File size: 39,892 Bytes
6370773 |
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 |
"""
Abstract base class for the various polynomial Classes.
The ABCPolyBase class provides the methods needed to implement the common API
for the various polynomial classes. It operates as a mixin, but uses the
abc module from the stdlib, hence it is only available for Python >= 2.6.
"""
import os
import abc
import numbers
from typing import Callable
import numpy as np
from . import polyutils as pu
__all__ = ['ABCPolyBase']
class ABCPolyBase(abc.ABC):
"""An abstract base class for immutable series classes.
ABCPolyBase provides the standard Python numerical methods
'+', '-', '*', '//', '%', 'divmod', '**', and '()' along with the
methods listed below.
.. versionadded:: 1.9.0
Parameters
----------
coef : array_like
Series coefficients in order of increasing degree, i.e.,
``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``, where
``P_i`` is the basis polynomials of degree ``i``.
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 the derived class domain.
window : (2,) array_like, optional
Window, see domain for its use. The default value is the
derived class window.
symbol : str, optional
Symbol used to represent the independent variable in string
representations of the polynomial expression, e.g. for printing.
The symbol must be a valid Python identifier. Default value is 'x'.
.. versionadded:: 1.24
Attributes
----------
coef : (N,) ndarray
Series coefficients in order of increasing degree.
domain : (2,) ndarray
Domain that is mapped to window.
window : (2,) ndarray
Window that domain is mapped to.
symbol : str
Symbol representing the independent variable.
Class Attributes
----------------
maxpower : int
Maximum power allowed, i.e., the largest number ``n`` such that
``p(x)**n`` is allowed. This is to limit runaway polynomial size.
domain : (2,) ndarray
Default domain of the class.
window : (2,) ndarray
Default window of the class.
"""
# Not hashable
__hash__ = None
# Opt out of numpy ufuncs and Python ops with ndarray subclasses.
__array_ufunc__ = None
# Limit runaway size. T_n^m has degree n*m
maxpower = 100
# Unicode character mappings for improved __str__
_superscript_mapping = str.maketrans({
"0": "⁰",
"1": "¹",
"2": "²",
"3": "³",
"4": "⁴",
"5": "⁵",
"6": "⁶",
"7": "⁷",
"8": "⁸",
"9": "⁹"
})
_subscript_mapping = str.maketrans({
"0": "₀",
"1": "₁",
"2": "₂",
"3": "₃",
"4": "₄",
"5": "₅",
"6": "₆",
"7": "₇",
"8": "₈",
"9": "₉"
})
# Some fonts don't support full unicode character ranges necessary for
# the full set of superscripts and subscripts, including common/default
# fonts in Windows shells/terminals. Therefore, default to ascii-only
# printing on windows.
_use_unicode = not os.name == 'nt'
@property
def symbol(self):
return self._symbol
@property
@abc.abstractmethod
def domain(self):
pass
@property
@abc.abstractmethod
def window(self):
pass
@property
@abc.abstractmethod
def basis_name(self):
pass
@staticmethod
@abc.abstractmethod
def _add(c1, c2):
pass
@staticmethod
@abc.abstractmethod
def _sub(c1, c2):
pass
@staticmethod
@abc.abstractmethod
def _mul(c1, c2):
pass
@staticmethod
@abc.abstractmethod
def _div(c1, c2):
pass
@staticmethod
@abc.abstractmethod
def _pow(c, pow, maxpower=None):
pass
@staticmethod
@abc.abstractmethod
def _val(x, c):
pass
@staticmethod
@abc.abstractmethod
def _int(c, m, k, lbnd, scl):
pass
@staticmethod
@abc.abstractmethod
def _der(c, m, scl):
pass
@staticmethod
@abc.abstractmethod
def _fit(x, y, deg, rcond, full):
pass
@staticmethod
@abc.abstractmethod
def _line(off, scl):
pass
@staticmethod
@abc.abstractmethod
def _roots(c):
pass
@staticmethod
@abc.abstractmethod
def _fromroots(r):
pass
def has_samecoef(self, other):
"""Check if coefficients match.
.. versionadded:: 1.6.0
Parameters
----------
other : class instance
The other class must have the ``coef`` attribute.
Returns
-------
bool : boolean
True if the coefficients are the same, False otherwise.
"""
if len(self.coef) != len(other.coef):
return False
elif not np.all(self.coef == other.coef):
return False
else:
return True
def has_samedomain(self, other):
"""Check if domains match.
.. versionadded:: 1.6.0
Parameters
----------
other : class instance
The other class must have the ``domain`` attribute.
Returns
-------
bool : boolean
True if the domains are the same, False otherwise.
"""
return np.all(self.domain == other.domain)
def has_samewindow(self, other):
"""Check if windows match.
.. versionadded:: 1.6.0
Parameters
----------
other : class instance
The other class must have the ``window`` attribute.
Returns
-------
bool : boolean
True if the windows are the same, False otherwise.
"""
return np.all(self.window == other.window)
def has_sametype(self, other):
"""Check if types match.
.. versionadded:: 1.7.0
Parameters
----------
other : object
Class instance.
Returns
-------
bool : boolean
True if other is same class as self
"""
return isinstance(other, self.__class__)
def _get_coefficients(self, other):
"""Interpret other as polynomial coefficients.
The `other` argument is checked to see if it is of the same
class as self with identical domain and window. If so,
return its coefficients, otherwise return `other`.
.. versionadded:: 1.9.0
Parameters
----------
other : anything
Object to be checked.
Returns
-------
coef
The coefficients of`other` if it is a compatible instance,
of ABCPolyBase, otherwise `other`.
Raises
------
TypeError
When `other` is an incompatible instance of ABCPolyBase.
"""
if isinstance(other, ABCPolyBase):
if not isinstance(other, self.__class__):
raise TypeError("Polynomial types differ")
elif not np.all(self.domain == other.domain):
raise TypeError("Domains differ")
elif not np.all(self.window == other.window):
raise TypeError("Windows differ")
elif self.symbol != other.symbol:
raise ValueError("Polynomial symbols differ")
return other.coef
return other
def __init__(self, coef, domain=None, window=None, symbol='x'):
[coef] = pu.as_series([coef], trim=False)
self.coef = coef
if domain is not None:
[domain] = pu.as_series([domain], trim=False)
if len(domain) != 2:
raise ValueError("Domain has wrong number of elements.")
self.domain = domain
if window is not None:
[window] = pu.as_series([window], trim=False)
if len(window) != 2:
raise ValueError("Window has wrong number of elements.")
self.window = window
# Validation for symbol
try:
if not symbol.isidentifier():
raise ValueError(
"Symbol string must be a valid Python identifier"
)
# If a user passes in something other than a string, the above
# results in an AttributeError. Catch this and raise a more
# informative exception
except AttributeError:
raise TypeError("Symbol must be a non-empty string")
self._symbol = symbol
def __repr__(self):
coef = repr(self.coef)[6:-1]
domain = repr(self.domain)[6:-1]
window = repr(self.window)[6:-1]
name = self.__class__.__name__
return (f"{name}({coef}, domain={domain}, window={window}, "
f"symbol='{self.symbol}')")
def __format__(self, fmt_str):
if fmt_str == '':
return self.__str__()
if fmt_str not in ('ascii', 'unicode'):
raise ValueError(
f"Unsupported format string '{fmt_str}' passed to "
f"{self.__class__}.__format__. Valid options are "
f"'ascii' and 'unicode'"
)
if fmt_str == 'ascii':
return self._generate_string(self._str_term_ascii)
return self._generate_string(self._str_term_unicode)
def __str__(self):
if self._use_unicode:
return self._generate_string(self._str_term_unicode)
return self._generate_string(self._str_term_ascii)
def _generate_string(self, term_method):
"""
Generate the full string representation of the polynomial, using
``term_method`` to generate each polynomial term.
"""
# Get configuration for line breaks
linewidth = np.get_printoptions().get('linewidth', 75)
if linewidth < 1:
linewidth = 1
out = pu.format_float(self.coef[0])
off, scale = self.mapparms()
scaled_symbol, needs_parens = self._format_term(pu.format_float,
off, scale)
if needs_parens:
scaled_symbol = '(' + scaled_symbol + ')'
for i, coef in enumerate(self.coef[1:]):
out += " "
power = str(i + 1)
# Polynomial coefficient
# The coefficient array can be an object array with elements that
# will raise a TypeError with >= 0 (e.g. strings or Python
# complex). In this case, represent the coefficient as-is.
try:
if coef >= 0:
next_term = "+ " + pu.format_float(coef, parens=True)
else:
next_term = "- " + pu.format_float(-coef, parens=True)
except TypeError:
next_term = f"+ {coef}"
# Polynomial term
next_term += term_method(power, scaled_symbol)
# Length of the current line with next term added
line_len = len(out.split('\n')[-1]) + len(next_term)
# If not the last term in the polynomial, it will be two
# characters longer due to the +/- with the next term
if i < len(self.coef[1:]) - 1:
line_len += 2
# Handle linebreaking
if line_len >= linewidth:
next_term = next_term.replace(" ", "\n", 1)
out += next_term
return out
@classmethod
def _str_term_unicode(cls, i, arg_str):
"""
String representation of single polynomial term using unicode
characters for superscripts and subscripts.
"""
if cls.basis_name is None:
raise NotImplementedError(
"Subclasses must define either a basis_name, or override "
"_str_term_unicode(cls, i, arg_str)"
)
return (f"·{cls.basis_name}{i.translate(cls._subscript_mapping)}"
f"({arg_str})")
@classmethod
def _str_term_ascii(cls, i, arg_str):
"""
String representation of a single polynomial term using ** and _ to
represent superscripts and subscripts, respectively.
"""
if cls.basis_name is None:
raise NotImplementedError(
"Subclasses must define either a basis_name, or override "
"_str_term_ascii(cls, i, arg_str)"
)
return f" {cls.basis_name}_{i}({arg_str})"
@classmethod
def _repr_latex_term(cls, i, arg_str, needs_parens):
if cls.basis_name is None:
raise NotImplementedError(
"Subclasses must define either a basis name, or override "
"_repr_latex_term(i, arg_str, needs_parens)")
# since we always add parens, we don't care if the expression needs them
return f"{{{cls.basis_name}}}_{{{i}}}({arg_str})"
@staticmethod
def _repr_latex_scalar(x, parens=False):
# TODO: we're stuck with disabling math formatting until we handle
# exponents in this function
return r'\text{{{}}}'.format(pu.format_float(x, parens=parens))
def _format_term(self, scalar_format: Callable, off: float, scale: float):
""" Format a single term in the expansion """
if off == 0 and scale == 1:
term = self.symbol
needs_parens = False
elif scale == 1:
term = f"{scalar_format(off)} + {self.symbol}"
needs_parens = True
elif off == 0:
term = f"{scalar_format(scale)}{self.symbol}"
needs_parens = True
else:
term = (
f"{scalar_format(off)} + "
f"{scalar_format(scale)}{self.symbol}"
)
needs_parens = True
return term, needs_parens
def _repr_latex_(self):
# get the scaled argument string to the basis functions
off, scale = self.mapparms()
term, needs_parens = self._format_term(self._repr_latex_scalar,
off, scale)
mute = r"\color{{LightGray}}{{{}}}".format
parts = []
for i, c in enumerate(self.coef):
# prevent duplication of + and - signs
if i == 0:
coef_str = f"{self._repr_latex_scalar(c)}"
elif not isinstance(c, numbers.Real):
coef_str = f" + ({self._repr_latex_scalar(c)})"
elif c >= 0:
coef_str = f" + {self._repr_latex_scalar(c, parens=True)}"
else:
coef_str = f" - {self._repr_latex_scalar(-c, parens=True)}"
# produce the string for the term
term_str = self._repr_latex_term(i, term, needs_parens)
if term_str == '1':
part = coef_str
else:
part = rf"{coef_str}\,{term_str}"
if c == 0:
part = mute(part)
parts.append(part)
if parts:
body = ''.join(parts)
else:
# in case somehow there are no coefficients at all
body = '0'
return rf"${self.symbol} \mapsto {body}$"
# Pickle and copy
def __getstate__(self):
ret = self.__dict__.copy()
ret['coef'] = self.coef.copy()
ret['domain'] = self.domain.copy()
ret['window'] = self.window.copy()
ret['symbol'] = self.symbol
return ret
def __setstate__(self, dict):
self.__dict__ = dict
# Call
def __call__(self, arg):
arg = pu.mapdomain(arg, self.domain, self.window)
return self._val(arg, self.coef)
def __iter__(self):
return iter(self.coef)
def __len__(self):
return len(self.coef)
# Numeric properties.
def __neg__(self):
return self.__class__(
-self.coef, self.domain, self.window, self.symbol
)
def __pos__(self):
return self
def __add__(self, other):
othercoef = self._get_coefficients(other)
try:
coef = self._add(self.coef, othercoef)
except Exception:
return NotImplemented
return self.__class__(coef, self.domain, self.window, self.symbol)
def __sub__(self, other):
othercoef = self._get_coefficients(other)
try:
coef = self._sub(self.coef, othercoef)
except Exception:
return NotImplemented
return self.__class__(coef, self.domain, self.window, self.symbol)
def __mul__(self, other):
othercoef = self._get_coefficients(other)
try:
coef = self._mul(self.coef, othercoef)
except Exception:
return NotImplemented
return self.__class__(coef, self.domain, self.window, self.symbol)
def __truediv__(self, other):
# there is no true divide if the rhs is not a Number, although it
# could return the first n elements of an infinite series.
# It is hard to see where n would come from, though.
if not isinstance(other, numbers.Number) or isinstance(other, bool):
raise TypeError(
f"unsupported types for true division: "
f"'{type(self)}', '{type(other)}'"
)
return self.__floordiv__(other)
def __floordiv__(self, other):
res = self.__divmod__(other)
if res is NotImplemented:
return res
return res[0]
def __mod__(self, other):
res = self.__divmod__(other)
if res is NotImplemented:
return res
return res[1]
def __divmod__(self, other):
othercoef = self._get_coefficients(other)
try:
quo, rem = self._div(self.coef, othercoef)
except ZeroDivisionError:
raise
except Exception:
return NotImplemented
quo = self.__class__(quo, self.domain, self.window, self.symbol)
rem = self.__class__(rem, self.domain, self.window, self.symbol)
return quo, rem
def __pow__(self, other):
coef = self._pow(self.coef, other, maxpower=self.maxpower)
res = self.__class__(coef, self.domain, self.window, self.symbol)
return res
def __radd__(self, other):
try:
coef = self._add(other, self.coef)
except Exception:
return NotImplemented
return self.__class__(coef, self.domain, self.window, self.symbol)
def __rsub__(self, other):
try:
coef = self._sub(other, self.coef)
except Exception:
return NotImplemented
return self.__class__(coef, self.domain, self.window, self.symbol)
def __rmul__(self, other):
try:
coef = self._mul(other, self.coef)
except Exception:
return NotImplemented
return self.__class__(coef, self.domain, self.window, self.symbol)
def __rdiv__(self, other):
# set to __floordiv__ /.
return self.__rfloordiv__(other)
def __rtruediv__(self, other):
# An instance of ABCPolyBase is not considered a
# Number.
return NotImplemented
def __rfloordiv__(self, other):
res = self.__rdivmod__(other)
if res is NotImplemented:
return res
return res[0]
def __rmod__(self, other):
res = self.__rdivmod__(other)
if res is NotImplemented:
return res
return res[1]
def __rdivmod__(self, other):
try:
quo, rem = self._div(other, self.coef)
except ZeroDivisionError:
raise
except Exception:
return NotImplemented
quo = self.__class__(quo, self.domain, self.window, self.symbol)
rem = self.__class__(rem, self.domain, self.window, self.symbol)
return quo, rem
def __eq__(self, other):
res = (isinstance(other, self.__class__) and
np.all(self.domain == other.domain) and
np.all(self.window == other.window) and
(self.coef.shape == other.coef.shape) and
np.all(self.coef == other.coef) and
(self.symbol == other.symbol))
return res
def __ne__(self, other):
return not self.__eq__(other)
#
# Extra methods.
#
def copy(self):
"""Return a copy.
Returns
-------
new_series : series
Copy of self.
"""
return self.__class__(self.coef, self.domain, self.window, self.symbol)
def degree(self):
"""The degree of the series.
.. versionadded:: 1.5.0
Returns
-------
degree : int
Degree of the series, one less than the number of coefficients.
Examples
--------
Create a polynomial object for ``1 + 7*x + 4*x**2``:
>>> poly = np.polynomial.Polynomial([1, 7, 4])
>>> print(poly)
1.0 + 7.0·x + 4.0·x²
>>> poly.degree()
2
Note that this method does not check for non-zero coefficients.
You must trim the polynomial to remove any trailing zeroes:
>>> poly = np.polynomial.Polynomial([1, 7, 0])
>>> print(poly)
1.0 + 7.0·x + 0.0·x²
>>> poly.degree()
2
>>> poly.trim().degree()
1
"""
return len(self) - 1
def cutdeg(self, deg):
"""Truncate series to the given degree.
Reduce the degree of the series to `deg` by discarding the
high order terms. If `deg` is greater than the current degree a
copy of the current series is returned. This can be useful in least
squares where the coefficients of the high degree terms may be very
small.
.. versionadded:: 1.5.0
Parameters
----------
deg : non-negative int
The series is reduced to degree `deg` by discarding the high
order terms. The value of `deg` must be a non-negative integer.
Returns
-------
new_series : series
New instance of series with reduced degree.
"""
return self.truncate(deg + 1)
def trim(self, tol=0):
"""Remove trailing coefficients
Remove trailing coefficients until a coefficient is reached whose
absolute value greater than `tol` or the beginning of the series is
reached. If all the coefficients would be removed the series is set
to ``[0]``. A new series instance is returned with the new
coefficients. The current instance remains unchanged.
Parameters
----------
tol : non-negative number.
All trailing coefficients less than `tol` will be removed.
Returns
-------
new_series : series
New instance of series with trimmed coefficients.
"""
coef = pu.trimcoef(self.coef, tol)
return self.__class__(coef, self.domain, self.window, self.symbol)
def truncate(self, size):
"""Truncate series to length `size`.
Reduce the series to length `size` by discarding the high
degree terms. The value of `size` must be a positive integer. This
can be useful in least squares where the coefficients of the
high degree terms may be very small.
Parameters
----------
size : positive int
The series is reduced to length `size` by discarding the high
degree terms. The value of `size` must be a positive integer.
Returns
-------
new_series : series
New instance of series with truncated coefficients.
"""
isize = int(size)
if isize != size or isize < 1:
raise ValueError("size must be a positive integer")
if isize >= len(self.coef):
coef = self.coef
else:
coef = self.coef[:isize]
return self.__class__(coef, self.domain, self.window, self.symbol)
def convert(self, domain=None, kind=None, window=None):
"""Convert series to a different kind and/or domain and/or window.
Parameters
----------
domain : array_like, optional
The domain of the converted series. If the value is None,
the default domain of `kind` is used.
kind : class, optional
The polynomial series type class to which the current instance
should be converted. If kind is None, then the class of the
current instance is used.
window : array_like, optional
The window of the converted series. If the value is None,
the default window of `kind` is used.
Returns
-------
new_series : series
The returned class can be of different type than the current
instance and/or have a different domain and/or different
window.
Notes
-----
Conversion between domains and class types can result in
numerically ill defined series.
"""
if kind is None:
kind = self.__class__
if domain is None:
domain = kind.domain
if window is None:
window = kind.window
return self(kind.identity(domain, window=window, symbol=self.symbol))
def mapparms(self):
"""Return the mapping parameters.
The returned values define a linear map ``off + scl*x`` that is
applied to the input arguments before the series is evaluated. The
map depends on the ``domain`` and ``window``; if the current
``domain`` is equal to the ``window`` the resulting map is the
identity. If the coefficients of the series instance are to be
used by themselves outside this class, then the linear function
must be substituted for the ``x`` in the standard representation of
the base polynomials.
Returns
-------
off, scl : float or complex
The mapping function is defined by ``off + scl*x``.
Notes
-----
If the current domain is the interval ``[l1, r1]`` and the window
is ``[l2, r2]``, then the linear mapping function ``L`` is
defined by the equations::
L(l1) = l2
L(r1) = r2
"""
return pu.mapparms(self.domain, self.window)
def integ(self, m=1, k=[], lbnd=None):
"""Integrate.
Return a series instance that is the definite integral of the
current series.
Parameters
----------
m : non-negative int
The number of integrations to perform.
k : array_like
Integration constants. The first constant is applied to the
first integration, the second to the second, and so on. The
list of values must less than or equal to `m` in length and any
missing values are set to zero.
lbnd : Scalar
The lower bound of the definite integral.
Returns
-------
new_series : series
A new series representing the integral. The domain is the same
as the domain of the integrated series.
"""
off, scl = self.mapparms()
if lbnd is None:
lbnd = 0
else:
lbnd = off + scl*lbnd
coef = self._int(self.coef, m, k, lbnd, 1./scl)
return self.__class__(coef, self.domain, self.window, self.symbol)
def deriv(self, m=1):
"""Differentiate.
Return a series instance of that is the derivative of the current
series.
Parameters
----------
m : non-negative int
Find the derivative of order `m`.
Returns
-------
new_series : series
A new series representing the derivative. The domain is the same
as the domain of the differentiated series.
"""
off, scl = self.mapparms()
coef = self._der(self.coef, m, scl)
return self.__class__(coef, self.domain, self.window, self.symbol)
def roots(self):
"""Return the roots of the series polynomial.
Compute the roots for the series. Note that the accuracy of the
roots decreases the further outside the `domain` they lie.
Returns
-------
roots : ndarray
Array containing the roots of the series.
"""
roots = self._roots(self.coef)
return pu.mapdomain(roots, self.window, self.domain)
def linspace(self, n=100, domain=None):
"""Return x, y values at equally spaced points in domain.
Returns the x, y values at `n` linearly spaced points across the
domain. Here y is the value of the polynomial at the points x. By
default the domain is the same as that of the series instance.
This method is intended mostly as a plotting aid.
.. versionadded:: 1.5.0
Parameters
----------
n : int, optional
Number of point pairs to return. The default value is 100.
domain : {None, array_like}, optional
If not None, the specified domain is used instead of that of
the calling instance. It should be of the form ``[beg,end]``.
The default is None which case the class domain is used.
Returns
-------
x, y : ndarray
x is equal to linspace(self.domain[0], self.domain[1], n) and
y is the series evaluated at element of x.
"""
if domain is None:
domain = self.domain
x = np.linspace(domain[0], domain[1], n)
y = self(x)
return x, y
@classmethod
def fit(cls, x, y, deg, domain=None, rcond=None, full=False, w=None,
window=None, symbol='x'):
"""Least squares fit to data.
Return a series instance that is the least squares fit to the data
`y` sampled at `x`. The domain of the returned instance can be
specified and this will often result in a superior fit with less
chance of ill conditioning.
Parameters
----------
x : array_like, shape (M,)
x-coordinates of the M sample points ``(x[i], y[i])``.
y : array_like, shape (M,)
y-coordinates of the M sample points ``(x[i], y[i])``.
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.
domain : {None, [beg, end], []}, optional
Domain to use for the returned series. If ``None``,
then a minimal domain that covers the points `x` is chosen. If
``[]`` the class domain is used. The default value was the
class domain in NumPy 1.4 and ``None`` in later versions.
The ``[]`` option was added in numpy 1.5.0.
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 weight ``w[i]`` applies to the unsquared
residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
chosen so that the errors of the products ``w[i]*y[i]`` all have
the same variance. When using inverse-variance weighting, use
``w[i] = 1/sigma(y[i])``. The default value is None.
.. versionadded:: 1.5.0
window : {[beg, end]}, optional
Window to use for the returned series. The default
value is the default class domain
.. versionadded:: 1.6.0
symbol : str, optional
Symbol representing the independent variable. Default is 'x'.
Returns
-------
new_series : series
A series that represents the least squares fit to the data and
has the domain and window specified in the call. If the
coefficients for the unscaled and unshifted basis polynomials are
of interest, do ``new_series.convert().coef``.
[resid, rank, sv, 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 `linalg.lstsq`.
"""
if domain is None:
domain = pu.getdomain(x)
if domain[0] == domain[1]:
domain[0] -= 1
domain[1] += 1
elif type(domain) is list and len(domain) == 0:
domain = cls.domain
if window is None:
window = cls.window
xnew = pu.mapdomain(x, domain, window)
res = cls._fit(xnew, y, deg, w=w, rcond=rcond, full=full)
if full:
[coef, status] = res
return (
cls(coef, domain=domain, window=window, symbol=symbol), status
)
else:
coef = res
return cls(coef, domain=domain, window=window, symbol=symbol)
@classmethod
def fromroots(cls, roots, domain=[], window=None, symbol='x'):
"""Return series instance that has the specified roots.
Returns a series representing the product
``(x - r[0])*(x - r[1])*...*(x - r[n-1])``, where ``r`` is a
list of roots.
Parameters
----------
roots : array_like
List of roots.
domain : {[], None, array_like}, optional
Domain for the resulting series. If None the domain is the
interval from the smallest root to the largest. If [] the
domain is the class domain. The default is [].
window : {None, array_like}, optional
Window for the returned series. If None the class window is
used. The default is None.
symbol : str, optional
Symbol representing the independent variable. Default is 'x'.
Returns
-------
new_series : series
Series with the specified roots.
"""
[roots] = pu.as_series([roots], trim=False)
if domain is None:
domain = pu.getdomain(roots)
elif type(domain) is list and len(domain) == 0:
domain = cls.domain
if window is None:
window = cls.window
deg = len(roots)
off, scl = pu.mapparms(domain, window)
rnew = off + scl*roots
coef = cls._fromroots(rnew) / scl**deg
return cls(coef, domain=domain, window=window, symbol=symbol)
@classmethod
def identity(cls, domain=None, window=None, symbol='x'):
"""Identity function.
If ``p`` is the returned series, then ``p(x) == x`` for all
values of x.
Parameters
----------
domain : {None, array_like}, optional
If given, the array must be of the form ``[beg, end]``, where
``beg`` and ``end`` are the endpoints of the domain. If None is
given then the class domain is used. The default is None.
window : {None, array_like}, optional
If given, the resulting array must be if the form
``[beg, end]``, where ``beg`` and ``end`` are the endpoints of
the window. If None is given then the class window is used. The
default is None.
symbol : str, optional
Symbol representing the independent variable. Default is 'x'.
Returns
-------
new_series : series
Series of representing the identity.
"""
if domain is None:
domain = cls.domain
if window is None:
window = cls.window
off, scl = pu.mapparms(window, domain)
coef = cls._line(off, scl)
return cls(coef, domain, window, symbol)
@classmethod
def basis(cls, deg, domain=None, window=None, symbol='x'):
"""Series basis polynomial of degree `deg`.
Returns the series representing the basis polynomial of degree `deg`.
.. versionadded:: 1.7.0
Parameters
----------
deg : int
Degree of the basis polynomial for the series. Must be >= 0.
domain : {None, array_like}, optional
If given, the array must be of the form ``[beg, end]``, where
``beg`` and ``end`` are the endpoints of the domain. If None is
given then the class domain is used. The default is None.
window : {None, array_like}, optional
If given, the resulting array must be if the form
``[beg, end]``, where ``beg`` and ``end`` are the endpoints of
the window. If None is given then the class window is used. The
default is None.
symbol : str, optional
Symbol representing the independent variable. Default is 'x'.
Returns
-------
new_series : series
A series with the coefficient of the `deg` term set to one and
all others zero.
"""
if domain is None:
domain = cls.domain
if window is None:
window = cls.window
ideg = int(deg)
if ideg != deg or ideg < 0:
raise ValueError("deg must be non-negative integer")
return cls([0]*ideg + [1], domain, window, symbol)
@classmethod
def cast(cls, series, domain=None, window=None):
"""Convert series to series of this class.
The `series` is expected to be an instance of some polynomial
series of one of the types supported by by the numpy.polynomial
module, but could be some other class that supports the convert
method.
.. versionadded:: 1.7.0
Parameters
----------
series : series
The series instance to be converted.
domain : {None, array_like}, optional
If given, the array must be of the form ``[beg, end]``, where
``beg`` and ``end`` are the endpoints of the domain. If None is
given then the class domain is used. The default is None.
window : {None, array_like}, optional
If given, the resulting array must be if the form
``[beg, end]``, where ``beg`` and ``end`` are the endpoints of
the window. If None is given then the class window is used. The
default is None.
Returns
-------
new_series : series
A series of the same kind as the calling class and equal to
`series` when evaluated.
See Also
--------
convert : similar instance method
"""
if domain is None:
domain = cls.domain
if window is None:
window = cls.window
return series.convert(domain, cls, window)
|