#
##
## This file is part of pyFormex 2.0 (Mon Sep 14 12:29:05 CEST 2020)
## pyFormex is a tool for generating, manipulating and transforming 3D
## geometrical models by sequences of mathematical operations.
## Home page: http://pyformex.org
## Project page: http://savannah.nongnu.org/projects/pyformex/
## Copyright 2004-2020 (C) Benedict Verhegghe (benedict.verhegghe@ugent.be)
## Distributed under the GNU General Public License version 3 or later.
##
## This program is free software: you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation, either version 3 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program. If not, see http://www.gnu.org/licenses/.
##
"""A collection of numerical utilities.
This module contains a large collection of numerical utilities.
Many of them are related to processing arrays. Some are similar
to existing NumPy functions but offer some extended functionality.
Note
----
While these functions were historically developed for pyFormex,
this module only depends on numpy and can be used outside
of pyFormex without changes.
"""
import warnings
import numpy as np
# NOTE: this overrides the Python builtin function abs. This is no
# problem because np.abs works just like abs on a simple values and arrays.
# But do not be tempted to override Python's sum with np.sum, because
# these work very differently on multidimensional arrays and have other
# optional arguments.
from numpy import (pi, sin, cos, tan, arcsin, arccos, arctan, # noqa: F401
arctan2, sqrt, abs)
_attributes_ = ['Float', 'Int', 'DEG', 'RAD', 'golden_ratio']
# default float and int types
Float = np.float32
Int = np.int32
# Some constants
DEG = np.pi/180.
"""float: multiplier to transform degrees to radians = pi/180."""
RAD = 1.
"""float: multiplier to transform radians to radians"""
golden_ratio = 0.5 * (1.0 + sqrt(5.))
"""golden ratio is defined as 0.5 * (1.0 + sqrt(5.))"""
#
# Functions for checking types, values and array dimensions
###########################################################
[docs]def isInt(obj):
"""Test if an object is an integer number.
Returns
-------
bool
True if the object is a single integer number (a Python ``int``
or a ``numpy.integer`` type), else False.
Examples
--------
>>> isInt(1)
True
>>> isInt(np.arange(3)[1])
True
"""
return isinstance(obj, (int, np.integer))
[docs]def isFloat(obj):
"""Test if an object is a floating point number.
Returns
-------
bool
True if the object is a single floating point number (a Python
``float`` or a ``numpy.floating`` type), else False.
Examples
--------
>>> isFloat(1.)
True
>>> isFloat(np.array([1,2],dtype=np.float32)[1])
True
"""
return isinstance(obj, (float, np.floating))
[docs]def isNum(obj):
"""Test if an object is an integer or a floating point number.
Returns
-------
bool
True if the object is a single integer or floating point
number, else False. The type of the object can be either
a Python ``int`` or ``float`` or a numpy ``integer`` or ``floating``.
Examples
--------
>>> isNum(1)
True
>>> isNum(1.0)
True
>>> isNum(np.array([1,2],dtype=np.int32)[1])
True
>>> isNum(np.array([1,2],dtype=np.float32)[1])
True
"""
return isInt(obj) or isFloat(obj)
[docs]def checkInt(value, min=None, max=None):
"""Check that a value is an int in the range min..max.
Parameters
----------
value: int-like
The value to check.
min: int, optional
If provided, minimal value to be accepted.
max: int, optional
If provided, maximal value to be accepted.
Returns
-------
checked_int: int
An integer not exceeding the provided boundaries.
Raises
------
ValueError:
If the value is not convertible to an integer type or exceeds one
of the specified boundaries.
Examples
--------
>>> checkInt(1)
1
>>> checkInt(1,min=0,max=1)
1
>>> checkInt('2',min=0)
2
"""
try:
a = int(value)
if min is not None and a < min:
raise ValueError
if max is not None and a > max:
raise ValueError
return a
except ValueError:
raise ValueError(
f"Expected an int in the range({min}, {max}), got: {value}")
[docs]def checkFloat(value, min=None, max=None):
"""Check that a value is a float in the range min..max.
Parameters
----------
value: float-like
The value to check
min: float-like, optional
If provided, minimal value to be accepted.
max: float-like, optional
If provided, maximal value to be accepted.
Returns
-------
checked_float: float
A float not exceeding the provided boundaries.
Raises
------
ValueError:
If the value is not convertible to a float type or exceeds one of the
specified boundaries.
Examples
--------
>>> checkFloat(1)
1.0
>>> checkFloat(1,min=0,max=1)
1.0
>>> checkFloat('2',min=0)
2.0
"""
try:
a = float(value)
if min is not None and a < min:
raise ValueError
if max is not None and a > max:
raise ValueError
return a
except Exception:
raise ValueError(
f"Expected a float in the range({min}, {max}), got: {value}")
[docs]def checkBroadcast(shape1, shape2):
"""Check that two array shapes are broadcast compatible.
In many numerical operations, NumPy will automatically broadcast
arrays of different shapes to a single shape, if they have broadcast
compatible shapes. Two array shapes are broadcast compatible if, in
all the last dimensions that exist in both arrays, either the shape
of both arrays has the same length, or one of the shapes has a length
1.
Parameters
----------
shape1: tuple of ints
Shape of first array
shape2: tuple of ints
Shape of second array
Returns
-------
tuple of ints
The broadcasted shape of the arrays.
Raises
------
ValueError: Shapes are not broadcast compatible
If the two shapes can not be broadcast to a single one.
Examples
--------
>>> checkBroadcast((8,1,6,1),(7,1,5))
(8, 7, 6, 5)
>>> checkBroadcast((5,4),(1,))
(5, 4)
>>> checkBroadcast((5,4),(4,))
(5, 4)
>>> checkBroadcast((15,3,5),(15,1,5))
(15, 3, 5)
>>> checkBroadcast((15,3,5),(3,5))
(15, 3, 5)
>>> checkBroadcast((15,3,5),(3,1))
(15, 3, 5)
>>> checkBroadcast((7,1,5),(8,1,6,1))
(8, 7, 6, 5)
"""
len1, len2 = len(shape1), len(shape2)
if len1 < len2:
shape1, shape2 = shape2, shape1
len1, len2 = len2, len1
shape = list(shape1[:len1-len2])
for n1, n2 in zip(shape1[len1-len2:], shape2):
if n1 == 1 or n2 == 1 or n1==n2:
shape.append(max(n1, n2))
else:
raise ValueError("Shapes are not broadcast compatible")
return tuple(shape)
[docs]def checkArray(a, shape=None, kind=None, allow=None, size=None, ndim=None,
bcast=None, subok=False, addaxis=False):
"""Check that an array a has the correct shape, type and/or size.
Parameters
----------
a: :term:`array_like`
An instance of a numpy.ndarray or a subclass thereof, or anything
that can be converted into a numpy array.
shape: tuple of ints, optional
If provided, the shape of the array should match this value
along each axis for which a nonzero value is specified. The
length of the shape tuple should also match (unless addaxis=True
is provided, see below).
kind: dtype.kind character code, optional
If provided, the array's dtype.kind should match this value,
or one of the values in ``allow``, if provided.
allow: string of dtype.kind character codes, optional
If provided, and ``kind`` is also specified, any of the specified
array types will also be accepted if it is convertible to the
specified ``kind``. See also Notes below.
size: int, optional
If provided, the total array size should match this value.
ndim: int, optional
If provided the input array should have exactly ``ndim`` dimensions
(unless addaxis=True is provided, see below).
bcast: tuple of ints, optional
If provided, the array's shape should be broadcast comaptible with
the specified shape.
subok: bool, optional
If True, the returned array is of the same class as the input array
``a``, if possible. If False (default), the returned array is always
of the base type numpy.ndarray.
addaxis: bool, optional
If False (default), and either ndim or shape are specified,
the input array should have precisely the number of dimensions
specified by ndim or the length of shape.
If True, an input array with less dimensions will automatically
be transformed by adding length 1 axes at the start of the shape
tuple until the correct dimension is reached.
Returns
-------
array
The checked_array is equivalent to the input data. It has the same
contents and shape. It also has the same type, unless ``kind`` is
is provided, in which case the result is converted to this type.
If ``subok=True`` was provided, the returned array will be of the
same array subclass as the input ``a``, if possible.
Raises
------
ValueError: invalid array
The input data failed for one of more of the tests requested by the
provided parameters.
Notes
-----
Currently, the only allowed conversion from an ``allow`` type to ``kind``
type, is to 'f'. Thus specifiying ``kind='f', allow='i'`` will accept
integer input but return float32 output.
See Also
--------
:func:`checkArray1D`
Examples
--------
>>> checkArray([1,2])
array([1, 2])
>>> checkArray([1,2],shape=(2,))
array([1, 2])
>>> checkArray([[1,2],[3,4]],shape=(2,-1))
array([[1, 2], [3, 4]])
>>> checkArray([1,2],kind='i')
array([1, 2])
>>> checkArray([1,2],kind='f',allow='i')
array([ 1., 2.])
>>> checkArray([1,2],size=2)
array([1, 2])
>>> checkArray([1,2],ndim=1)
array([1, 2])
>>> checkArray([[1,2],[3,4]],bcast=(3,1,2))
array([[1, 2], [3, 4]])
>>> checkArray([[1,2],[3,4]],ndim=3,addaxis=True)
array([[[1, 2],
[3, 4]]])
>>> checkArray([[1,2],[3,4]],shape=(-1,-1,2),addaxis=True)
array([[[1, 2],
[3, 4]]])
"""
a = np.asanyarray(a) if subok else np.asarray(a)
if ndim is None and shape is not None:
ndim = len(shape)
if ndim is not None:
if a.ndim != ndim:
if addaxis and a.ndim < ndim:
while a.ndim < ndim:
a = np.expand_dims(a, axis=0)
else:
raise ValueError(
f"Nonmatching ndim: expected {ndim}, got {a.ndim}")
if size is not None:
if a.size != size:
raise ValueError(
f"Nonmatching size: expected {size}, got {a.size}")
if shape is not None:
shape = np.asarray(shape)
w = np.where(shape >= 0)[0]
if (np.asarray(a.shape)[w] != shape[w]).any():
raise ValueError(
f"Nonmatching shape: expected {shape}, got {a.shape}")
if kind is not None:
if a.dtype.kind != kind:
if allow is not None and a.dtype.kind in allow:
if kind == 'f':
a = a.astype(Float)
elif kind == 'i':
a = a.astype(Int)
else:
raise ValueError(
f"Nonmatching kind: expected {kind}, got {a.dtype.kind}")
if kind == 'f':
# FORCE TO float32, otherwise we break some opengl function
# e.g. setTriade
# Leave this until we have a proper dtype==Float checking
# everywhere
a = a.astype(Float)
if bcast is not None:
checkBroadcast(a.shape, bcast)
return a
[docs]def checkArray1D(a, kind=None, allow=None, size=None):
"""Check and force an array to be 1D.
This is equivalent to calling :func:`checkArray` without the ``shape`` and
``ndim`` parameters, and then turning the result into a 1D array.
Parameters
----------
See :func:`checkArray`.
Returns
-------
1D array
The checked_array holds the same data as the input, but the shape
is rveled to 1D. It also has the same type, unless ``kind`` is
is provided, in which case the result is converted to this type.
Examples
--------
>>> checkArray1D([[1,2],[3,4]],size=4)
array([1, 2, 3, 4])
"""
return checkArray(a, kind=kind, allow=allow, size=size).ravel()
[docs]def checkUniqueNumbers(nrs, nmin=0, nmax=None):
"""Check that an array contains a set of unique integers in a given range.
This functions tests that all integer numbers in the array are within the
range math:`nmin <= i < nmax`. Default range is [0,unlimited].
Parameters
----------
nrs: :term:`array_like`, int
Input array with integers to check against provided limits.
nmin: int or None, optional
If not None, no value in ``a`` should be lower than this.
nmax: int, optionallMinimum allowed value.
- `nmax`: maximum allowed value + 1! If set to None, the test is skipped.
If provided, no value in ``a`` should be higher than this.
Returns
-------
1D int array
Containing the sorted unique numbers from the input.
Raises
------
ValueError
If the numbers are not unique or some input value surpasses one of
the specified limmits.
Examples
--------
>>> checkUniqueNumbers([0,5,1,7,2])
array([0, 1, 2, 5, 7])
>>> checkUniqueNumbers([0,5,1,7,-2],nmin=None)
array([-2, 0, 1, 5, 7])
"""
nrs = np.asarray(nrs)
uniq = np.unique(nrs)
if (uniq.size != nrs.size
or (nmin is not None and uniq.min() < nmin)
or (nmax is not None and uniq.max() > nmax)):
raise ValueError("Values not unique or not in range")
return uniq
###########################################################################
##
## Some generic array functions
##
#########################
# Deprecated
def addAxis(a, axis, **kargs):
warnings.warn("The use of arraytools.addAxis is deprecated. Please use "
"numpy.expand_dims instead.")
return np.expand_dims(a, axis)
[docs]def growAxis(a, add, axis=-1, fill=0):
"""Increase the length of an array axis.
Parameters
----------
a: :term:`array_like`
The array in which to extend n axis.
add: int
The length over which the specified axis should grow.
If add<=0, the array is returned unchanged.
axis: int
Position of the target axis in the array. Default is last (-1).
fill: int or float
Value to set the new elements along the grown axis to.
Returns
-------
array
Same type and data as `a`, but length of specified axis has been
increased with a value `add` and the new elements are filled with
the value `fill`.
Raises
------
ValueError:
If the specified axis exceeds the array dimensions.
Examples
--------
>>> growAxis([[1,2,3],[4,5,6]],2)
array([[1, 2, 3, 0, 0],
[4, 5, 6, 0, 0]])
>>> growAxis([[1,2,3],[4,5,6]],1,axis=0,fill=-3)
array([[ 1, 2, 3],
[ 4, 5, 6],
[-3, -3, -3]])
>>> growAxis([[1,2,3],[4,5,6]],-1)
array([[1, 2, 3],
[4, 5, 6]])
"""
a = np.asarray(a)
if axis >= a.ndim or axis < -a.ndim:
raise ValueError("Array with ndim %s has no axis %s" % (a.ndim, axis))
if add <= 0:
return a
else:
missing = list(a.shape)
missing[axis] = add
return np.concatenate([a, fill * np.ones(missing, dtype=a.dtype)], axis=axis)
[docs]def reorderAxis(a, order, axis=-1):
"""Reorder the planes of an array along the specified axis.
Parameters
----------
a: :term:`array_like`
The array in which to reorder the elements.
order: int :term:`array_like` | str
Specifies how to reorder the elements. It can be an integer index
which should be a permutation of `arange(a.shape[axis])`. Each value
in the index specified the old index of the elements that should be
placed at its position. This is equivalent to `a.take(order,axis)`.
`order` can also be one of the following predefined sting values,
resulting in the corresponding renumbering scheme being generated:
- 'reverse': the elements along axis are placed in reverse order
- 'random': the elements along axis are placed in random order
axis: int
The axis of the array along which the elements are to be reordered.
Default is last (-1).
Returns
-------
array
Same type and data as `a`, but the element planes are along `axis`
have been reordered.
Examples
--------
>>> reorderAxis([[1,2,3],[4,5,6]], [2,0,1])
array([[3, 1, 2],
[6, 4, 5]])
"""
a = np.asarray(a)
n = a.shape[axis]
if order == 'reverse':
order = np.arange(n-1, -1, -1)
elif order == 'random':
order = np.random.permutation(n)
else:
order = np.asarray(order)
return a.take(order, axis)
[docs]def reverseAxis(a, axis=-1):
"""Reverse the order of the elements along an axis.
Parameters
----------
a: :term:`array_like`
The array in which to reorder the elements.
axis: int
The axis of the array along which the elements are to be reordered.
Default is last (-1).
Returns
-------
array
Same type and data as `a`, but the elements along `axis` are now
in reversed order.
Note
----
This function is especially useful if axis has a computed value.
If the axis is known in advance, it is more efficient to use
an indexing operation. Thus **reverseAxis(A,-1)** is equivalent to
**A[...,::-1]**.
Examples
--------
>>> A = np.array([[1,2,3],[4,5,6]])
>>> reverseAxis(A)
array([[3, 2, 1],
[6, 5, 4]])
>>> A[...,::-1]
array([[3, 2, 1],
[6, 5, 4]])
"""
return reorderAxis(a, 'reverse', axis)
[docs]def interleave(a, b):
"""Interleave two arrays along their first axis.
Parameters
----------
a: :term:`array_like`
First array
b: :term:`array_like`
Second array, with same type and shape as `a`, except that the first
dimension may be one less than that of `a`, and if data type of `b`
can be one that is convertible to that of `a`.
Returns
-------
array
An array with interleaved rows from `a` and `b`. The array has the
datatype of `a` and its first axis has the combined length of that
of `a` and `b`.
Examples
--------
>>> interleave(np.arange(4), 10*np.arange(3))
array([ 0, 0, 1, 10, 2, 20, 3])
>>> a = np.arange(8).reshape(2,4)
>>> print(interleave(a,10*a))
[[ 0 1 2 3]
[ 0 10 20 30]
[ 4 5 6 7]
[40 50 60 70]]
"""
a = np.asarray(a)
b = np.asarray(b)
if a.shape[0]-b.shape[0] not in (0, 1) or a.shape[1:] != b.shape[1:]:
raise ValueError("Array sizes not compatible for interleave")
c = np.empty((a.shape[0]+b.shape[0],)+a.shape[1:], dtype=a.dtype)
c[0::2] = a
c[1::2] = b
return c
[docs]def multiplex(a, n, axis, warn=True):
"""Multiplex an array over a length n in direction of a new axis.
Inserts a new axis in the array at the specified position and repeats
the data of the array `n` times in the direction of the new axis.
Parameters
----------
a: :term:`array_like`
The input array.
n: int
Number of times to repeat the data in direction of `axis`.
axis: int, optional
Position of the new axis in the expanded array. Should be
in the range -a.ndim..a.ndim.
Returns
-------
array
An array with n times the original data repeated in the direction
of the specified axis.
See Also
--------
repeatValues: Repeat values in a 1-dim array a number of times
Examples
--------
>>> a = np.arange(6).reshape(2,3)
>>> print(a)
[[0 1 2]
[3 4 5]]
>>> print(multiplex(a,4,-1))
[[[0 0 0 0]
[1 1 1 1]
[2 2 2 2]]
<BLANKLINE>
[[3 3 3 3]
[4 4 4 4]
[5 5 5 5]]]
>>> print(multiplex(a,4,-2))
[[[0 1 2]
[0 1 2]
[0 1 2]
[0 1 2]]
<BLANKLINE>
[[3 4 5]
[3 4 5]
[3 4 5]
[3 4 5]]]
"""
if warn and axis < 0:
warnings.warn('warn_multiplex_changed')
return np.expand_dims(a, axis).repeat(n, axis=axis)
[docs]def repeatValues(a, n):
"""Repeat values in a 1-dim array a number of times.
Parameters
----------
a: :term:`array_like`, 1-dim
The input array. Can be a list or a single element.
n: int :term:`array_like`, 1-dim
Number of times to repeat the corresponding value of ``a``.
If ``n`` has less elements than ``a``, it is reused until
the end of ``a`` is reached.
Returns
-------
array
An 1-dim array of the same dtype as ``a`` with the value ``a[i]``
repeated ``n[i]`` times.
See Also
--------
multiplex: Multiplex an array over a length n in direction of a new axis
Examples
--------
>>> repeatValues(2,3)
array([2, 2, 2])
>>> repeatValues([2,3],2)
array([2, 2, 3, 3])
>>> repeatValues([2,3,4],[2,3])
array([2, 2, 3, 3, 3, 4, 4])
>>> repeatValues(1.6,[3,5])
array([ 1.6, 1.6, 1.6])
"""
a = checkArray1D(a)
n = checkArray1D(n, kind='i')
n = np.resize(n, a.shape)
return np.concatenate([np.resize(ai, ni) for ai, ni in zip(a, n)])
[docs]def concat(al, axis=0):
"""Smart array concatenation ignoring empty arrays.
Parameters
----------
al: list of arrays
All arrays should have same shape except for the length of
the concatenation axis, or be empty arrays.
axis: int
The axis along which the arrays are concatenated.
Returns
-------
:array
The concatenation of all non-empty arrays in the list,
or an empty array if all arrays in the list are empty.
Note
----
This is just like numpy.concatenate, but allows empty arrays
in the list and silently ignores them.
Examples
--------
>>> concat([np.array([0,1]),np.array([]),np.array([2,3])])
array([0, 1, 2, 3])
"""
al = [a for a in al if a.size > 0]
if len(al) > 0:
return np.concatenate(al, axis=axis)
else:
return np.array([])
[docs]def splitrange(n, nblk):
"""Split a range of integers 0..n in almost equal sized slices.
Parameters
----------
n: int
Highest integer value in the range.
nblk: int
Number of blocks to split into. Should be <= n to allow splitting.
Returns
-------
: 1-dim int array
If nblk <= n, returns the boundaries that divide the integers
in the range 0..n in nblk almost equal slices.
The outer boundaries 0 and n are included, so the length of the
array is nblk+1.
If nblk >= n, returns range(n+1), thus all slices have length 1.
Examples
--------
>>> splitrange(7,3)
array([0, 2, 5, 7])
"""
if n > nblk:
ndata = (np.arange(nblk+1) * n * 1.0 / nblk).round().astype(int)
else:
ndata = np.arange(n+1)
return ndata
[docs]def splitar(a, nblk, axis=0, close=False):
"""Split an array in nblk subarrays along a given axis.
Parameters
----------
a: :term:`array_like`
Array to be divided in subarrays.
nblk: int
Number of subarrays to obtain. The subarrays will be
of almost the same size.
axis: int:
Axis along which to split the array (default 0)
close: bool
If True, the last item of each block will be repeated
as the first item of the next block.
Returns
-------
: list of arrays
A list of subarrays obtained by splitting a along the specified axis.
All arrays have almost the same shape. The number of arrays is equal
to nblk, unless nblk is larger than a.shape[axis], in which case a
a list with only the original array is returned.
Examples
--------
>>> splitar(np.arange(7),3)
[array([0, 1]), array([2, 3, 4]), array([5, 6])]
>>> splitar(np.arange(7),3,close=True)
[array([0, 1, 2]), array([2, 3, 4]), array([4, 5, 6])]
>>> X = np.array([[0.,1.,2.],[3.,4.,5.]])
>>> splitar(X,2)
[array([[ 0., 1., 2.]]), array([[ 3., 4., 5.]])]
>>> splitar(np.matrix(X),2,axis=-1)
[matrix([[ 0., 1.],
[ 3., 4.]]), matrix([[ 2.],
[ 5.]])]
>>> splitar(X,3)
[array([[ 0., 1., 2.],
[ 3., 4., 5.]])]
"""
a = np.asanyarray(a).swapaxes(axis, 0)
na = a.shape[0]
if close:
na -= 1
if nblk > na:
return [a]
ndata = splitrange(na, nblk)
k = 1 if close else 0
return [a[i:j+k].swapaxes(0, axis) for i, j in zip(ndata[:-1], ndata[1:])]
[docs]def minmax(a, axis=-1):
"""Compute the minimum and maximum along an axis.
Parameters
----------
a: :term:`array_like`
The data array for which to compute the minimum and maximum.
axis: int
The array axis along which to compute the minimum and maximum.
Returns
-------
: array
The array has the same dtype as `a`. It also has the same shape,
except for the specified axis, which will have a length of 2.
The first value along this axis holds the minimum value of the input,
the second holds the maximum value.
Examples
--------
>>> a = np.array([[ [1.,0.,0.], [0.,1.,0.] ],
... [ [2.,0.,0.], [0.,2.,0.] ] ])
>>> print(minmax(a,axis=1))
[[[ 0. 0. 0.]
[ 1. 1. 0.]]
<BLANKLINE>
[[ 0. 0. 0.]
[ 2. 2. 0.]]]
"""
return np.stack([a.min(axis=axis), a.max(axis=axis)], axis=axis)
[docs]def stretch(a, min=None, max=None, axis=None):
"""Scale the values of an array to fill a given range.
Parameters
----------
a: :term:`array_like`, int or float
Input data.
min: int or float, optional
The targeted minimum value in the array. Same type as `a`.
If not provided, the minimum of a is used.
max: int or float, optional
The targeted maximum value in the array. Same type as `a`.
If not provided, the maximum of a is used.
axis: int, optional
If provided, each slice along the specified axis is
independently scaled.
Returns
-------
: array
Array of the same type and size as the input array, but in which
the values have been linearly scaled to fill the specified range.
Examples
--------
>>> stretch([1.,2.,3.],min=0,max=1)
array([ 0. , 0.5, 1. ])
>>> A = np.arange(6).reshape(2,3)
>>> stretch(A,min=20,max=30)
array([[20, 22, 24],
[26, 28, 30]])
>>> stretch(A,min=20,max=30,axis=1)
array([[20, 25, 30],
[20, 25, 30]])
>>> stretch(A,max=30)
array([[ 0, 6, 12],
[18, 24, 30]])
>>> stretch(A,min=2,axis=1)
array([[2, 4, 5],
[2, 4, 5]])
>>> stretch(A.astype(Float),min=2,axis=1)
array([[ 2. , 3.5, 5. ],
[ 2. , 3.5, 5. ]])
"""
a = np.asarray(a)
atype = a.dtype
if min is None:
min = a.min()
if max is None:
max = a.max()
if not min < max:
raise ValueError('max must be larger than min in `rng` parameter.')
amin = a.min(axis=axis)
amax = a.max(axis=axis)
if axis is not None:
amin = np.expand_dims(amin, axis)
amax = np.expand_dims(amax, axis)
sc = amax-amin
if atype.kind == 'i':
sc = sc.astype(Float)
b = (a-amin) / sc * (max-min) + min
if atype.kind == 'i':
b = b.round()
return b.astype(atype)
[docs]def stringar(s, a):
"""Nicely format a string followed by an array.
Parameters
----------
s: str
String to appear before the formatted array
a: array
Array to be formatted after the string, with proper vertical
alignment
Returns
-------
: str
A multiline string where the first line consists of the
string s and the first line of the formatted array, and the next lines
hold the remainder of the array lines, properly indented to align with
the first line of the array.
Examples
--------
>>> print(stringar("Indented array: ",np.arange(4).reshape(2,2)))
Indented array: [[0 1]
[2 3]]
"""
s = str(s)
n = len(s)
repl = ' '*n
return s + str(a).replace('\n', '\n'+repl)
[docs]def array2str(a):
"""String representation of an array.
This creates a string representation of an array. It is visually
equivalent with numpy.ndarray.__repr__ without the dtype, except
for 'uint.' types.
Note
----
This function can be used to set the default string representation
of numpy arrays, using the following::
import numpy as np
np.set_string_function(array2str)
To reset it to the default, do::
np.set_string_function(None)
Because this reference manual was created with the default numpy
routine replaced with ours, you will never see the dtype, except
for uint types. See also the examples below.
Parameters
----------
a: array
Any :class:`numpy.ndarray` object.
Returns
-------
The string representation of the array as created by its
``__repr__`` method, except that the ``dtype`` is left away.
Examples
--------
>>> a = np.arange(5).astype(np.int8)
>>> print(array2str(a))
array([0, 1, 2, 3, 4])
>>> a
array([0, 1, 2, 3, 4])
Reset the numpy string function to its default.
>>> np.set_string_function(None)
>>> a
array([0, 1, 2, 3, 4], dtype=int8)
Change back to ours.
>>> np.set_string_function(array2str)
>>> a
array([0, 1, 2, 3, 4])
"""
import re
from numpy.core import arrayprint
return re.sub(r", dtype=[^u]\w*", "", arrayprint.array_repr(a))
[docs]def printar(s, a):
"""Print a string followed by a vertically aligned array.
Parameters
----------
s: str
String to appear before the formatted array
a: array
Array to be formatted after the string, with proper vertical
alignment
Note
----
This is a shorthand for ``print(stringar(s,a))``.
Examples
--------
>>> printar("Indented array: ",np.arange(4).reshape(2,2))
Indented array: [[0 1]
[2 3]]
"""
print(stringar(s, a))
[docs]def writeArray(fil, array, sep=' '):
"""Write an array to an open file.
This uses :func:`numpy.tofile` to write an array to an open file.
Parameters
----------
fil: file or str
Open file object or filename.
array: :term:`array_like`
The array to write to the file.
sep: str
If empty, the array is written in binary mode.
If not empty, the array is written in text mode, with this
string as separator between the elements.
See also
--------
readArray
"""
array.tofile(fil, sep=sep)
[docs]def readArray(fil, dtype, shape, sep=' '):
"""Read data for an array with known size and type from an open file.
This uses :func:`numpy.fromfile` to read an array with known shape and
data type from an open file.
Parameters
----------
fil: file or str
Open file object or filename.
dtype: data-type
Data type of the array to be read.
shape: tuple of ints
The shape of the array to be read.
sep: str
If not empty, the array is read in text mode, with this
string as separator between the elements.
If empty, the array is read in binary mode and an extra '\\n'
after the data will be stripped off
See Also
--------
writeArray
"""
shape = np.asarray(shape)
size = shape.prod()
data = np.fromfile(fil, dtype=dtype, count=size, sep=sep).reshape(shape)
if sep == '':
pos = fil.tell()
byte = fil.read(1)
if not ord(byte) == 10:
# not a newline: push back
fil.seek(pos)
return data
[docs]def powers(x, n):
"""Compute all the powers of x from zero up to n.
Parameters
----------
x: int, float or array (int,float)
The number or numbers to be raised to the specified powers.
n: int
Maximal power to raise the numbers to.
Returns
-------
powers: list
A list of numbers or arrays of the same shape and type as the input.
The list contains ``N+1`` items, being the input raised to the powers
in ``range(n+1)``.
Examples
--------
>>> powers(2,5)
[1, 2, 4, 8, 16, 32]
>>> powers(np.array([1.0,2.0]),5)
[array([ 1., 1.]), array([ 1., 2.]), array([ 1., 4.]), \
array([ 1., 8.]), array([ 1., 16.]), array([ 1., 32.])]
"""
return [x ** i for i in range(n+1)]
###########################################################################
##
## some math functions
##
#########################
# Convenience functions: trigonometric functions with argument in degrees
[docs]def sind(arg, angle_spec=DEG):
"""Return the sine of an angle in degrees.
Parameters
----------
arg: float number or array
Angle(s) for which the sine is to be returned.
By default, angles are specified in degrees (see ``angle_spec``).
angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float
Multiplier to apply to ``arg`` before taking the sine.
The default multiplier DEG makes the argument
being intrepreted as an angle in degrees. Use RAD when angles
are specified in radians.
Returns
-------
:float number or array
The sine of the input angle(s)
See also
--------
cosd
tand
arcsind
arccosd
arctand
arctand2
Examples
--------
>>> print(sind(30), sind(pi/6,RAD))
0.5 0.5
>>> sind(np.array([0.,30.,45.,60.,90.]))
array([ 0. , 0.5 , 0.71, 0.87, 1. ])
"""
return np.sin(arg*angle_spec)
[docs]def cosd(arg, angle_spec=DEG):
"""Return the cosine of an angle in degrees.
Parameters
----------
arg: float number or array
Angle(s) for which the cosine is to be returned.
By default, angles are specified in degrees (see ``angle_spec``).
angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float
Multiplier to apply to ``arg`` before taking the sine.
The default multiplier DEG makes the argument
being intrepreted as an angle in degrees. Use RAD when angles
are specified in radians.
Returns
-------
:float number or array
The cosine of the input angle(s)
See also
--------
sind
tand
arcsind
arccosd
arctand
arctand2
Examples
--------
>>> print(cosd(60), cosd(pi/3,RAD))
0.5 0.5
"""
return np.cos(arg*angle_spec)
[docs]def tand(arg, angle_spec=DEG):
"""Return the tangens of an angle in degrees.
Parameters
----------
arg: float number or array
Angle(s) for which the tangens is to be returned.
By default, angles are specified in degrees (see ``angle_spec``).
angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float
Multiplier to apply to ``arg`` before taking the sine.
The default multiplier DEG makes the argument
being intrepreted as an angle in degrees. Use RAD when angles
are specified in radians.
Returns
-------
:float number or array
The tangens of the input angle(s)
See also
--------
sind
cosd
arcsind
arccosd
arctand
arctand2
Examples
--------
>>> print(tand(45), tand(pi/4,RAD))
1.0 1.0
"""
return np.tan(arg*angle_spec)
[docs]def arcsind(arg, angle_spec=DEG):
"""Return the angle whose sine is equal to the argument.
Parameters
----------
arg: float number or array, in the range -1.0 to 1.0.
Value(s) for which to return the arcsine.
angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float, nonzero.
Divisor applied to the resulting angles before returning.
The default divisor DEG makes the angles be returned in
degrees. Use RAD to get angles in radians.
Returns
-------
:float number or array
The angle(s) for which the input value(s) is/are the cosine.
The default ``angle_spec=DEG`` returns values in the range
-90 to +90.
See also
--------
sind
cosd
tand
arccosd
arctand
arctand2
Examples
--------
>>> print(f"{arcsind(0.5):.1f} {arcsind(1.0,RAD):.4f}")
30.0 1.5708
>>> arcsind(-1)
-90.0
>>> arcsind(1)
90.0
"""
return np.arcsin(arg)/angle_spec
[docs]def arccosd(arg, angle_spec=DEG):
"""Return the angle whose cosine is equal to the argument.
Parameters
----------
arg: float number or array, in the range -1.0 to 1.0.
Value(s) for which to return the arccos.
angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float, nonzero.
Divisor applied to the resulting angles before returning.
The default divisor DEG makes the angles be
returned in degrees. Use RAD to get angles in radians.
Returns
-------
:float number or array
The angle(s) for which the input value(s) is/are the cosine.
The default ``angle_spec=DEG`` returns values in the range
0 to 180.
See also
--------
sind
cosd
tand
arcsind
arctand
arctand2
Examples
--------
>>> print(f"{arccosd(0.5):.1f} {arccosd(-1.0,RAD):.4f}")
60.0 3.1416
>>> arccosd(np.array([-1,0,1]))
array([ 180., 90., 0.])
"""
return np.arccos(arg)/angle_spec
[docs]def arctand(arg, angle_spec=DEG):
"""Return the angle whose tangens is equal to the argument.
Parameters
----------
arg: float number or array.
Value(s) for which to return the arctan.
angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float, nonzero.
Divisor applied to the resulting angles before returning.
The default divisor DEG makes the angles be
returned in degrees. Use RAD to get angles in radians.
Returns
-------
:float number or array
The angle(s) for which the input value(s) is/are the tangens.
The default ``angle_spec=DEG`` returns values in the range
-90 to +90.
See also
--------
sind
cosd
tand
arcsind
arccosd
arctand2
Examples
--------
>>> print(f"{arctand(1.0):.1f} {arctand(-1.0,RAD):.4f}")
45.0 -0.7854
>>> arctand(np.array([-np.inf,-1,0,1,np.inf]))
array([-90., -45., 0., 45., 90.])
"""
return np.arctan(arg)/angle_spec
[docs]def arctand2(sin, cos, angle_spec=DEG):
"""Return the angle whose sine and cosine values are given.
Parameters
----------
sin: float number or array with same shape as ``cos``.
Sine value(s) for which to return the corresponding angle.
cos: float number or array with same shape as ``sin``
Cosine value(s) for which to return the corresponding angle.
angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float, nonzero.
Divisor applied to the resulting angles before returning.
The default divisor DEG makes the angles be returned in degrees.
Use RAD to get angles in radians.
Returns
-------
:float number or array with same shape as ``sin`` and ``cos``.
The angle(s) for which the input value(s) are the sine and
cosine.
The default ``angle_spec=DEG`` returns values in the range
[-180, 180].
Note
----
The input values ``sin`` and ``cos`` are not restricted to the [-1.,1.]
range. The returned angle is that for which the tangens is given by
``sin/cos``, but with a sine and cosine that have the same sign as the
``sin`` and ``cos`` values.
See also
--------
sind
cosd
tand
arcsind
arccosd
arctand
Examples
--------
>>> print(f"{arctand2(0.0,-1.0):.1f} "
... f"{arctand2(-sqrt(0.5),-sqrt(0.5),RAD):.4f}")
180.0 -2.3562
>>> arctand2(np.array([0., 1., 0., -1.]), np.array([1., 0., -1., 0.]))
array([ 0., 90., 180., -90.])
>>> arctand2(2.,2.)
45.0
"""
return arctan2(sin, cos)/angle_spec
[docs]def niceLogSize(f):
"""Return an integer estimate of the magnitude of a float number.
Parameters
----------
f: float
Value for which the integer magnitude has to be computed. The
sign of the value is disregarded.
Returns
-------
:int
An integer magnitude estimator for the input.
Note
----
The returned value is the smallest integer ``e`` such that
``10**e > abs(f)``.
If positive, it is equal to the number of digits
before the decimal point; if negative, it is equal to the number of
leading zeros after the decimal point.
See also
--------
nicenumber
Examples
--------
>>> print([ niceLogSize(a) for a in [1.3, 35679.23, 0.4, 0.0004567, -1.3] ])
[1, 5, 0, -3, 1]
"""
return int(np.ceil(np.log10(abs(f))))
[docs]def niceNumber(f, round=np.ceil):
"""Return a nice number close to abs(f).
A nice number is a number which only has only one significant digit
(in the decimal system).
Parameters
----------
f: float
A float number to approximate with a nice number. The sign of ``f``
is disregarded.
round: callable
A function that rounds a float to the nearest integer. Useful
functions are ``ceil``, ``floor`` and ``round`` from either
NumPy or Python's math module. Default is ``numpy.ceil``.
Returns
-------
:float
A float value close to the input value, but having only
a single decimal digit.
Examples
--------
>>> numbers = [ 0.0837, 0.867, 8.5, 83.7, 93.7]
>>> [ str(niceNumber(f)) for f in numbers ]
['0.09', '0.9', '9.0', '90.0', '100.0']
>>> [ str(niceNumber(f,round=np.floor)) for f in numbers ]
['0.08', '0.8', '8.0', '80.0', '90.0']
>>> [ str(niceNumber(f,round=np.round)) for f in numbers ]
['0.08', '0.9', '8.0', '80.0', '90.0']
"""
fa = abs(f)
s = "%.1e" % fa
m, n = s.split('e')
m = int(round(float(m)))
n = int(n)
return m*10.**n
[docs]def isqrt(n):
"""Compute the square root of an integer number.
Parameters
----------
n: int
An integer number that is a perfect square.
Returns
-------
:int
The square root from the input number
Raises
------
ValueError:
If the input integer is not a perfect square.
Examples
--------
>>> isqrt(36)
6
"""
import math
i = int(math.sqrt(n) + 0.5)
if i*i != n:
raise ValueError('Input is not a perfect square: %s' % n)
return i
###########################################################################
##
## Vector operations
##
#########################
[docs]def dotpr(A, B, axis=-1):
"""Return the dot product of vectors of A and B in the direction of axis.
Parameters
----------
A: float :term:`array_like`
Array containing vectors in the direction of axis.
B: float :term:`array_like`
Array containing vectors in the direction of axis. Same shape
as A, or broadcast-compatible.
axis: int
Axis of A and B in which direction the vectors are layed out.
Default is the last axis. A and B should have the same length
along this axis.
Returns
-------
float array, shape as A and B with axis direction removed.
The elements contain the dot product of the vectors of A and B at
that position.
Note
----
This multiplies the elements of the A and B and then sums them
in the direction of the specified axis.
Examples
--------
>>> A = np.array( [[1.0, 1.0], [1.0,-1.0], [0.0, 5.0]] )
>>> B = np.array( [[5.0, 3.0], [2.0, 3.0], [1.33,2.0]] )
>>> print(dotpr(A,B))
[ 8. -1. 10.]
>>> print(dotpr(A,B,0))
[ 7. 10.]
"""
A = np.asarray(A)
B = np.asarray(B)
return (A*B).sum(axis)
[docs]def length(A, axis=-1):
"""Returns the length of the vectors of A in the direction of axis.
Parameters
----------
A: float :term:`array_like`
Array containing vectors in the direction of axis.
axis: int
Axis of A in which direction the vectors are layed out.
Default is the last axis. A and B shoud have the same length
along this axis.
Returns
-------
: float array, shape of A with axis direction removed.
The elements contain the length of the vector in A at that position.
Note
----
This is equivalent with ``sqrt(dotpr(A,A))``.
Examples
--------
>>> A = np.array( [[1.0, 1.0], [1.0,-1.0], [0.0, 5.0]] )
>>> print(length(A))
[ 1.41 1.41 5. ]
>>> print(length(A,0))
[ 1.41 5.2 ]
"""
A = np.asarray(A)
return sqrt((A*A).sum(axis))
[docs]def normalize(A, axis=-1, on_zeros='n', return_length=False, ignore_zeros=False):
"""Normalize the vectors of A in the direction of axis.
Parameters
----------
A: float :term:`array_like`
Array containing vectors in the direction of axis.
axis: int
Axis of A in which direction the vectors are layed out.
on_zeros: 'n', 'e' or 'i'
Specifies how to treat occurrences of zero length vectors (having
all components equal to zero):
- 'n': return a vector of nan values
- 'e': raise a ValueError
- 'i': ignore zero vectors and return them as such.
return_length: bool
If True, also returns also the length of the original vectors.
ignore_zeros: bool
(Deprecated) Equivalent to specifying ``on_zeros='i'``.
Returns
-------
norm: float array
Array with same shape as A but where each vector along axis
has been rescaled so that its length is 1.
len: float array, optional
Array with shape like A but with axis removed. The length of
the original vectors in the direction of axis. Only returned
if ``return_length=True`` provided.
Raises
------
ValueError: Can not normalize zero-length vector
If any of the vectors of B is a zero vector.
Examples
--------
>>> A = np.array( [[3.0, 3.0], [4.0,-3.0], [0.0, 0.0]] )
>>> print(normalize(A))
[[ 0.71 0.71]
[ 0.8 -0.6 ]
[ nan nan]]
>>> print(normalize(A,on_zeros='i'))
[[ 0.71 0.71]
[ 0.8 -0.6 ]
[ 0. 0. ]]
>>> print(normalize(A,0))
[[ 0.6 0.71]
[ 0.8 -0.71]
[ 0. 0. ]]
>>> n,l = normalize(A,return_length=True)
>>> print(n)
[[ 0.71 0.71]
[ 0.8 -0.6 ]
[ nan nan]]
>>> print(l)
[ 4.24 5. 0. ]
"""
if ignore_zeros:
on_zeros = 'i'
A = np.asarray(A)
Al = length(A, axis)
if on_zeros != 'n':
if (Al == 0.).any():
if on_zeros=='i':
Al[Al==0.] = 1.
else:
raise ValueError("Can not normalize zero-length vector.")
with np.errstate(divide='ignore', invalid='ignore'):
res = A / np.expand_dims(Al, axis)
if return_length:
return res, Al
else:
return res
[docs]def projection(A, B, axis=-1):
"""Return the (signed) length of the projection of vectors of A on B.
Parameters
----------
A: float :term:`array_like`
Array containing vectors in the direction of axis.
B: float :term:`array_like`
Array containing vectors in the direction of axis. Same shape
as A, or broadcast-compatible.
axis: int
Axis of A and B in which direction the vectors are layed out.
Default is the last axis. A and B should have the same length
along this axis.
Returns
-------
: float array, shape as A and B with axis direction removed.
The elements contain the length of the projections of vectors of A
on the directions of the corresponding vectors of B.
Raises
------
ValueError: Can not normalize zero-length vector
If any of the vectors of B is a zero vector.
Note
----
This returns ``dotpr(A,normalize(B))``.
Examples
--------
>>> A = [[2.,0.],[1.,1.],[0.,1.]]
>>> projection(A,[1.,0.])
array([ 2., 1., 0.])
>>> projection(A,[1.,1.])
array([ 1.41, 1.41, 0.71])
>>> projection(A,[[1.],[1.],[0.]],axis=0)
array([ 2.12, 0.71])
"""
return dotpr(A, normalize(B, axis=axis, on_zeros='e'), axis=axis)
[docs]def parallel(A, B, axis=-1):
"""Return the component of vector of A that is parallel to B.
Parameters
----------
A, B: float :term:`array_like`
Broadcast compatible arrays containing vectors in the direction of
axis.
axis: int
Axis of A and B in which direction the vectors are layed out.
Default is the last axis. A and B should have the same dimension
along this axis.
Returns
-------
: float array, same shape as A and B.
The vectors in the axis direction are the vectors of A projected
on the direction of the corresponding vectors of B.
See also
--------
orthog
Examples
--------
>>> A = [[2.,0.],[1.,1.],[0.,1.]]
>>> parallel(A,[1.,0.])
array([[ 2., 0.],
[ 1., 0.],
[ 0., 0.]])
>>> parallel(A,A)
array([[ 2., 0.],
[ 1., 1.],
[ 0., 1.]])
>>> parallel(A,[[1.],[1.],[0.]],axis=0)
array([[ 1.5, 0.5],
[ 1.5, 0.5],
[ 0. , 0. ]])
"""
Bn = normalize(B, axis=axis, on_zeros='e')
p = dotpr(A, Bn, axis=axis)
return np.expand_dims(p, axis) * Bn
[docs]def orthog(A, B, axis=-1):
"""Return the component of vector of A that is orthogonal to B.
Parameters
----------
A: float :term:`array_like`
Array containing vectors in the direction of axis.
B: float :term:`array_like`
Array containing vectors in the direction of axis. Same shape
as A, or broadcast-compatible.
axis: int
Axis of A and B in which direction the vectors are layed out.
Default is the last axis. A and B should have the same length
along this axis.
Returns
-------
: float array, same shape as A and B.
The vectors in the axis direction are the components of the vectors
of A orthogonal to the direction of the corresponding vectors of B.
See also
--------
parallel
Examples
--------
>>> A = [[2.,0.],[1.,1.],[0.,1.]]
>>> orthog(A,[1.,0.])
array([[ 0., 0.],
[ 0., 1.],
[ 0., 1.]])
>>> orthog(A,[[1.],[1.],[0.]],axis=0)
array([[ 0.5, -0.5],
[-0.5, 0.5],
[ 0. , 1. ]])
"""
return A - parallel(A, B, axis=axis)
[docs]def inside(p, mi, ma):
"""Return true if point p is inside bbox defined by points mi and ma.
Parameters
----------
p: float :term:`array_like` with shape (ndim,)
Point to test against the boundaries.
mi: float :term:`array_like` with shape (ndim,)
Minimum values for the components of p
ma: float :term:`array_like` with shape (ndim,)
Maximum values for the components of p
Returns
-------
:bool
True is all components are inside the specified limits, limits
included. This means that the n-dimensional point p lies within
the n-dimensional rectangular bounding box defined by the two
n-dimensional points (mi,ma).
Examples
--------
>>> inside([0.5,0.5],[0.,0.],[1.,1.])
True
>>> inside([0.,1.],[0.,0.],[1.,1.])
True
>>> inside([0.,1.1],[0.,0.],[1.,1.])
False
"""
p = np.asarray(p)
return (p >= mi).all() and (p <= ma).all()
[docs]def unitVector(v):
"""Return a unit vector in the direction of v.
Parameters
----------
v: a single integer or a (3,) shaped float :term:`array_like`
If an int, it specifies one of the global axes (0,1,2).
Else, it is a vector in 3D space.
Returns
-------
: (3,) shaped float array
A unit vector along the specified direction.
Examples
--------
>>> unitVector(1)
array([ 0., 1., 0.])
>>> unitVector([0.,3.,4.])
array([ 0. , 0.6, 0.8])
"""
if isInt(v):
if v not in range(3):
raise ValueError("v should be one of 0, 1 or 2")
u = np.zeros((3), dtype=Float)
u[v] = 1.0
else:
v = checkArray(v, shape=(3,), kind='f', allow='i')
u = normalize(v, on_zeros='e')
return u
[docs]def rotationMatrix(angle, axis=None, angle_spec=DEG):
"""Create a 2D or 3D rotation matrix over angle, optionally around axis.
Parameters
----------
angle: float
Rotation angle, by default in degrees.
axis: int or (3,) float :term:`array_like`, optional
If not provided, a 2D rotation matrix is returned.
If provided, it specifies the rotation axis in a 3D world. It is either
one of 0,1,2, specifying a global axis, or a vector with 3 components
specifying an axis through the origin. The returned matrix is 3D.
angle_spec: float, DEG or RAD, optional
The default (DEG) interpretes the angle in degrees. Use RAD to
specify the angle in radians.
Returns
-------
float array
Rotation matrix which will rotate a vector over the specified angle.
Shape is (3,3) if axis is specified, or (2,2) if not.
See also
--------
rotationMatrix3: subsequent rotation around 3 axes
rotmat: rotation matrix specified by three points in space
trfmat: transformation matrix to transform 3 points
rotMatrix: rotation matrix transforming global axis 0 into a given vector
rotMatrix2: rotation matrix that transforms one vector into another
Examples
--------
>>> rotationMatrix(30.,1)
array([[ 0.87, 0. , -0.5 ],
[ 0. , 1. , 0. ],
[ 0.5 , 0. , 0.87]])
>>> rotationMatrix(45.,[1.,1.,0.])
array([[ 0.85, 0.15, -0.5 ],
[ 0.15, 0.85, 0.5 ],
[ 0.5 , -0.5 , 0.71]])
"""
a = angle*angle_spec
c = cos(a)
s = sin(a)
if axis is None:
f = [[c, s], [-s, c]]
elif np.array(axis).size == 1:
f = np.zeros((3, 3))
axes = list(range(3))
i, j, k = axes[axis:]+axes[:axis]
f[i][i] = 1.0
f[j][j] = c
f[j][k] = s
f[k][j] = -s
f[k][k] = c
else:
X, Y, Z = unitVector(axis)
t = 1.-c
f = [[t*X*X + c, t*X*Y + s*Z, t*X*Z - s*Y],
[t*Y*X - s*Z, t*Y*Y + c, t*Y*Z + s*X],
[t*Z*X + s*Y, t*Z*Y - s*X, t*Z*Z + c]]
return np.array(f)
[docs]def rotationMatrix3(rx, ry, rz, angle_spec=DEG):
"""Create a rotation matrix defined by three angles.
This applies successive rotations about the 0, 1 and 2 axes, over
the angles rx, ry and rz, respectively. These angles are also known
as the cardan angles.
Parameters
----------
rx: float
Rotation angle around the 0 axis.
ry: float
Rotation angle around the 1 axis.
rz: float
Rotation angle around the 2 axis.
angle_spec: float, DEG or RAD, optional
The default (DEG) interpretes the angles in degrees. Use RAD to
specify the angle in radians.
Returns
-------
: float array (3,3)
Rotation matrix that performs the combined rotation equivalent
to subsequent rotations around the three global axes.
See Also
--------
rotationMatrix: rotation matrix specified by an axis and angle
cardanAngles: find cardan angles that produce a given rotation matrix
Examples
--------
>>> rotationMatrix3(60,45,30)
array([[ 0.61, 0.35, -0.71],
[ 0.28, 0.74, 0.61],
[ 0.74, -0.57, 0.35]])
"""
Rx = rotationMatrix(rx, 0, angle_spec=angle_spec)
Ry = rotationMatrix(ry, 1, angle_spec=angle_spec)
Rz = rotationMatrix(rz, 2, angle_spec=angle_spec)
return np.dot(Rx, np.dot(Ry, Rz))
[docs]def cardanAngles(R, angle_spec=DEG):
"""Compute cardan angles from rotation matrix
Computes the angles over which to rotate subsequently around
the 0-axis, the 1-axis and the 2-axis to obtain the rotation
corresponding to the given rotation matrix.
Parameters
----------
R: (3,3) float :term:`array_like`
Rotation matrix for post multiplication (see Notes)
angle_spec: :py:attr:`DEG`, :py:attr:`RAD` or float, nonzero.
Divisor applied to the resulting angles before returning.
The default divisor DEG makes the angles be returned in
degrees. Use RAD to get angles in radians.
Returns
-------
(rx,ry,rz): tuple of floats
The three rotation angles around that when applied subsequently
around the global 0, 1 and 2 axes, yield the same rotation
as the input. The default angle_spec=DEG returns the angles
in degrees.
Notes
-----
The returned angles are but one of many ways to obtain a given
rotation by three subsequent rotations around frame axes. Look
on the web for 'Euler angles' to get comprehensive information.
Different sets of angles can be obtained depending on the sequence
of rotation axes used, and whether fixed axes (extrinsic) or
rotated axes (intrinsic) are used in subsequent rotations.
The here obtained 'cardan' angles are commonly denoted as a zy'x''
system with intrinsic angles or xyz with extrinsic angles. It is the
latter angles that are returned.
Because pyFormex stores rotation matrices as post-multiplication
matrices (to be applied on row-vectors), the combined rotation
around first the 0-axis, then the 1-axis and finally the 2-axis,
is found as the matrix product Rx.Ry.Rz. (Traditionally,
vectors were often written as column matrices, and rotation
matrices were pre-multiplication matrices, so the subsequent
rotation matrices would have to be multiplied in reverse order.)
Even if one chooses a single frame system for the subsequent
rotations, the resulting angles are not unique. There are infinitely
many sets of angles that will result in the same rotation matrix.
The implementation here results in angles rx and rz in the range
[-pi,pi], while the angle ry will be in [-pi/2,pi/2]. Even then,
there remain infinite solutions in the case where the elements
R[0,2] == R[2,0] equal +1 or -1 (ry = +pi/2 or -pi/2). The result
will then be the solution with rx==0.
Examples
--------
>>> print("%8.2f "*3 % cardanAngles(rotationMatrix3(60,45,30)))
60.00 45.00 30.00
>>> print("%8.2f "*3 % cardanAngles(rotationMatrix3(0,90,77)))
0.00 90.00 77.00
>>> print("%8.2f "*3 % cardanAngles(rotationMatrix3(0,-90,30)))
0.00 -90.00 30.00
But:
>>> print("%8.2f "*3 % cardanAngles(rotationMatrix3(20,-90,30)))
0.00 -90.00 50.00
"""
if abs(R[0, 2]) < 1.:
theta = -arcsin(R[0, 2])
if theta < -pi/2:
theta = pi - theta
c = cos(theta)
psi = arctan2(R[1, 2]/c, R[2, 2]/c)
phi = arctan2(R[0, 1]/c, R[0, 0]/c)
else:
psi = 0.
if R[0, 2] < 0.:
theta = pi/2
phi = psi - arctan2(R[1, 0], R[2, 0])
else:
theta = -pi/2
phi = psi + arctan2(-R[1, 0], -R[2, 0])
return psi/DEG, theta/DEG, phi/DEG
[docs]def rotmat(x):
"""Create a rotation matrix defined by 3 points in space.
Parameters
----------
x: :term:`array_like` (3,3)
The rows contain the coordinates in 3D space of three non-colinear
points x0, x1, x2.
Returns
-------
rotmat: matrix(3,3)
Rotation matrix which transforms the global axes
into a new (orthonormal) coordinate system with the
following properties:
- the origin is at point x0,
- the 0 axis is along the direction x1-x0
- the 1 axis is in the plane (x0,x1,x2) with x2 lying
at the positive side.
Notes
-----
The rows of the rotation matrix represent the unit vectors
of the resulting coordinate system.
The coodinates in the rotated axes of any point are
obtained by the reverse transformation, i.e. multiplying the
point with the transpose of the rotation matrix.
See also
--------
rotationMatrix: rotation matrix specified by angle and axis
trfmat: transformation matrices defined by 2 sets of 3 points
rotMatrix: rotation matrix transforming global axis 0 into a given vector
rotMatrix2: rotation matrix that transforms one vector into another
Examples
--------
>>> rotmat([[0,0,0],[1,0,0],[0,1,0]])
array([[ 1., 0., 0.],
[ 0., 1., 0.],
[ 0., 0., 1.]])
>>> rotmat(np.eye(3,3))
array([[-0.71, 0.71, 0. ],
[-0.41, -0.41, 0.82],
[ 0.58, 0.58, 0.58]])
>>> s,c = sind(30),cosd(30)
>>> R = rotmat([[0,0,0],[c,s,0],[0,1,0]])
>>> print(R)
[[ 0.87 0.5 0. ]
[-0.5 0.87 0. ]
[ 0. -0. 1. ]]
>>> B = np.array([[2.,0.,0.],[3*s,3*c,3]])
>>> D = np.dot(B,R) # Rotate some vectors with the matrix R
>>> print(D)
[[ 1.73 1. 0. ]
[-0. 3. 3. ]]
"""
x = checkArray(x, shape=(3, 3), kind='f', allow='i')
u = normalize(x[1]-x[0])
v = normalize(x[2]-x[0])
v = normalize(orthog(v, u))
w = np.cross(u, v) # is orthog and normalized
m = np.row_stack([u, v, w])
return m
[docs]def trfmat(x, y):
"""Find the transformation matrices from 3 points x into y.
Constructs the rotation matrix and translation vector that will
transform the points x thus that:
- point x0 coincides with point y0,
- line x0,x1 coincides with line y0,y1
- plane x0,x1,x2 coincides with plane y0,y1,y2
Parameters
----------
x: float :term:`array_like` (3,3)
Original coordinates of three non-colinear points.
y: float :term:`array_like` (3,3)
Final coordinates of the three points.
Returns
-------
rot: float array (3,3)
The rotation matrix for the transformation x to y.
trf:
float array(3,)
The translation vector for the transformation x to y, Obviously,
this is equal to y0-x0.
The rotation is to be applied first and should be around the first
point x0. The full transformation of a Coords object is thus obtained
by ``(coords-x0)*rot+trl+x0 = coords*rot+(trl+x0-x0*rot)``.
Examples
--------
>>> R,T = trfmat(np.eye(3,3), [[0,0,0],[1,0,0],[0,1,0]])
>>> print(R)
[[-0.71 -0.41 0.58]
[ 0.71 -0.41 0.58]
[ 0. 0.82 0.58]]
>>> print(T)
[ 0.71 0.41 -0.58]
"""
# rotation matrices for both systems
r1 = rotmat(x)
r2 = rotmat(y)
# combined rotation matrix
r = np.dot(r1.transpose(), r2)
# translation vector (in a rotate first operation
t = y[0] - np.dot(x[0], r)
return r, t
[docs]def rotMatrix(u, w=[0., 0., 1.]):
# TODO: we could allow here a None value for w, like in rotMatrix2
"""Create a rotation matrix that rotates global axis 0 to a given vector.
Parameters
----------
u: (3,) :term:`array_like`
Vector specifying the direction to which the global axis 0 should
be rotated by the returned rotation matrix.
w: (3,) :term:`array_like`
Vector that is not parallel to u. This vector is used to uniquely
define the resulting rotation. It will be equivalen to rotating
first around ``w``, until the target ``u`` lies in the plane
of the rotated axis 0 and ``w``, then rotated in that plane until
the rotated axis 0 coincides with ``u``. See also Note.
If a parallel w is provided, it will be replaced with a non-parallel
one.
Returns
-------
: float array (3,3)
Rotation matrix that transforms a vector [1.,0.,0.] into ``u``.
The returned matrix should be used in postmultiplication to the
coordinates.
See Also
--------
rotMatrix2: rotation matrix that transforms one vector into another
rotationMatrix: rotation matrix specified by an axis and angle
rotmat: rotation matrix specified by three points in space
trfmat: rotation and translation matrix that transform three points
Examples
--------
>>> rotMatrix([1,0,0])
array([[ 1., 0., 0.],
[ 0., 1., 0.],
[ 0., 0., 1.]])
>>> rotMatrix([0,1,0])
array([[ 0., 1., 0.],
[-1., 0., 0.],
[ 0., -0., 1.]])
>>> rotMatrix([0,0,1])
array([[ 0., 0., 1.],
[ 1., 0., 0.],
[ 0., 1., 0.]])
>>> rotMatrix([0,1,1])
array([[ 0. , 0.71, 0.71],
[-1. , 0. , 0. ],
[ 0. , -0.71, 0.71]])
>>> rotMatrix([1,0,1])
array([[ 0.71, 0. , 0.71],
[ 0. , 1. , 0. ],
[-0.71, 0. , 0.71]])
>>> rotMatrix([1,1,0])
array([[ 0.71, 0.71, 0. ],
[-0.71, 0.71, 0. ],
[ 0. , -0. , 1. ]])
>>> rotMatrix([1,1,1])
array([[ 0.58, 0.58, 0.58],
[-0.71, 0.71, 0. ],
[-0.41, -0.41, 0.82]])
>>> np.dot([1,0,0], rotMatrix([1,1,1]))
array([ 0.58, 0.58, 0.58])
"""
u = unitVector(u)
w = unitVector(w)
v = np.cross(w, u)
if length(v) == 0:
# u and w are parallel
x, y, z = u
w = np.array([0., 1., 0.]) if x==0. and y==0. else np.array([-y, x, 0.])
v = np.cross(w, u)
v = unitVector(v)
w = unitVector(np.cross(u, v))
m = np.row_stack([u, v, w])
return m
[docs]def rotMatrix2(vec1, vec2, upvec=None):
"""Create a rotation matrix that rotates a vector vec1 to vec2.
Parameters
----------
vec1: (3,) :term:`array_like`
Original vector.
vec2: (3,) :term:`array_like`
Direction of ``vec1`` after rotation.
upvec: (3,) :term:`array_like`, optional
If provided, the rotation matrix will be such that the plane of
vec2 and the rotated upvec will be parallel to the original upvec.
If not provided, the rotation matrix will perform a rotation
around the normal to the plane ov the two vectors.
Returns
-------
: float array (3,3)
Rotation matrix that transforms a vector [1.,0.,0.] into ``u``.
The returned matrix should be used in postmultiplication to the
coordinates.
See Also
--------
rotMatrix: rotation matrix transforming global axis 0 into a given vector
rotationMatrix: rotation matrix specified by an axis and angle
rotmat: rotation matrix specified by three points in space
trfmat: rotation and translation matrix that transform three points
Examples
--------
>>> rotMatrix2([1,0,0],[1,0,0])
array([[ 1., 0., 0.],
[ 0., 1., 0.],
[ 0., 0., 1.]])
>>> rotMatrix2([1,0,0],[0,1,0])
array([[ 0., 1., 0.],
[-1., 0., 0.],
[ 0., 0., 1.]])
>>> rotMatrix2([1,0,0],[0,0,1])
array([[ 0., 0., 1.],
[ 0., 1., 0.],
[-1., 0., 0.]])
>>> rotMatrix2([1,0,0],[0,1,1])
array([[ 0. , 0.71, 0.71],
[-0.71, 0.5 , -0.5 ],
[-0.71, -0.5 , 0.5 ]])
>>> rotMatrix2([1,0,0],[1,0,1])
array([[ 0.71, 0. , 0.71],
[ 0. , 1. , 0. ],
[-0.71, 0. , 0.71]])
>>> rotMatrix2([1,0,0],[1,1,0])
array([[ 0.71, 0.71, 0. ],
[-0.71, 0.71, 0. ],
[ 0. , 0. , 1. ]])
>>> rotMatrix2([1,0,0],[1,1,1])
array([[ 0.58, 0.58, 0.58],
[-0.58, 0.79, -0.21],
[-0.58, -0.21, 0.79]])
>>> rotMatrix2([1,0,0],[1,0,0],[0,0,1])
array([[ 1., 0., 0.],
[ 0., 1., 0.],
[ 0., 0., 1.]])
>>> rotMatrix2([1,0,0],[0,1,0],[0,0,1])
array([[ 0., 1., 0.],
[-1., 0., 0.],
[ 0., 0., 1.]])
>>> rotMatrix2([1,0,0],[0,0,1],[0,0,1])
array([[ 0., 0., 1.],
[ 1., 0., 0.],
[ 0., 1., 0.]])
>>> rotMatrix2([1,0,0],[0,1,1],[0,0,1])
array([[ 0. , 0.71, 0.71],
[-1. , 0. , 0. ],
[ 0. , -0.71, 0.71]])
>>> rotMatrix2([1,0,0],[1,0,1],[0,0,1])
array([[ 0.71, 0. , 0.71],
[ 0. , 1. , 0. ],
[-0.71, 0. , 0.71]])
>>> rotMatrix2([1,0,0],[1,1,0],[0,0,1])
array([[ 0.71, 0.71, 0. ],
[-0.71, 0.71, 0. ],
[ 0. , 0. , 1. ]])
>>> rotMatrix2([1,0,0],[1,1,1],[0,0,1])
array([[ 0.58, 0.58, 0.58],
[-0.71, 0.71, 0. ],
[-0.41, -0.41, 0.82]])
"""
vec1 = checkArray(vec1, shape=(3,), kind='f', allow='i')
vec2 = checkArray(vec2, shape=(3,), kind='f', allow='i')
if upvec is None:
upvec = np.cross(vec1, vec2)
if length(upvec) == 0.: # vec1 and vec2 are parallel
return np.eye(3, 3, dtype=Float)
mat1 = rotMatrix(vec1, upvec)
mat2 = rotMatrix(vec2, upvec)
mat = np.dot(mat1.T, mat2)
return mat
[docs]def abat(a, b):
"""Compute the matrix product a * b * at.
Parameters
----------
a: :term:`array_like`, 2-dim
Array with shape (m,n).
b: :term:`array_like`, 2-dim
Array with square shape (n,n).
Returns
-------
:array
Array with shape (m,m) holding the matrix product a * b * at.
See Also
--------
atba
Examples
--------
>>> abat([[1],[2]],[[3]])
array([[ 3, 6],
[ 6, 12]])
>>> abat([[1,2]],[[0,1],[2,3]])
array([[18]])
"""
a = checkArray(a, ndim=2)
b = checkArray(b, shape=(a.shape[1], a.shape[1]))
return np.dot(np.dot(a, b), a.T)
[docs]def atba(a, b):
"""Compute the matrix product at * b * a
Parameters
----------
a: :term:`array_like`, 2-dim
Array with shape (n,m).
b: :term:`array_like`, 2-dim
Array with square shape (n,n).
Returns
-------
:array
Array with shape (m,m) holding the matrix product at * b * a.
Note
----
This multiplication typically occurs when rotating a symmetric tensor b
to axes defined by the rotation matrix a.
See Also
--------
abat
Examples
--------
>>> atba([[1,2]],[[3]])
array([[ 3, 6],
[ 6, 12]])
>>> atba([[1],[2]],[[0,1],[2,3]])
array([[18]])
"""
a = checkArray(a, ndim=2)
b = checkArray(b, shape=(a.shape[0], a.shape[0]))
return np.dot(np.dot(a.T, b), a)
[docs]def horner(a, u):
"""Compute the value of a polynom using Horner's rule.
Parameters
----------
a: float :term:`array_like` (n+1,nd)
``nd``-dimensional coefficients of a polynom of degree ``n``
in a scalar variable ``u``. The coefficients are in order of
increasing degree.
u: float :term:`array_like` (nu)
Parametric values where the polynom is to be evaluated.
Returns
-------
:float array(nu,nd)
The nd-dimensional values of the polynom at the specified `nu`
parameter values.
Examples
--------
>>> print(horner([[1.,1.,1.],[1.,2.,3.]],[0.5,1.0]))
[[ 1.5 2. 2.5]
[ 2. 3. 4. ]]
"""
a = checkArray(a, ndim=2, kind='f', allow='i')
u = checkArray(u, ndim=1, kind='f', allow='i').reshape(-1, 1)
c = a[-1]
for i in range(-2, -1-len(a), -1):
c = c * u + a[i]
return c
[docs]def solveMany(A, B):
"""Solve many systems of linear equations.
Parameters
----------
A: float :term:`array_like` (nsys,ndof,ndof)
Coefficient matrices for nsys systems of ndof linear equations
in ndof unknowns.
B: float :term:`array_like` (nsys,ndof,nrhs)
Right hand sides for the nsys systems of linear equations in ndof
unkn owns.
Each of the nsys systems is solved simultaneously for nrhs right
hand sides.
Returns
-------
X: float array (nsys,ndof,nrhs)
The set of values X(nsys,ndof,nrhs) that solve the systems of linear
equations A @ X = B, where @ is the Python matrix multiplication
operator. Thus for each set of submatrices A[i], B[i], X[i], the
normal matrix multiplication holds: A[i] . X[i] = B[i].
Notes
-----
For values of ndof >= 4, a general linear system soultion method is used.
For values 1, 2 or 3 however, a direct solution method is used which is
much faster.
Examples
--------
This example creates random systems and random values for the variables,
then computes the right hand sides, and solves the equations. Finally
the found solution is compared with the original values of the unknowns.
>>> def solveRandom(nsys, ndof, nrhs):
... A = np.random.rand(nsys, ndof, ndof)
... X = np.random.rand(nsys, ndof, nrhs)
... B = np.stack([np.dot(a,x) for a,x in zip(A,X)])
... Y = solveMany(A,B)
... return np.allclose(X,Y,atol=1.e-4)
>>> nsys, nrhs = 10, 5
>>> print([solveRandom(nsys,ndof,nrhs) for ndof in [1,2,3,4]])
[True, True, True, True]
"""
B = checkArray(B, ndim=3, kind='f', allow='i')
nsys, ndof, nrhs = B.shape
A = checkArray(A, shape=(nsys,ndof,ndof), kind='f', allow='i')
if ndof < 4:
if ndof == 1:
X = B / A
else:
AA = np.expand_dims(A,3)
BB = np.expand_dims(B,2)
if ndof == 2:
N = np.cross(AA[:, :, 0], AA[:, :, 1], axis=1)
AS = np.roll(AA, -1, axis=2)
AS[:, :, 1] *= -1.
X = np.cross(BB, AS, axis=1) / N[:,np.newaxis]
elif ndof == 3:
C = np.cross(np.roll(AA, -1, axis=2),
np.roll(AA, -2, axis=2), axis=1)
N = dotpr(AA[:, :, 0], C[:, :, 0], axis=1)
X = dotpr(BB, C, axis=1) / N[:,np.newaxis]
else:
X = np.stack([np.linalg.solve(A[i], B[i]) for i in range(nsys)])
return X
[docs]def cubicEquation(a, b, c, d):
"""Solve a cubiq equation using a direct method.
Given a polynomial equation of the third degree with real coefficients::
a*x**3 + b*x**2 + c*x + d = 0
Such an equation (with a non-zero) always has exactly three roots, with
some possibly being complex, or identical.
This function computes all three roots of the equation and returns full
information about their nature, multiplicity and sorting order.
It uses scaling of the variables to enhance the accuracy.
Parameters
----------
a: float
Coefficient of the third degree term.
b: float
Coefficient of the second degree term.
c: float
Coefficient of the first degree term.
d: float
Constant in the third degree polynom.
Returns
-------
r1: float
First real root of the cubiq equation
r2: float
Second real root of the cubiq equation or real part of the
complex conjugate second and third root.
r3: float
Third real root of the cubiq equation or imaginary part of the
complex conjugate second and third root.
kind: int
A value specifying the nature and ordering of the roots:
====== ============================================================
kind roots
====== ============================================================
0 three real roots r1 < r2 < r3
1 three real roots r1 < r2 = r3
2 three real roots r1 = r2 < r3
3 three real roots r1 = r2 = r3
4 one real root r1 and two complex conjugate roots with real
part r2 and imaginary part r3; the complex roots are thus:
r2+i*r3 en r2-i*r3, where i=sqrt(-1).
====== ============================================================
Raises
------
ValueError:
If the coefficient a==0 and the equation reduces to a second degree.
Examples
--------
>>> cubicEquation(1.,-6.,11.,-6.)
(array([ 1., 2., 3.]), 0)
>>> cubicEquation(1.,-2.,1.,0.)
(array([-0., 1., 1.]), 1)
>>> cubicEquation(1.,-5.,8.,-4.)
(array([ 1., 2., 2.]), 1)
>>> cubicEquation(1.,-4.,5.,-2.)
(array([ 1., 1., 2.]), 2)
>>> cubicEquation(1.,-3.,3.,-1.)
(array([ 1., 1., 1.]), 3)
>>> cubicEquation(1.,-1.,1.,-1.)
(array([ 1., 0., 1.]), 4)
>>> cubicEquation(1.,-3.,4.,-2.)
(array([ 1., 1., 1.]), 4)
"""
#
# BV: We should return the solution of a second degree equation if a==0
#
if a == 0.0:
raise ValueError("Coefficient a of cubiq equation should not be 0")
e3 = 1./3.
pie = np.pi*2.*e3
r = b/a
s = c/a
t = d/a
# scale variable
sc = max(abs(r), sqrt(abs(s)), abs(t)**e3)
sc = 10**(int(np.log10(sc)))
r = r/sc
s = s/sc/sc
t = t/sc/sc/sc
gc = max(abs(r), abs(s), abs(t)) * 1.e-8
rx = r*e3
p3 = (s-r*rx)*e3
q2 = rx**3-rx*s/2.+t/2.
q2s = q2*q2
p3c = p3**3
som = q2s+p3c
if abs(som) < gc:
# two equal real roots
ic = 1
u = -q2
r1 = np.sign(u) * abs(u)**e3
r2 = -r1-rx
r3 = r2
r1 = r1+r1-rx
if abs(r1-r2) < gc:
ic = 3
if r1 > r2:
ic = 2
r3, r1 = r1, r2
roots = np.array([r1, r2, r3])
elif som < 0.0:
# 3 different roots
ic = 0
rt = sqrt(-p3c)
roots = np.array([-rx] * 3)
if abs(rt) > gc:
phi = arccos(-q2/rt)*e3
rt = 2.*sqrt(-p3)
roots += rt*cos(phi + np.array([0., +pie, -pie]))
# sort the 3 roots
roots.sort()
if roots[1] == roots[2]:
ic += 1
if roots[1] == roots[0]:
ic += 2
else: # som > 0.0
# 1 real and 2 complex conjugate roots
ic = 4
som = sqrt(som)
u = -q2+som
u = np.sign(u) * abs(u)**e3
v = -q2-som
v = np.sign(v) * abs(v)**e3
r1 = u+v
r2 = -r1/2-rx
r3 = (u-v)*sqrt(3.)/2.
r1 = r1-rx
roots = np.array([r1, r2, r3])
# scale and return values
roots *= sc
return roots, ic
###########################################################################
##
## Operations on integer arrays
##
#########################
[docs]def renumberIndex(index):
"""Renumber an index sequentially.
Given a one-dimensional integer array with only non-negative values,
and `nval` being the number of different values in it, and you want to
replace its elements with values in the range `0..nval`, such that
identical numbers are always replaced with the same number and the
new values at their first occurrence form an increasing sequence `0..nval`.
This function will give you the old numbers corresponding with each
position `0..nval`.
Parameters
----------
index: 1-dim int :term:`array_like`
Array with non-negative integer values.
Returns
-------
: int array
A 1-dim int array with length equal to `nval`, where `nval`
is the number of different values in `index`. The elements are
the original values corresponding to the new values `0..nval`.
See Also
--------
inverseUniqueIndex: get the inverse mapping.
Examples
--------
>>> renumberIndex([0,5,2,2,6,0])
array([0, 5, 2, 6])
>>> inverseUniqueIndex(renumberIndex([0,5,2,2,6,0]))[[0,5,2,2,6,0]]
array([0, 1, 2, 2, 3, 0])
"""
un, pos = np.unique(index, True)
srt = pos.argsort()
old = un[srt]
return old
[docs]def inverseUniqueIndex(index):
"""Inverse an index.
Given a 1-D integer array with *unique* non-negative values, and
`max` being the highest value in it, this function returns the position
in the array of the values `0..max`. Values not occurring in input index
get a value -1 in the inverse index.
Parameters
----------
index: 1-dim int :term:`array_like`
Array with non-negative values, which all have to be unique.
It's highest value is `max = index.max()`.
Returns
-------
1-dim int array
Array with length `max+1`, with the position in `index` of each
of the values `0..max`, or -1 if the value does not occur in
`index`.
Note
----
This is a low level function that does not check whether the input
has indeed all unique values.
The inverse index translates the unique index numbers in a
sequential index, so that
``inverseUniqueIndex(index)[index] == np.arange(1+index.max())``.
Examples
--------
>>> inverseUniqueIndex([0,5,2,6])
array([ 0, -1, 2, -1, -1, 1, 3])
>>> inverseUniqueIndex([0,5,2,6])[[0,5,2,6]]
array([0, 1, 2, 3])
"""
index = checkArray(index, ndim=1, kind='i')
if np.unique(index).size != index.size:
raise ValueError("The array does not contain unique values")
inv = -np.ones(index.max()+1, dtype=index.dtype)
inv[index] = np.arange(index.size, dtype=index.dtype)
return inv
[docs]def cumsum0(a):
"""Cumulative sum of a list of numbers preprended with a 0.
Parameters
----------
a: :term:`array_like`, int
List of integers to compute the cumulative sum for.
Returns
-------
: array, int
Array with ``len(a)+1`` integers holding the cumulative sum of the
integers from ``a`` with a 0 prepended.
Examples
--------
>>> cumsum0([2,4,3])
array([0, 2, 6, 9])
A common use case is when concatenating some blocks of different length.
If the list `a` holds the length of each block, the cumsum0(a) holds
the start and end of each block in the concatenation.
>>> L = [ [0,1], [2,3,4,5], [6], [7,8,9] ]
>>> n = cumsum0([len(i) for i in L])
>>> print(n)
[ 0 2 6 7 10]
>>> C = np.concatenate(L)
>>> print(C)
[0 1 2 3 4 5 6 7 8 9]
>>> for i,j in zip(n[:-1],n[1:]):
... print("%s:%s = %s" % (i,j,C[i:j]))
...
0:2 = [0 1]
2:6 = [2 3 4 5]
6:7 = [6]
7:10 = [7 8 9]
"""
return np.cumsum([0]+a)
[docs]def multiplicity(a):
"""Return the multiplicity of the numbers in an array.
Parameters
----------
a: :term:`array_like`, 1-dim
The data array, will be flattened if it is not 1-dim.
Returns
-------
mult: 1-dim int array
The multiplicity of the unique values in a
uniq: 1-dim array
Array of same type as a, with the sorted list of unique values in a.
Examples
--------
>>> multiplicity([0,1,4,3,1,4,3,4,3,3])
(array([1, 2, 4, 3]), array([0, 1, 3, 4]))
>>> multiplicity([[1.0, 0.0, 0.5],[0.2,0.5,1.0]])
(array([1, 1, 2, 2]), array([ 0. , 0.2, 0.5, 1. ]))
"""
a = checkArray1D(a)
bins = np.unique(a)
if bins.size > 0:
mult, b = np.histogram(a, bins=np.concatenate([bins, [max(a)+1]]))
else:
mult = bins
return mult, bins
[docs]def subsets(a):
"""Split an array of integers into subsets.
The subsets of an integer array are sets of elements
with the same value.
Parameters
----------
a: int :term:`array_like`, 1-dim
Array with integer values to be split in subsets
Returns
-------
val: array of ints
The unique values in ``a``, sorted in increasing order.
ind: :class:`varray.Varray`
The Varray has the same number of rows as the number of values in
``ind``. Each row contains the indices in a of the elements with
the corresponding value in ``val``.
Examples
--------
>>> A = [0,1,4,3,1,4,3,4,3,3]
>>> val,ind = subsets(A)
>>> print(val)
[0 1 3 4]
>>> print(ind)
Varray (4,4)
[0]
[1 4]
[3 6 8 9]
[2 5 7]
<BLANKLINE>
The inverse of ``ind`` can be used to restore A from val.
>>> inv = ind.inverse().data
>>> print(inv)
[0 1 3 2 1 3 2 3 2 2]
>>> (val[inv] == A).all()
True
"""
from pyformex.varray import Varray
a = checkArray(a, ndim=1, kind='i')
val = np.unique(a)
ind = Varray([np.where(a==v)[0] for v in val])
return val, ind
[docs]def sortSubsets(a, w=None):
"""Sort subsets of an integer array a.
Subsets of an array are the sets of elements with equal values.
By default the subsets are sorted according to decreasing number
of elements in the set, or if a weight for each element is provided,
according to decreasing sum of weights in the set.
Parameters
----------
a: 1-dim int :term:`array_like`
Input array containing non-negative integer sets to be sorted.
w: 1-dim int or float :term:`array_like`, optional
If provided, it should have the same length as a. Each element of a
will be attributed the corresponding weight.
Specifying no weigth is equivalent to giving all elements the same
weight.
Returns
-------
: int array
Array with same size as a, specifying for each element of a
the index of its subset in the sorted list of subsets.
Examples
--------
>>> sortSubsets([0,1,3,2,1,3,2,3,2,2])
array([3, 2, 1, 0, 2, 1, 0, 1, 0, 0])
>>> sortSubsets([0,1,4,3,1,4,3,4,3,3])
array([3, 2, 1, 0, 2, 1, 0, 1, 0, 0])
>>> sortSubsets([0,1,4,3,1,4,3,4,3,3],w=[9,8,7,6,5,4,3,2,1,0])
array([3, 1, 0, 2, 1, 0, 2, 0, 2, 2])
"""
a = checkArray(a, ndim=1, kind='i')
# Make sure we have unique numbers
a = inverseUniqueIndex(renumberIndex(a))[a]
if w is None:
h, u = multiplicity(a)
else:
w = checkArray(w, shape=a.shape, kind='f', allow='i')
u = np.unique(a)
h = [w[a==j].sum() for j in u]
srt = np.argsort(h)[::-1]
inv = inverseUniqueIndex(srt)
return inv[a]
[docs]def collectOnLength(items, return_index=False):
"""Separate items in a list according to their length.
The length of all items in the list are determined and
the items are put in separate lists according to their length.
Parameters
----------
items: list
A list of any items that can be accepted
as parameter of the len() function.
return_index: bool
If True, also return an index with the positions of the equal
length items in the original iterable.
Returns
-------
col: dict
A dict whose keys are the item lengths and values are lists of
items with this length.
ind: dict, optional
A dict with the same keys as ``col``, and the values being a
list of indices in the list where the corresponding item
of ``col`` appeared.
Examples
--------
>>> collectOnLength(['a','bc','defg','hi','j','kl'])
{1: ['a', 'j'], 2: ['bc', 'hi', 'kl'], 4: ['defg']}
>>> collectOnLength(['a','bc','defg','hi','j','kl'],return_index=True)[1]
{1: [0, 4], 2: [1, 3, 5], 4: [2]}
"""
val, ind = subsets([len(e) for e in items])
col = {}
index = {}
for v, i in zip(val, ind):
col[v] = [items[j] for j in i]
if return_index:
index[v] = i.tolist()
if return_index:
return col, index
else:
return col
[docs]def complement(index, n=-1):
"""Return the complement of an index in a range(0,n).
The complement is the list of numbers from the range(0,n) that are
not included in the index.
Parameters
----------
index: 1-dim int or bool :term:`array_like`
If integer, the array contains non-negative numbers in the range(0,n)
and the return value will be the numbers in range(0,n) not included
in index.
If boolean, False value flag elements to be included (having a value
True) in the output.
n: int
Upper limit for the range of numbers. If `index` is of
type integer and `n` is not specified or is negative, it will be set
equal to the largest number in `index` plus 1. If `index` is of type
boolean and `n` is larger than the length of `index`, `index` will be
padded with `False` values until length `n`.
Returns
-------
: 1-dim array, type int or bool.
The output array has the same dtype as the input.
If `index` is integer: it is an array with the numbers from
range(0,n) that are not included in `index`. If `index` is boolean,
it is the negated input, padded to or cut at length `n`.
Examples
--------
>>> print(complement([0,5,2,6]))
[1 3 4]
>>> print(complement([0,5,2,6],10))
[1 3 4 7 8 9]
>>> print(complement([False,True,True,True],6))
[ True False False False True True]
"""
index = np.asarray(index)
if index.dtype == bool:
m = index.shape[0]
if n > m:
comp = np.ones(n, dtype=bool)
comp[:m] = ~index
else:
comp = ~index[:n]
else:
if n < 0:
n = max(n, 1+index.max())
comp = np.delete(np.arange(n), index)
return comp
[docs]def sortByColumns(a):
"""Sort an array on all its columns, from left to right.
The rows of a 2-dimensional array are sorted, first on the first
column, then on the second to resolve ties, etc..
Parameters
----------
a: :term:`array_like`, 2-dim
The array to be sorted
Returns
-------
: int array, 1-dim
Index specifying the order in which the rows have to
be taken to obtain an array sorted by columns.
Examples
--------
>>> sortByColumns([[1,2],[2,3],[3,2],[1,3],[2,3]])
array([0, 3, 1, 4, 2])
"""
a = checkArray(a, ndim=2)
keys = [a[:, i] for i in range(a.shape[1]-1, -1, -1)]
return np.lexsort(keys)
[docs]def minroll(a):
"""Roll a 1-D array to get the lowest values in front
If the lowest value occurs more than once, the one with the
lowest next value is choosen, etcetera.
Parameters
----------
a: array, 1-dim
The array to roll
Returns
-------
m: int
The index of the element that should be put in front. This
means that the ``np.roll(a,-m)`` gives the rolled array with
the lowest elements in front.
Examples
--------
>>> minroll([1,3,5,1,2,6])
3
>>> minroll([0,0,2,0,0,1])
3
>>> minroll([0,0,0,0,0,0])
0
"""
a = checkArray(a, ndim=1)
m = np.argmin(a)
w = np.where(a==a[m])[0]
m = w[0]
for k in w[1:]:
d = np.roll(a, -m) - np.roll(a, -k)
u = np.where(d!=0)[0]
if len(u) > 0 and d[u[0]] > 0:
m = k
return m
[docs]def isroll(a, b):
"""Check that two 1-dim arrays can be rolled into eachother
Parameters
----------
a: array, 1-dim
The first array
b: array, 1-dim
The second array, same length and dtype as a to be non-trivial.
Returns
-------
m: int
The number of positions (non-negative) that b has to be rolled
to be equal to a, or -2 if the two arrays have a different length,
or -1 if their elements are not the same or not in the same order.
Examples
--------
>>> isroll(np.array([1,2,3,4]), np.array([2,3,4,1]))
1
>>> isroll(np.array([1,2,3,4]), np.array([2,3,1,4]))
-1
>>> isroll(np.array([1,2,3,4]), np.array([3,2,1,4]))
-1
>>> isroll(np.array([1,2,3,4]), np.array([1,2,3]))
-2
"""
a = np.asarray(a)
b = np.asarray(b)
if a.size != b.size:
return -2
for i in range(len(b)):
if (a==np.roll(b, i)).all():
return i
return -1
[docs]def findEqualRows(a, permutations='none'):
"""Find equal rows in a 2-dim array.
Parameters
----------
a: :term:`array_like`, 2-dim
The array in which to find the equal rows.
permutations: str
Defines which permutations of the row data are allowed while still
considering the rows equal. Possible values are:
- 'none': no permutations are allowed: rows must match the same data
at the same positions. This is the default;
- 'roll': rolling is allowed. Rows that can be transformed into
each other by rolling are considered equal;
- 'all': any permutation of the same data will be considered an
equal row.
Returns
-------
ind: 1-dim int array
A row index sorting the rows in such order that equal rows are grouped
together.
ok: 1-dim bool array
An array flagging the rows in the order of ``index`` with True if
it is the first row of a group of equal rows, or with False if the
row is equal to the previous.
Notes
-----
This function provides the functionality for detecting equal rows,
but is seldomly used directly. There are wrapper functions providing more
practical return values. See below.
See Also
--------
equalRows: return the indices of the grouped equal rows
uniqueRows: return the indices of the unique rows
uniqueRowsIndex: like uniqueRows, but also returns index for all rows
Examples
--------
>>> print(*findEqualRows([[1,2],[2,3],[3,2],[1,3],[2,3]]))
[0 3 1 4 2] [ True True True False True]
>>> print(*findEqualRows([[1,2],[2,3],[3,2],[1,3],[2,3]],permutations='all'))
[0 3 1 2 4] [ True True True False False]
>>> print(*findEqualRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]]))
[0 3 2 1] [ True False True True]
>>> print(*findEqualRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='all'))
[0 1 2 3] [ True False False False]
>>> print(*findEqualRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='roll'))
[0 2 3 1] [ True False False True]
"""
a = checkArray(a, ndim=2)
if permutations == 'all':
# Sort the rows
a = a.copy()
a.sort(axis=1)
elif permutations == 'roll':
# Roll the rows until smallest is in front
a = np.stack([np.roll(ai, -minroll(ai)) for ai in a])
ind = sortByColumns(a) # groups the equal rows together
a = a.take(ind, axis=0)
ok = (a != np.roll(a, 1, axis=0)).any(axis=1)
ok[0] = True
return ind, ok
[docs]def equalRows(a, permutations='none'):
"""Return equal rows in a 2-dim array.
Parameters: see :meth:`findEqualRows`
Returns
-------
V: :class:`varray.Varray`
A Varray where each row contains a list of the row numbers
from a that are considered equal. The entries in each row are
sorted and the rows are sorted according to their first element.
Notes
-----
The return Varray holds a lot of information:
- ``V.col(0)`` gives the indices of the unique rows.
- ``complement(V.col(0),len(a))`` gives the indices of duplicate rows.
- ``V.col(0)[V.lengths==1]`` gives the indices of rows without duplicate.
- ``Va.inverse().data`` gives an index into the unique
rows for each of the rows of ``a``.
See Also
--------
findEqualRows: sorts and detects equal rows
uniqueRows: return the indices of the unique rows
uniqueRowsIndex: like uniqueRows, but also returns index for all rows
Examples
--------
>>> equalRows([[1,2],[2,3],[3,2],[1,3],[2,3]])
Varray([[0], [1, 4], [2], [3]])
>>> equalRows([[1,2],[2,3],[3,2],[1,3],[2,3]],permutations='all')
Varray([[0], [1, 2, 4], [3]])
>>> equalRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]])
Varray([[0, 3], [1], [2]])
>>> equalRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='all')
Varray([[0, 1, 2, 3]])
>>> equalRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='roll')
Varray([[0, 2, 3], [1]])
"""
from pyformex import varray
ind, ok = findEqualRows(a, permutations=permutations)
w = np.where(ok)[0]
w = np.concatenate([w, [len(ind)]])
v = [ind[np.arange(i, j)] for i, j in zip(w[:-1], w[1:])]
v = sorted(v, key=lambda a: a.min())
return varray.Varray(v)
[docs]def uniqueRows(a, permutations='none'):
"""Find the unique rows of a 2-D array.
Parameters: see :meth:`findEqualRows`
Returns
-------
uniq: 1-dim int array
Contains the indices of the unique rows in `a`.
See Also
--------
equalRows: return the indices of the grouped equal rows
uniqueRowsIndex: like uniqueRows, but also returns index for all rows
Examples
--------
>>> uniqueRows([[1,2],[2,3],[3,2],[1,3],[2,3]])
array([0, 1, 2, 3])
>>> uniqueRows([[1,2],[2,3],[3,2],[1,3],[2,3]],permutations='all')
array([0, 1, 3])
>>> uniqueRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]])
array([0, 1, 2])
>>> uniqueRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='all')
array([0])
>>> uniqueRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='roll')
array([0, 1])
>>> uniqueRows([[1,2,3],[3,2,1],[2,3,1],[1,2,3]])
array([0, 1, 2])
"""
ind, ok = findEqualRows(a, permutations=permutations)
return np.sort(ind[ok])
[docs]def uniqueRowsIndex(a, permutations='none'):
"""Return the unique rows of a 2-D array and an index for all rows.
Parameters
----------
a: :term:`array_like`, 2-dim
Array from which to find the unique rows.
permutations: bool
If True, rows which are permutations of the same data are considered
equal. The default is to consider permutations as different.
roll: bool
If True, rows which can be rolled into the same contents are
considered equal.
Returns
-------
uniq: 1-dim int array
Contains the indices of the unique rows in `a`.
The order of the elements in `uniq` is determined by the sorting
procedure: in the current implementation this is :func:`sortByColumns`.
If `permutations==True`, `a` is sorted along its last axis -1 before
calling this sorting function. If `roll=True`, the rows of ``a``
are rolled to put the lowest values at the front.
ind: 1-dim int array
For each row of `a`, holds the index in `uniq` where the row with
the same data is found.
See Also
--------
equalRows: return the indices of the grouped equal rows
uniqueRows: return the indices of the unique rows
Examples
--------
>>> print(*uniqueRowsIndex([[1,2],[2,3],[3,2],[1,3],[2,3]]))
[0 1 2 3] [0 1 2 3 1]
>>> print(*uniqueRowsIndex([[1,2],[2,3],[3,2],[1,3],[2,3]],permutations='all'))
[0 1 3] [0 1 1 2 1]
>>> print(*uniqueRowsIndex([[1,2,3],[3,2,1],[2,3,1],[1,2,3]]))
[0 1 2] [0 1 2 0]
>>> print(*uniqueRowsIndex([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='all'))
[0] [0 0 0 0]
>>> print(*uniqueRowsIndex([[1,2,3],[3,2,1],[2,3,1],[1,2,3]],permutations='roll'))
[0 1] [0 1 0 0]
>>> print(*uniqueRowsIndex([[1,2,3],[3,2,1],[2,3,1],[1,2,3]]))
[0 1 2] [0 1 2 0]
"""
Va = equalRows(a, permutations=permutations)
return Va.col(0), Va.inverse().data
[docs]def argNearestValue(values, target):
"""Return the index of the item nearest to target.
Find in a list of floats the position of the value nearest to
the target value.
Parameters
----------
values: list
List of float values.
target: float
Target value to look up in list.
Returns
-------
:int
The index in `values` of the float value that is closest
to `target`.
See Also
--------
nearestValue
Examples
--------
>>> argNearestValue([0.1,0.5,0.9],0.7)
1
"""
v = np.array(values).ravel()
c = v - target
return np.argmin(c*c)
[docs]def nearestValue(values, target):
"""Return the float nearest to target.
Find in a list of floats the value that is closest to
the target value.
Parameters
----------
values: list
List of float values.
target: float
Target value to look up in list.
Returns
-------
:float
The value from the list that is closest to `target`.
See Also
--------
argNearestValue
Examples
--------
>>> nearestValue([0.1,0.5,0.9],0.7)
0.5
"""
return values[argNearestValue(values, target)]
[docs]def inverseIndex(a, sort=True):
"""Create the inverse of a 2D-index array.
A 2D-index array is a 2D integer array where only the nonnegative values.
are relevant. Negative values are flagging a non-existent element. This
allows for rows with different number of entries.
While in most practical cases all (non-negative) values in a row are
unique, this is not a requirement.
Parameters
----------
a: :term:`varray_like`.
The input index table. This can be anything that is acceptable as
data for the Varray constructor.
sort: bool.
If True, the values on each row of the returned index are sorted.
The default (False) will leave the values in the order obtained
by the algorithm, which depends on Python/numpy sorting.
Returns
-------
inv: :class:`numpy.ndarray`
The inverse index as an array. Each row ``i`` of the inverse index
contains the numbers of the rows of the input in which a value ``i``
appeared, and padded with -1 values to make all rows equal length.
With ``sort=True``, the values on each row are guaranteed to be sorted.
See also
--------
Varray.inverse Return the inverse as a Varray
Note
----
If the same value occurs multiple times on the same row of the input,
the inverse index will also contain repeated row numbers for that value.
Examples
--------
>>> A = np.array([[0,1],[0,2],[1,2],[0,3]])
>>> print(A)
[[0 1]
[0 2]
[1 2]
[0 3]]
>>> inv = inverseIndex(A)
>>> print(inv)
[[ 0 1 3]
[-1 0 2]
[-1 1 2]
[-1 -1 3]]
The inverse of the inverse returns the original:
>>> (inverseIndex(inv) == A).all()
True
"""
from pyformex import varray
return varray.Varray(a).inverse(sort=sort).toArray()
[docs]def findFirst(target, values):
"""Find first position of values in target.
Find the first position in the array `target` of all the elements
in the array `values`.
Parameters
----------
target: 1-dim int array
Integer array with all non-negative values. If not 1-dim, it
will be flattened.
values: 1-dim int array
Array with values to look up in target. If not 1-dim, it
will be flattened.
Returns
-------
: int array
Array with same size as `values`. For each element in `values`,
the return array contains the position of that value
in the flattened `target`, or -1 if that number does not occur
in `target`.
If an element from `values` occurs more than once in `target`, it is
currently undefined which of those positions is returned.
Note
----
After ``m = findIndex(target,values)`` the equality
``target[m] == values`` holds for all the non-negative positions of `m`.
Examples
--------
>>> A = np.array([1,3,4,5,7,3,8,9])
>>> B = np.array([0,7,1,3])
>>> ind = findFirst(A,B)
>>> print(ind)
[-1 4 0 1]
>>> (A[ind[ind>=0]] == B[ind>=0]).all()
True
"""
from pyformex import varray
target = checkArray1D(target, kind='i').reshape(-1, 1)
values = checkArray1D(values, kind='i')
inv = varray.Varray(target).inverse(sort=True)
return np.array([inv[i][0] if i < inv.nrows and len(inv[i]) > 0 else -1
for i in values])
[docs]def matchLine2(edges, edges1):
"""Match Line elems in a given Connectivity.
Find the rows in edges that have the same nodes as the rows of edges1.
Parameters
----------
edges: int :term:`array_like`
An int array (nedges,2), e.g. a Line2 :class:`Connectivity`.
edges1: int :term:`array_like`
An int array (nedges1,2), e.g. a Line2 :class:`Connectivity`.
Returns
-------
int array
An int array (nedges1,) specifying for each row of edges1 which row
of edges contains the same two nodes (in any order). Rows that do
not occur in edges get a value -1. If multiple rows are matching,
the first one is returned.
"""
e = np.sort(edges, axis=-1)
e1 = np.sort(edges1, axis=-1)
maxv = max(e.max(), e1.max()) + 1
e = maxv*e[:, 0] + e[:, 1]
e1 = maxv*e1[:, 0] + e1[:, 1]
return findFirst(e, e1)
[docs]def findAll(target, values):
"""Find all locations of values in target.
Find the position in the array `target` of all occurrences of
the elements in the array `values`.
Parameters
----------
target: 1-dim int array
Integer array with all non-negative values. If not 1-dim, it
will be flattened.
values: 1-dim int array
Array with values to look up in target. If not 1-dim, it
will be flattened.
Returns
-------
: list of int arrays.
For each element in values, an array is returned with the indices
in target of the elements with the same value.
See Also
--------
findFirst
Examples
--------
>>> gid = np.array([ 2, 1, 1, 6, 6, 1 ])
>>> values = np.array([ 1, 2, 6 ])
>>> print(findAll(gid,values))
[array([1, 2, 5]), array([0]), array([3, 4])]
"""
return [np.where(target==i)[0] for i in values]
[docs]def groupArgmin(val, gid):
"""Compute the group minimum.
Computes the minimum value per group of a set of values tagged with
a group number.
Parameters
----------
val: 1-dim array
Data values
gid: 1-dim int :term:`array_like`
Array with same length as val, containing the group identifiers.
Returns
-------
ugid: 1-dim int array
(ngrp,) shaped array with unique group identifiers.
minpos: 1-dim int array
(ngrp,) shaped array giving the position in `val` of the minimum
of all values with the corresponding group identifier in `ugid`.
The minimum values corresponding to the groups in `ugid` can be
obtained with ``val[minpos]``.
Examples
--------
>>> val = np.array([ 0.0, 1.0, 2.0, 3.0, 4.0, -5.0 ])
>>> gid = np.array([ 2, 1, 1, 6, 6, 1 ])
>>> print(groupArgmin(val,gid))
(array([1, 2, 6]), array([5, 0, 3]))
"""
ugid = np.unique(gid)
pos = findAll(gid, ugid)
minid = np.hstack([val[ind].argmin() for ind in pos])
minpos = np.hstack([ind[k] for ind, k in zip(pos, minid)])
return ugid, minpos
[docs]def collectRowsByColumnValue(a, col):
"""Collects rows of a 2D array by common value in a specified column.
Parameters
----------
a: 2-dim :term:`array_like`
Any 2-dim array.
col: int
Column number on which values to collect the rows.
Returns
-------
dict
A dict where the keys are the unique values of the specified
column of the array `a`. The values are int arrays with the
indices of the rows that have the key value in their `col`
column.
Examples
--------
>>> a = np.array([[0,0], [1,1], [1,0], [0,1], [4,0]])
>>> print(a)
[[0 0]
[1 1]
[1 0]
[0 1]
[4 0]]
>>> d = collectRowsByColumnValue(a,0)
>>> print(d)
{0: array([0, 3]), 1: array([1, 2]), 4: array([4])}
>>> d = collectRowsByColumnValue(a,1)
>>> print(d)
{0: array([0, 2, 4]), 1: array([1, 3])}
"""
edgset = np.unique(a[:, col])
d = {}
for e in edgset:
d[e] = np.where(a[:, col]==e)[0]
return d
###########################################################
# Working with sets of vectors
###########################
[docs]def vectorPairAreaNormals(vec1, vec2):
"""Compute area of and normals on parallellograms formed by vec1 and vec2.
Parameters
----------
vec1: (3,) or (n,3) shaped float :term:`array_like`
Array with 1 or n vectors in 3D space.
vec2: (3,) or (n,3) shaped float :term:`array_like`
Array with 1 or n vectors in 3D space.
Returns
-------
area: (n,) shaped float array
The area of the parallellograms formed by the vectors vec1 and vec2.
normal: (n,3) shaped float array
The unit length vectors normal to each vector pair (vec1,vec2).
Note
----
This first computes the cross product of vec1 and vec2, which is a normal
vector with length equal to the area. Then :func:`normalize` produces
the required results.
Note that where two vectors are parallel, an area zero results and
an axis with components NaN.
See Also
--------
vectorPairNormals: only returns the normal vectors
vectorPairArea: only returns the area
Examples
--------
>>> a = np.array([[3.,4,0],[1,0,0],[1,-2,1]])
>>> b = np.array([[1.,3.,0],[1,0,1],[-2,4,-2]])
>>> l,v = vectorPairAreaNormals(a,b)
>>> print(l)
[ 5. 1. 0.]
>>> print(v)
[[ 0. 0. 1.]
[ 0. -1. 0.]
[ nan nan nan]]
"""
vec1 = checkArray(vec1, kind='f', allow='i')
vec2 = checkArray(vec2, kind='f', allow='i')
normal = np.cross(vec1.reshape(-1, 3), vec2.reshape(-1, 3))
return normalize(normal, return_length=True)[::-1]
[docs]def vectorPairNormals(vec1, vec2):
"""Create unit vectors normal to vec1 and vec2.
Parameters
----------
vec1: (3,) or (n,3) shaped float :term:`array_like`
Array with 1 or n vectors in 3D space.
vec2: (3,) or (n,3) shaped float :term:`array_like`
Array with 1 or n vectors in 3D space.
Returns
-------
normal: (n,3) shaped float array
The unit length vectors normal to each vector pair (vec1,vec2).
See Also
--------
vectorPairAreaNormals: returns the normals and the area between vectors
vectorPairArea: only returns the area between vectors
Examples
--------
>>> a = np.array([[3.,4,0],[1,0,0],[1,-2,1]])
>>> b = np.array([[1.,3.,0],[1,0,1],[-2,4,-2]])
>>> v = vectorPairNormals(a,b)
>>> print(v)
[[ 0. 0. 1.]
[ 0. -1. 0.]
[ nan nan nan]]
"""
return vectorPairAreaNormals(vec1, vec2)[1]
[docs]def vectorPairArea(vec1, vec2):
"""Compute area of the parallellogram formed by a vector pair vec1,vec2.
Parameters
----------
vec1: (3,) or (n,3) shaped float :term:`array_like`
Array with 1 or n vectors in 3D space.
vec2: (3,) or (n,3) shaped float :term:`array_like`
Array with 1 or n vectors in 3D space.
Returns
-------
area: (n,) shaped float array
The area of the parallellograms formed by the vectors vec1 and vec2.
See Also
--------
vectorPairAreaNormals: returns the normals and the area between vectors
vectorPairNormals: only returns the normal vectors
Examples
--------
>>> a = np.array([[3.,4,0],[1,0,0],[1,-2,1]])
>>> b = np.array([[1.,3.,0],[1,0,1],[-2,4,-2]])
>>> l = vectorPairArea(a,b)
>>> print(l)
[ 5. 1. 0.]
"""
vec1 = checkArray(vec1, kind='f', allow='i')
vec2 = checkArray(vec2, kind='f', allow='i')
normal = np.cross(vec1.reshape(-1, 3), vec2.reshape(-1, 3))
return length(normal)
[docs]def vectorTripleProduct(vec1, vec2, vec3):
"""Compute triple product vec1 . (vec2 x vec3).
Parameters
----------
(vec1, vec2, vec3): three (3,) or (n,3) shaped float :term:`array_like`
Three arrays with same shape holding 1 or n vectors in 3D space.
Returns
-------
: (n,) shaped float array
The triple product of each set of corresponding vectors from
vec1, vec2, vec3.
Note
----
The triple product is the dot product of the first vector(s) and
the normal(s) on the second and third vector(s).
This is also twice the volume of the parallellepid formed by
the 3 vectors.
If vec1 has a unit length, the result is also the area of the parallellogram
(vec2,vec3) projected in the direction vec1.
This is functionally equivalent with `dotpr(vec1, np.cross(vec2, vec3))`
but is implemented in a more efficient way, using the determinant formula.
Examples
--------
>>> vectorTripleProduct([[1.,0.,0.],[2.,0.,0.]],
... [[1.,1.,0.],[2.,2.,0.]],
... [[1.,1.,1.],[2.,2.,2.]])
array([ 1., 8.])
"""
vec1 = np.asarray(vec1)
vec2 = np.asarray(vec2)
vec3 = np.asarray(vec3)
a11 = vec1[..., 0]
a12 = vec1[..., 1]
a13 = vec1[..., 2]
a21 = vec2[..., 0]
a22 = vec2[..., 1]
a23 = vec2[..., 2]
a31 = vec3[..., 0]
a32 = vec3[..., 1]
a33 = vec3[..., 2]
return a11*(a22*a33-a32*a23) + a12*(a23*a31-a33*a21) + a13*(a21*a32-a22*a31)
[docs]def vectorPairCosAngle(v1, v2):
"""Return the cosinus of the angle between the vectors v1 and v2.
vec1 and vec2 are (n,3) shaped arrays holding collections of vectors.
The result is an (n) shaped array with the cosinus of the angle between
each pair of vectors (vec1,vec2).
"""
v1 = np.asarray(v1)
v2 = np.asarray(v2)
cos = dotpr(v1, v2) / sqrt(dotpr(v1, v1)*dotpr(v2, v2))
# clip to [-1.,1.] in case of rounding errors
return cos.clip(min=-1., max=1.)
[docs]def vectorPairAngle(v1, v2):
"""Return the angle (in radians) between the vectors v1 and v2.
vec1 and vec2 are (n,3) shaped arrays holding collections of vectors.
The result is an (n) shaped array with the angle between
each pair of vectors (vec1,vec2).
>>> vectorPairAngle([1,0,0],[0,1,0]) / DEG
90.0
>>> vectorPairAngle([[1,0,0],[0,1,0]],[[1,1,0],[1,1,1]]) / DEG
array([ 45. , 54.74])
"""
return arccos(vectorPairCosAngle(v1, v2))
[docs]def det2(a):
"""Compute the determinant of 2x2 matrices.
Parameters
----------
a: int or float :term:`array_like` (...,2,2)
Array containing one or more (2,2) square matrices.
Returns
-------
: int or float number or array(...)
The determinant(s) of the matrices. The result has the same type
as the input array.
Note
----
This method is faster than the generic numpy.linalg.det.
See Also
--------
det3: determinant of (3,3) matrices
det4: determinant of (4,4) matrices
numpy.linalg.det: determinant of any size matrix
Examples
--------
>>> det2([[1,2],[2,1]])
-3
>>> det2([[[1,2],[2,1]],[[4,2],[1,3]]])
array([-3, 10])
"""
a = np.asarray(a)
a11 = a[..., 0, 0]
a12 = a[..., 0, 1]
a21 = a[..., 1, 0]
a22 = a[..., 1, 1]
return a11*a22 - a12*a21
[docs]def det3(a):
"""Compute the determinant of 3x3 matrices.
Parameters
----------
a: int or float :term:`array_like` (...,3,3)
Array containing one or more (3,3) square matrices.
Returns
-------
: int or float number or array(...)
The determinant(s) of the matrices. The result has the same type
as the input array.
Note
----
This method is faster than the generic numpy.linalg.det.
See Also
--------
det2: determinant of (2,2) matrices
det4: determinant of (4,4) matrices
numpy.linalg.det: determinant of any size matrix
Examples
--------
>>> det3([[1,2,3],[2,2,2],[3,2,1]])
0
>>> det3([[[1.,0.,0.],[1.,1.,0.],[1.,1.,1.]],
... [[2.,0.,0.],[2.,2.,0.],[2.,2.,2.]]])
array([ 1., 8.])
"""
a = np.asarray(a)
return vectorTripleProduct(a[..., 0, :], a[..., 1, :], a[..., 2, :])
[docs]def det4(a):
"""Compute the determinant of 4x4 matrices.
Parameters
----------
a: int or float :term:`array_like` (...,4,4)
Array containing one or more (4,4) square matrices.
Returns
-------
: int or float number or array(...)
The determinant(s) of the matrices. The result has the same type
as the input array.
Note
----
This method is faster than the generic numpy.linalg.det.
See Also
--------
det2: determinant of (2,2) matrices
det3: determinant of (3,3) matrices
numpy.linalg.det: determinant of any size matrix
Examples
--------
>>> det4([[[1.,0.,0.,0.],[1.,1.,0.,0.],[1.,1.,1.,0.],[1.,1.,1.,1.]],
... [[2.,0.,0.,0.],[2.,2.,0.,0.],[2.,2.,2.,0.],[2.,2.,2.,2.]]])
array([ 1., 16.])
"""
a = np.asarray(a)
a00 = a[..., 0, 0]
a01 = a[..., 0, 1]
a02 = a[..., 0, 2]
a03 = a[..., 0, 3]
m00 = det3(a[..., 1:, [1, 2, 3]])
m01 = det3(a[..., 1:, [0, 2, 3]])
m02 = det3(a[..., 1:, [0, 1, 3]])
m03 = det3(a[..., 1:, [0, 1, 2]])
return a00*m00 - a01*m01 + a02*m02 - a03*m03
[docs]def percentile(values, perc=[25., 50., 75.], wts=None):
"""Return percentiles of a set of values.
A percentiles is the value such that at least a given percent of the
values is lower or equal than the value.
Parameters
----------
values: 1-dim int or float :term:`array_like`
The set of values for which to compute the percentiles.
perc: 1-dim int or float :term:`array_like`
One or multiple percentile values to compute. All values should be
in the range [0,100]. By default, the quartiles are computed.
wts: 1-dim array
Array with same shape as values and all positive values. These are
weights to be assigned to the values.
Returns
-------
: 1-dim float array
Array with the percentile value(s) that is/are greater or equal than
`perc` percent of `values`. If the result lies between two items of
`values`, it is obtained by interpolation.
Examples
--------
>>> percentile(np.arange(100),[10,50,90])
array([ 9., 49., 89.])
>>> percentile([1,1,1,1,1,2,2,2,3,5])
array([ 1., 1., 2.])
"""
values = checkArray1D(values)
perc = checkArray1D(perc)
if perc.min() < 0. or perc.max() > 100.:
raise ValueError("Percentiles should be between 0 and 100, got %s." % perc)
if wts is not None and wts.min() <= 0.:
raise ValueError("Weights should be positive, got %s." % wts.min())
ind = values.argsort()
values = values[ind]
if wts is None:
wts = np.resize(1., values.shape)
else:
wts = wts[ind]
wts = wts.cumsum()
w = perc /100. * (wts[-1])
ind = wts.searchsorted(w)
val = np.where(ind>0,
values[ind-1] + (w-wts[ind-1]) / (wts[ind]-wts[ind-1])
* (values[ind]-values[ind-1]),
values[0])
return val
[docs]def histogram2(a, bins, range=None):
"""Compute the histogram of a set of data.
This is similar to the numpy histogram function, but also returns
the bin index for each individual entry in the data set.
Parameters
----------
a: :term:`array_like`
Input data. The histogram is computed over the flattened array.
bins: int or sequence of scalars.
If `bins` is an int, it defines the number of equal-width bins
in the given range (nbins).
If `bins` is a sequence, it defines the bin edges, including the
rightmost edge, allowing for non-uniform bin widths. The number of
bins (nbins) is then equal to `len(bins) - 1`.
A value `v` will be sorted in bin `i` if
`bins[i] <= v < bins[i+1]`, except for the last bin, which will also
contain the values equal to the right bin edge.
range`: (float, float), optional.
The lower and upper range of the bins.
If not provided, range is simply (a.min(), a.max()). Values outside the
range are ignored. This parameter is ignored if bins is a sequence.
Returns
-------
hist: int array
The number of elements from `a` sorted in each of the bins.
ind: list of ``nbins`` int arrays
Each array holds the indices the elements sorted in the
corresponding bin.
bin_edges: float array
The array contains the ``len(hist)+1`` bin edges.
Example
-------
>>> hist,ind,bins = histogram2([1,2,3,4,2,3,1],[1,2,3,4,5])
>>> print(hist)
[2 2 2 1]
>>> for i in ind: print(i)
[0 6]
[1 4]
[2 5]
[3]
>>> print(bins)
[1 2 3 4 5]
>>> hist,bins = np.histogram([1,2,3,4,2,3,1],5)
>>> print(hist)
[2 2 0 2 1]
>>> hist,ind,bins = histogram2([1,2,3,4,2,3,1],5)
>>> print(hist)
[2 2 0 2 1]
>>> for i in ind: print(i)
[0 6]
[1 4]
[]
[2 5]
[3]
"""
a = np.asarray(a)
if np.iterable(bins):
bins = np.asarray(bins)
if (np.diff(bins) < 0).any():
raise ValueError("`bins` must increase monotonically.")
else:
if not isInt(bins) or bins < 1:
raise ValueError("`bins` should be a positive integer.")
if range is None:
if a.size == 0:
# handle empty arrays. Can't determine range, so use 0-1.
range = (0, 1)
else:
range = (a.min(), a.max())
mn, mx = [mi+0.0 for mi in range]
if mn == mx:
mn -= 0.5
mx += 0.5
bins = 1
else:
mn, mx = range
if mn >= mx:
raise ValueError('max must be larger than min in `range` parameter.')
bins = np.linspace(mn, mx, bins+1, endpoint=True)
# Make sure to include the right edge of the last bin
mx = bins[-1]
bins[-1] = np.nextafter(mx, mx+0.5)
d = np.digitize(a, bins)
ind = [np.where(d==i)[0] for i in np.arange(1, len(bins))]
hist = np.asarray([i.size for i in ind])
return hist, ind, bins
[docs]def movingView(a, size):
"""Create a moving view along the first axis of an array.
A moving view of an array is a view stacking a sequence of subarrays
with fixed size along the 0 axis of the array, where each next subarray
shifts one position down along the 0 axis.
Parameters
----------
a: :term:`array_like`
Array for which to create a moving view.
size : int
Size of the moving view: this is the number of rows to include
in the subarray.
Returns
-------
: view of the array ``a``
The view of the original array has an extra first axis with length
``1 + a.shape[0] - size``, a second axis with length ``size``, and
the remaining axes have the same length as those in ``a``.
Note
----
While this function limits the moving view to the direction of the 0 axis,
using swapaxes(0,axis) allows to create moving views over any axis.
See Also
--------
movingAverage: compute moving average values along axis 0
Examples
--------
>>> x=np.arange(10).reshape((5,2))
>>> print(x)
[[0 1]
[2 3]
[4 5]
[6 7]
[8 9]]
>>> print(movingView(x, 3))
[[[0 1]
[2 3]
[4 5]]
<BLANKLINE>
[[2 3]
[4 5]
[6 7]]
<BLANKLINE>
[[4 5]
[6 7]
[8 9]]]
Calculate rolling sum of first axis:
>>> print(movingView(x, 3).sum(axis=0))
[[ 6 9]
[12 15]
[18 21]]
"""
from numpy.lib import stride_tricks
if size < 1:
raise ValueError("`size` must be at least 1.")
if size > a.shape[0]:
raise ValueError("`size` is too long.")
shape = (size, a.shape[0] - size + 1) + a.shape[1:]
strides = (a.strides[0],) + a.strides
return stride_tricks.as_strided(a, shape=shape, strides=strides)
[docs]def movingAverage(a, n, m0=None, m1=None):
"""Compute the moving average along the first axis of an array.
Parameters
----------
a: :term:`array_like`
The array to be averaged.
n: int
Sample length along axis 0 over which to compute the average.
m0: int, optional
If provided, the first data row of ``a`` will be prepended to ``a``
this number of times.
m1 : int, optional
If provided, the last data row of ``a`` will be appended to ``a``
this number of times.
Returns
-------
: float array
Array containing the moving average over data sets of length ``n``
along the first axis of ``a``.
The array has a shape like ``a`` except for its first axis,
which may have a different length.
If neither m0 nor m1 are set, the first axis will have a length of
1 + a.shape[0] - n.
If both m0 and m1 are given, the first axis will have a length of
1 + a.shape[0] - n + m0 + m1.
If either m0 or m1 are set and the other not, the missing value m0
or m1 will be computed thus that the return array has a first axis
with length a.shape[0].
Examples
--------
>>> x=np.arange(10).reshape((5,2))
>>> print(x)
[[0 1]
[2 3]
[4 5]
[6 7]
[8 9]]
>>> print(movingAverage(x,3))
[[ 2. 3.]
[ 4. 5.]
[ 6. 7.]]
>>> print(movingAverage(x,3,2))
[[ 0. 1. ]
[ 0.67 1.67]
[ 2. 3. ]
[ 4. 5. ]
[ 6. 7. ]]
"""
if m0 is None and m1 is None:
ae = a
else:
if m0 is None:
m0 = n-1 - m1
elif m1 is None:
m1 = n-1 - m0
if m0 < 0 or m1 < 0:
raise ValueError("Invalid value a m0 or m1")
ae = [a[:1]] * m0 + [a] + [a[-1:]] * m1
ae = np.concatenate(ae, axis=0)
return movingView(ae, n).mean(axis=0)
[docs]def randomNoise(shape, min=0.0, max=1.0):
"""Create an array with random float values between min and max
Parameters
----------
shape: tuple of ints
Shape of the array to create.
min: float
Minimum value of the random numbers.
max: float
Maximum value of the random numbers.
Returns
-------
: float array
An array of the requested shape filled with random numbers
in the specified range.
Examples
--------
>>> x = randomNoise((3,4))
>>> x.shape == (3,4)
True
>>> (x >= 0.0).all()
True
>>> (x <= 1.0).all()
True
"""
return np.random.random(shape) * (max-min) + min
[docs]def stuur(x, xval, yval, exp=2.5):
"""Returns a (non)linear response on the input x.
xval and yval should be lists of 3 values:
``[xmin,x0,xmax], [ymin,y0,ymax]``.
Together with the exponent exp, they define the response curve
as function of x. With an exponent > 0, the variation will be
slow in the neighbourhood of (x0,y0).
For values x < xmin or x > xmax, the limit value ymin or ymax
is returned.
Examples
--------
>>> x = unitDivisor(4)
>>> x
array([ 0. , 0.25, 0.5 , 0.75, 1. ])
>>> np.array([stuur(xi, (0.,0.5,1.0), (0.,0.5,1.0) ) for xi in x])
array([ 0. , 0.41, 0.5 , 0.59, 1. ])
"""
xmin, x0, xmax = xval
ymin, y0, ymax = yval
if x < xmin:
return ymin
elif x < x0:
xr = float(x-x0) / (xmin-x0)
return y0 + (ymin-y0) * xr**exp
elif x < xmax:
xr = float(x-x0) / (xmax-x0)
return y0 + (ymax-y0) * xr**exp
else:
return ymax
[docs]def unitDivisor(div):
"""Divide a unit interval in equal parts.
This function is intended to be used by interpolation functions
that accept an input as either an int or a list of floats.
Parameters
----------
div: int, or list of floats in the range [0.0, 1.0].
If it is an integer, it specifies the number of equal sized parts
in which the interval [0.0, 1.0] is to be divided.
If a list of floats, its values should be monotonuously increasing
from 0.0 to 1.0. The values are returned unchanged.
Returns
-------
: 1-dim float array
The float values that border the parts of the interval.
If `div` is a an integer, returns the floating point values
dividing the unit interval in div equal parts. If `div` is a list,
just returns `div` as a 1D array.
Examples
--------
>>> unitDivisor(4)
array([ 0. , 0.25, 0.5 , 0.75, 1. ])
>>> unitDivisor([0., 0.3, 0.7, 1.0])
array([ 0. , 0.3, 0.7, 1. ])
"""
if isInt(div):
div = np.arange(div+1) / float(div)
else:
div = checkArray1D(div, kind='f', allow='i')
return div
[docs]def unitAttractor(x, e0=0., e1=0.):
"""Moves values in the range 0..1 closer to or away from the limits.
Parameters
----------
x: float :term:`array_like`
Values in the range 0.0 to 1.0, to be pulled to/pushed from ends.
e0: float
Attractor force to the start of the interval (0.0). A negative
value will push the values away from this point.
e1: float
Attractor force to the end of the interval (1.0). A negative
value will push the values away from this point.
Note
----
This function is usually called from the :func:`seed` function,
passing an initially uniformly distributed set of points.
Examples
--------
>>> np.set_printoptions(precision=4)
>>> print(unitAttractor([0.,0.25,0.5,0.75,1.0], 2.))
[ 0. 0.0039 0.0625 0.3164 1. ]
>>> np.set_printoptions(precision=2)
>>> unitAttractor([0.,0.25,0.5,0.75,1.0])
array([ 0. , 0.25, 0.5 , 0.75, 1. ])
"""
x = np.asarray(x)
e0 = 2**e0
e1 = 2**e1
at0 = lambda x, e: x**e # noqa: E731
at1 = lambda x, e: 1.-(1.-x)**e # noqa: E731
return 0.5 * (at1(at0(x, e0), e1) + at0(at1(x, e1), e0))
[docs]def seed(n, e0=0., e1=0.):
# TODO: this function could be merged into smartSeed
"""Create a list of seed values.
A seed list is a list of float values in the range 0.0 to 1.0.
It can be used to subdivide a line segment or to seed nodes
along lines for meshing purposes.
This function divides the unit interval in `n` parts, resulting
in `n+1` seed values. While the intervals are by default of equal
length, the `e0` and `e1` can be used to create unevenly spaced
seed values.
Parameters
----------
n: int
Positive integer: the number of elements (yielding `n+1`
parameter values).
e0: float
Attractor force at the start of the interval. A value larger
than zero will attract the points closer to 0.0, while
a negative value will repulse them.
e1: float
Attractor force at the end of the interval. A value larger
than zero will attract the points closer to 1.0, while
a negative value will repulse them.
Returns
-------
float arraya list of `n+1` float values in the range 0.0 to 1.0.
The values are in ascending order, starting with 0.0 and ending with 1.0.
See Also
--------
seed1: attractor at one end and equidistant points at the other.
smartSeed: similar function accepting a variety of input.
Examples
--------
>>> np.set_printoptions(precision=4)
>>> print(seed(5,2.,2.))
[ 0. 0.0639 0.3362 0.6638 0.9361 1. ]
>>> np.set_printoptions(precision=2)
>>> for e0 in [ 0., 0.1, 0.2, 0.5, 1.0]: print(seed(5,e0))
[ 0. 0.2 0.4 0.6 0.8 1. ]
[ 0. 0.18 0.37 0.58 0.79 1. ]
[ 0. 0.16 0.35 0.56 0.77 1. ]
[ 0. 0.1 0.27 0.49 0.73 1. ]
[ 0. 0.04 0.16 0.36 0.64 1. ]
"""
x = np.arange(n+1) * 1. / n
if e0 != 0. or e1 != 0.:
x = unitAttractor(x, e0, e1)
return x
[docs]def seed1(n, nuni=0, e0=0.):
"""Create a list of seed values.
A seed list is a list of float values in the range 0.0 to 1.0.
It can be used to subdivide a line segment or to seed nodes
along lines for meshing purposes.
This function divides the unit interval in `n` parts, resulting
in `n+1` seed values. While the intervals are by default of equal
length, the `nuni` and `e0` can be used to create unevenly spaced
seed values.
Parameters
----------
n: int
The number of intervals in which to divide the range. This will
yield `n+1` parameter values.
nuni: 0..n-1
The number of intervals at the end of the range that
will have equal length. If n < 2, this function is equivalent
with `seed(n,e0,0.0)`.
e0: float
Attractor for the start of the range.
A value larger than zero will attract the points closer to the
startpoint, while a negative value will repulse them.
Returns
-------
float array
A list of `n+1` float values in the range 0.0 to 1.0.
The values are in ascending order, starting with 0.0 and ending with 1.0.
See Also
--------
`seed`: an analogue function with attractors at both ends of the range.
Examples
--------
>>> np.set_printoptions(precision=4)
>>> S = seed1(5,0,1.)
>>> print(S)
[ 0. 0.04 0.16 0.36 0.64 1. ]
>>> print(S[1:]-S[:-1])
[ 0.04 0.12 0.2 0.28 0.36]
>>> S = seed1(5,2,1.)
>>> print(S)
[ 0. 0.0435 0.1739 0.3913 0.6957 1. ]
>>> print(S[1:]-S[:-1])
[ 0.0435 0.1304 0.2174 0.3043 0.3043]
>>> np.set_printoptions(precision=2)
"""
if nuni < 2:
seeds = seed(n, e0, 0.)
else:
n0 = n-nuni+1
seeds = seed(n0, e0, 0.)
length = seeds[-1] - seeds[-2]
seeds2 = seeds[-1] + np.arange(1, nuni) * length
seeds = np.concatenate([seeds, seeds2])
seeds /= float(seeds[-1])
return seeds
[docs]def smartSeed(n):
"""Create a list of seed values.
Like the :func:`seed` function, this function creates a list of float
values in the range 0.0 to 1.0. It accepts however a variety of inputs,
making it the prefered choice when it is not known in advance how the
user wants to control the seeds: automatically created or self specified.
Parameters
----------
n: int, tuple or float :term:`seed`
Action depends on the argument:
- if an int, returns ``seed(n)``,
- if a tuple (n,), (n,e0) or (n,e0,e1): returns ``seed(*n)``,
- if a float array-like, it is normally a sorted list of float
values in the range 0.0 to 1.0: the values are returned
unchanged in an array.
Returns
-------
float array
The values created depending on the input argument.
Examples
--------
>>> np.set_printoptions(precision=4)
>>> print(smartSeed(5))
[ 0. 0.2 0.4 0.6 0.8 1. ]
>>> print(smartSeed((5,2.,1.)))
[ 0. 0.01 0.1092 0.3701 0.7504 1. ]
>>> print(smartSeed([0.0,0.2,0.3,0.4,0.8,1.0]))
[ 0. 0.2 0.3 0.4 0.8 1. ]
>>> np.set_printoptions(precision=2)
"""
if isInt(n):
return seed(n)
elif isinstance(n, tuple):
return seed(*n)
elif isinstance(n, (list, np.ndarray)):
return np.asarray(n)
else:
raise ValueError("Expected an integer, tuple or list; got %s = %s" % (type(n), n))
[docs]def gridpoints(seed0, seed1=None, seed2=None):
"""Create weights for 1D, 2D or 3D element coordinates.
Parameters
----------
seed0: int or list of floats
Subdivisions along the first parametric direction
seed1: int or list of floats
Subdivisions along the second parametric direction
seed2: int or list of floats
Subdivisions along the third parametric direction
If these parameters are integer values the divisions will be equally
spaced between 0 and 1.
Examples
--------
>>> gridpoints(4)
array([ 0. , 0.25, 0.5 , 0.75, 1. ])
>>> gridpoints(4,2)
array([[ 1. , 0. , 0. , 0. ],
[ 0.75, 0.25, 0. , 0. ],
[ 0.5 , 0.5 , 0. , 0. ],
[ 0.25, 0.75, 0. , 0. ],
[ 0. , 1. , 0. , 0. ],
[ 0.5 , 0. , 0. , 0.5 ],
[ 0.38, 0.12, 0.12, 0.38],
[ 0.25, 0.25, 0.25, 0.25],
[ 0.12, 0.38, 0.38, 0.12],
[ 0. , 0.5 , 0.5 , 0. ],
[ 0. , 0. , 0. , 1. ],
[ 0. , 0. , 0.25, 0.75],
[ 0. , 0. , 0.5 , 0.5 ],
[ 0. , 0. , 0.75, 0.25],
[ 0. , 0. , 1. , 0. ]])
"""
if seed0 is not None:
if isInt(seed0):
seed0 = seed(seed0)
sh = 1
pts = np.asarray(seed0)
if seed1 is not None:
if isInt(seed1):
seed1 = seed(seed1)
sh = 4
x1 = np.asarray(seed0)
y1 = np.asarray(seed1)
x0 = 1.-x1
y0 = 1.-y1
pts = np.dstack([np.outer(y0, x0), np.outer(y0, x1),
np.outer(y1, x1), np.outer(y1, x0)])
if seed2 is not None:
if isInt(seed2):
seed2 = seed(seed2)
sh = 8
z1 = np.asarray(seed2)
z0 = 1.-z1
pts = np.dstack([np.dstack([np.outer(pts[:, :, ipts], zz)
for ipts in range(pts.shape[2])])
for zz in [z0, z1]])
return pts.reshape(-1, sh).squeeze()
[docs]def nodalSum(val, elems, nnod=-1):
"""Compute the nodal sum of values defined on element nodes.
Parameters
----------
val: float array (nelems,nplex,nval)
Array with ``nval`` values at ``nplex`` nodes of ``nelems`` elements.
elems: int array (nelems,nplex)
The node indices of the elements.
nnod: int, optional
If provided, the length of the output arrays will be set to
this value. It should be higher than the highest node number
appering in elems. The default will set it automatically to
``elems.max() + 1``.
Returns
-------
sum: float array (nnod, nval)
The sum of all the values at the same node.
cnt: int array (nnod)
The number of values summed at each node.
See Also
--------
nodalAvg: compute the nodal average of values defined on element nodes
"""
from pyformex.lib import misc
if val.ndim != 3:
val = val.reshape(val.shape+(1,))
if elems.shape != val.shape[:2]:
raise RuntimeError(f"shapes of elems ({elems.shape}) and val"
f"({val.shape[:2]}) do not match")
val = val.astype(Float)
elems = elems.astype(Int)
return misc.nodalSum(val, elems, nnod)
[docs]def nodalAvg(val, elems, nnod=-1):
"""Compute the nodal average of values defined on element nodes.
Parameters
----------
val: float array (nelems,nplex,nval)
Array with ``nval`` values at ``nplex`` nodes of ``nelems`` elements.
elems: int array (nelems,nplex)
The node indices of the elements.
nnod: int, optional
If provided, the length of the output arrays will be set to
this value. It should be higher than the highest node number
appering in elems. The default will set it automatically to
``elems.max() + 1``.
Returns
-------
avg: float array (nnod, nval)
The average of all the values at the same node.
See Also
--------
nodalSum: compute the nodal sum of values defined on element nodes
"""
sum, cnt = nodalSum(val, elems, nnod)
return sum/cnt[:, np.newaxis]
[docs]def fmtData1d(data, npl=8, sep=', ', linesep='\n', fmt=str):
"""Format data in lines with maximum npl items.
Formats a list or array of data items in groups containing
a maximum number of items. The data items are converted to
strings using the `fmt` function, concatenated
in groups of `npl` items using `sep` as a separator between them.
Finally, the groups are concatenated with a `linesep` separator.
Parameters
----------
data: list or array.
List or array with data. If an array, if will be flattened.
npl: int
Maximum number of items per group. Items will be concatenated
groups of this number of items. The last group
may contain less items.
sep: str
Separator to add between individual items in a group.
linesep: str
Separator to add between groups. The default (newline) will
put each group of `npl` items on a separate line.
fmt: callable
Used to convert a single item to a string. Default
is the Python built-in string converter.
Returns
-------
: str
Multiline string with the formatted data.
Examples
--------
>>> print(fmtData1d(np.arange(10)))
0, 1, 2, 3, 4, 5, 6, 7
8, 9
>>> print(fmtData1d([1.25, 3, 'no', 2.50, 4, 'yes'],npl=3))
1.25, 3, no
2.5, 4, yes
>>> myformat = lambda x: "%10s" % str(x)
>>> print(fmtData1d([1.25, 3, 'no', 2.50, 4, 'yes'],npl=3,fmt=myformat))
1.25, 3, no
2.5, 4, yes
"""
if isinstance(data, np.ndarray):
data = data.flat
return linesep.join([sep.join(map(fmt, data[i:i+npl]))
for i in range(0, len(data), npl)])
# These are retained for compatibility, but deprecated
def trfMatrix(*args, **kargs):
warnings.warn("warn_trfMatrix_deprecated")
return trfmat(*args, **kargs)
def vectorRotation(*args, **kargs):
warnings.warn("warn_vectorRotation_deprecated")
return rotMatrix2(*args, **kargs)
def rotationAnglesFromMatrix(*args, **kargs):
warnings.warn("warn_rotationAnglesFromMatrix_deprecated")
return cardanAngles(*args, **kargs)
def findIndex(*args, **kargs):
warnings.warn("warn_findIndex_deprecated")
return findFirst(*args, **kargs)
def groupPositions(*args, **kargs):
warnings.warn("warn_groupPositions_deprecated")
return findAll(*args, **kargs)
def vectorLength(*args, **kargs):
warnings.warn("warn_vectorLength_deprecated")
return length(*args, **kargs)
def vectorNormalize(A):
warnings.warn("warn_vectorNormalize_deprecated")
v, ll = normalize(A, return_length=True)
return ll, v
# Removed
# def matchIndex(*args,**kargs):
# warnings.warn("warn_matchIndex_deprecated")
# return findFirst(*args,**kargs)
# End
