate (default: first derivative).
This number includes the function value as 0th derivative.
Returns
-------
d : ndarray
Derivative interpolated at the x-points. Shape of `d` is
determined by replacing the interpolation axis in the
original array with the shape of `x`.
"""
x, x_shape = self._prepare_x(x)
y = self._evaluate_derivatives(x, der+1, all_lower=False)
return self._finish_y(y, x_shape)
def _evaluate_derivatives(self, x, der=None, all_lower=True):
# NB: der here is not the order of the highest derivative;
# instead, it is the size of the derivatives matrix that
# would be returned with all_lower=True, including the
# '0th' derivative (the undifferentiated function).
# E.g. to evaluate the 5th derivative alone, call
# _evaluate_derivatives(x, der=6, all_lower=False).
if (not all_lower) and (x.size == 0 or self.r == 0):
return np.zeros((0, self.r), dtype=self.dtype)
if (not all_lower) and der == 1:
return self._evaluate(x)
if (not all_lower) and (der > self.n):
return np.zeros((len(x), self.r), dtype=self.dtype)
if der is None:
der = self.n
if all_lower and (x.size == 0 or self.r == 0):
return np.zeros((der, len(x), self.r), dtype=self.dtype)
if self._diff_cij is None:
# c[i,j] = xi[i] - xi[j]
c = self.xi[:, np.newaxis] - self.xi
# avoid division by 0 (diagonal entries are so far zero by construction)
np.fill_diagonal(c, 1)
# c[i,j] = (w[j] / w[i]) / (xi[i] - xi[j]) (equation 9.4)
c = self.wi/ (c * self.wi[..., np.newaxis])
# fill in correct diagonal entries: each column sums to 0
np.fill_diagonal(c, 0)
# calculate diagonal
# c[j,j] = -sum_{i != j} c[i,j] (equation 9.5)
d = -c.sum(axis=1)
# c[i,j] = l_j(x_i)
np.fill_diagonal(c, d)
self._diff_cij = c
if self._diff_baryint is None:
# initialise and cache derivative interpolator and cijs;
# reuse weights wi (which depend only on interpolation points xi),
# to avoid unnecessary re-computation
self._diff_baryint = BarycentricInterpolator(xi=self.xi,
yi=self._diff_cij @ self.yi,
wi=self.wi)
self._diff_baryint._diff_cij = self._diff_cij
if all_lower:
# assemble matrix of derivatives from order 0 to order der-1,
# in the format required by _Interpolator1DWithDerivatives.
cn = np.zeros((der, len(x), self.r), dtype=self.dtype)
for d in range(der):
cn[d, :, :] = self._evaluate_derivatives(x, d+1, all_lower=False)
return cn
# recursively evaluate only the derivative requested
return self._diff_baryint._evaluate_derivatives(x, der-1, all_lower=False)
def barycentric_interpolate(xi, yi, x, axis=0, *, der=0, rng=None):
"""
Convenience function for polynomial interpolation.
Constructs a polynomial that passes through a given set of points,
then evaluates the polynomial. For reasons of numerical stability,
this function does not compute the coefficients of the polynomial.
This function uses a "barycentric interpolation" method that treats
the problem as a special case of rational function interpolation.
This algorithm is quite stable, numerically, but even in a world of
exact computation, unless the `x` coordinates are chosen very
carefully - Chebyshev zeros (e.g., cos(i*pi/n)) are a good choice -
polynomial interpolation itself is a very ill-conditioned process
due to the Runge phenomenon.
Parameters
----------
xi : array_like
1-D array of x coordinates of the points the polynomial should
pass through
yi : array_like
The y coordinates of the points the polynomial should pass through.
x : scalar or array_like
Point or points at which to evaluate the interpolant.
axis : int, optional
Axis in the `yi` array corresponding to the x-coordinate values.
der : int or list or None, optional
How many derivatives to evaluate, or None for all potentially
nonzero derivatives (that is, a number equal to the number
of points), or a list of derivatives to evaluate. This number
includes the function value as the '0th' derivative.
rng : `numpy.random.Generator`, optional
Pseudorandom number generator state. When `rng` is None, a new
`numpy.random.Generator` is created using entropy from the
operating system. Types other than `numpy.random.Generator` are
passed to `numpy.random.default_rng` to instantiate a ``Generator``.
Returns
-------
y : scalar or array_like
Interpolated values. Shape is determined by replacing
the interpolation axis in the original array with the shape of `x`.
See Also
--------
BarycentricInterpolator : Barycentric interpolator
Notes
-----
Construction of the interpolation weights is a relatively slow process.
If you want to call this many times with the same xi (but possibly
varying yi or x) you should use the class `BarycentricInterpolator`.
This is what this function uses internally.
Examples
--------
We can interpolate 2D observed data using barycentric interpolation:
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.interpolate import barycentric_interpolate
>>> x_observed = np.linspace(0.0, 10.0, 11)
>>> y_observed = np.sin(x_observed)
>>> x = np.linspace(min(x_observed), max(x_observed), num=100)
>>> y = barycentric_interpolate(x_observed, y_observed, x)
>>> plt.plot(x_observed, y_observed, "o", label="observation")
>>> plt.plot(x, y, label="barycentric interpolation")
>>> plt.legend()
>>> plt.show()
"""
P = BarycentricInterpolator(xi, yi, axis=axis, rng=rng)
if der == 0:
return P(x)
elif _isscalar(der):
return P.derivative(x, der=der)
else:
return P.derivatives(x, der=np.amax(der)+1)[der]
|