General purpose functions

In this section, we present an overview of functions for working with interval quantities as well as some functions for creating interval objects.

Run the following commands to connect the necessary modules

>>> import intvalpy as ip
>>> import numpy as np

Converting data to interval type

def asinterval(a)

To convert the input data to the interval type, use the asinterval function:

Parameters:

  • aint, float, array_like

    Input data in any form that can be converted to an interval data type. These include int, float, list and ndarrays.

Returns:

  • outInterval

    The conversion is not performed if the input is already of type Interval. Otherwise an object of interval type is returned.

Examples:

>>> ip.asinterval(3)
'[3, 3]'
>>> ip.asinterval([1/2, ip.Interval(-2, 5), 2])
Interval(['[0.5, 0.5]', '[-2, 5]', '[2, 2]'])

Interval scatterplot

Математическая диаграмма, изображающая значения двух переменных в виде брусов на декартовой плоскости.

Parameters:

  • xInterval

    Интервальный вектор положения данных на оси OX.

  • yInterval

    Интервальный вектор положения данных на оси OY.

  • title: str, optional

    Верхняя легенда графика.

  • color: str, optional

    Цвет отображения брусов.

  • alpha: float, optional

    Прозрачность брусов.

  • s: float, optional

    Насколько велики точки вершин.

  • size: tuple, optional

    Размер отрисовочного окна.

  • save: bool, optional

    Если значение True, то график сохраняется.

Returns:

  • out: None

    A scatterplot is displayed.

Examples:

>>> x = ip.Interval(np.array([1.06978355, 1.94152571, 1.70930717, 2.94775725, 4.55556349, 6, 6.34679035, 6.62305275]), \
>>>                 np.array([1.1746937 , 2.73256075, 1.95913956, 3.61482169, 5.40818299, 6, 7.06625362, 7.54738552]))
>>> y = ip.Interval(np.array([0.3715678 , 0.37954135, 0.38124681, 0.39739009, 0.42010472, 0.45, 0.44676075, 0.44823645]), \
>>>                 np.array([0.3756708 , 0.4099036 , 0.3909104 , 0.42261893, 0.45150898, 0.45, 0.47255936, 0.48118948]))
>>> ip.scatter_plot(x, y)

Intersection of intervals

Функция intersection осуществляет пересечение интервальных данных. В случае, если на вход поданы массивы, то осуществляется покомпонентное пересечение.

Parameters:
A, B: Interval

В случае, если операнды не являются интервальным типом, то они преобразуются функцией asinterval.

Returns:
out: Interval

Возвращается массив пересечённых интервалов. Если некоторые интервалы не пересекаются, то на их месте выводится интервал Interval(float('-inf'), float('-inf')).

Примеры:

>>> import intvalpy as ip
>>> f = ip.Interval([-3., -6., -2.], [0., 5., 6.])
>>> s = ip.Interval(-1, 10)
>>> ip.intersection(f, s)
interval(['[-1.0, 0.0]', '[-1.0, 5.0]', '[-1.0, 6.0]'])

>>> f = ip.Interval([-3., -6., -2.], [0., 5., 6.])
>>> s = -2
>>> ip.intersection(f, s)
interval(['[-2.0, -2.0]', '[-2.0, -2.0]', '[-2.0, -2.0]'])

>>> f = ip.Interval([-3., -6., -2.], [0., 5., 6.])
>>> s = ip.Interval([ 2., -8., -6.], [6., 7., 0.])
>>> ip.intersection(f, s)
interval(['[-inf, -inf]', '[-6.0, 5.0]', '[-2.0, 0.0]'])

Distance

def dist(x, y, order=float(„inf“))

To calculate metrics or multimetrics in interval spaces, the dist function is provided. The mathematical formula for distance is given as follows: distorder = (sumij ||xij - yij ||order )1/order.

It is important to note that this formula involves an algebraic difference, not the usual interval difference.

Parameters:

  • a, bInterval

    The intervals between which you need to calculate the distance. In the case of multidimensional operands a multimetric is calculated.

  • orderint, optional

    The order of the metric is set. By default, setting is Chebyshev distance.

Returns:

  • out: float

    The distance between the input operands is returned.

Examples:

>>> f = ip.Interval([
        [[0, 1], [2, 3]],
        [[4, 5], [6, 7]],
    ])
>>> s = ip.Interval([
        [[1, 2], [3, 4]],
        [[5, 6], [7, 8]],
    ])
>>> ip.dist(f, s)
1.0

The detailed information about various metrics can be found in the referenced monograph.

Zero intervals

def zeros(shape)

To create an interval array where each element is point and equal to zero, the function zeros is provided:

Parameters:

  • shapeint, tuple

    Shape of the new interval array, e.g., (2, 3) or 4.

Returns:

  • outInterval

    An interval array of zeros with a given shape

Examples:

>>> ip.zeros((2, 3))
Interval([['[0, 0]', '[0, 0]', '[0, 0]'],
          ['[0, 0]', '[0, 0]', '[0, 0]']])
>>> ip.zeros(4)
Interval(['[0, 0]', '[0, 0]', '[0, 0]', '[0, 0]'])

Identity interval matrix

def eye(N, M=None, k=0)

Return a 2-D interval array with ones on the diagonal and zeros elsewhere.

Parameters:

  • Nint

    Shape of the new interval array, e.g., (2, 3) or 4.

  • Mint, optional

    Number of columns in the output. By default, M = N.

  • kint, optional

    Index of the diagonal: 0 refers to the main diagonal, a positive value refers to an upper diagonal, and a negative value to a lower diagonal. By default, k = 0.

Returns:

  • outInterval of shape (N, M)

    An interval array where all elements are equal to zero, except for the k-th diagonal, whose values are equal to one.

Examples:

>>> ip.eye(3, M=2, k=-1)
Interval([['[0, 0]', '[0, 0]'],
          ['[1, 1]', '[0, 0]'],
          ['[0, 0]', '[1, 1]']])

Diagonal of the interval matrix

def diag(v, k=0)

Extract a diagonal or construct a diagonal interval array.

Parameters:

  • vInterval

    If v is a 2-D interval array, return a copy of its k-th diagonal. If v is a 1-D interval array, return a 2-D interval array with v on the k-th diagonal.

  • kint, optional

    Diagonal in question. Use k>0 for diagonals above the main diagonal, and k<0 for diagonals below the main diagonal. By default, k=0.

Returns:

  • outInterval

    The extracted diagonal or constructed diagonal interval array.

Examples:

>>> A, b = ip.Shary(3)
>>> ip.diag(A)
Interval(['[2, 3]', '[2, 3]', '[2, 3]'])
>>> ip.diag(b)
Interval([['[-2, 2]', '[0, 0]', '[0, 0]'],
          ['[0, 0]', '[-2, 2]', '[0, 0]'],
          ['[0, 0]', '[0, 0]', '[-2, 2]']])

Elementary mathematical functions

This section presents the basic elementary mathematical functions that are most commonly encountered in various kinds of applied problems.

The square root

def sqrt(x)

Interval enclosure of the square root intrinsic over an interval.

Parameters:

  • xInterval

    The values whose square-roots are required.

Returns:

  • outInterval

    An array of the same shape as x, containing the interval enclosure of the square root of each element in x.

Examples:

>>> f = ip.Interval([[-3, -1], [-3, 2], [0, 4]])
>>> ip.sqrt(f)
Interval(['[nan, nan]', '[0, 1.41421]', '[0, 2]'])

The exponent

def exp(x)

Interval enclosure of the exponential intrinsic over an interval.

Parameters:

  • xInterval

    The values to take the exponent from.

Returns:

  • outInterval

    An array of the same shape as x, containing the interval enclosure of the exponential of each element in x.

Examples:

>>> f = ip.Interval([[-3, -1], [-3, 2], [0, 4]])
>>> ip.exp(f)
Interval(['[0.0497871, 0.367879]', '[0.0497871, 7.38906]', '[1, 54.5982]'])

The natural logarithm

def log(x)

Interval enclosure of the natural logarithm intrinsic over an interval.

Parameters:

  • xInterval

    The values to take the natural logarithm from.

Returns:

  • outInterval

    An array of the same shape as x, containing the interval enclosure of the natural logarithm of each element in x.

Examples:

>>> f = ip.Interval([[-3, -1], [-3, 2], [1, 4]])
>>> ip.log(f)
Interval(['[nan, nan]', '[-inf, 0.693147]', '[0, 1.38629]'])

The sine function

def sin(x)

Interval enclosure of the sin intrinsic over an interval.

Parameters:

  • xInterval

    The values to take the sin from.

Returns:

  • outInterval

    An array of the same shape as x, containing the interval enclosure of the sin of each element in x.

Examples:

>>> f = ip.Interval([[-3, -1], [-3, 2], [0, 4]])
>>> ip.sin(f)
Interval(['[-1, -0.14112]', '[-1, 1]', '[-0.756802, 1]'])

The cosine function

def cos(x)

Interval enclosure of the cos intrinsic over an interval.

Parameters:

  • xInterval

    The values to take the cos from.

Returns:

  • outInterval

    An array of the same shape as x, containing the interval enclosure of the cos of each element in x.

Examples:

>>> f = ip.Interval([[-3, -1], [-3, 2], [0, 4]])
>>> ip.cos(f)
Interval(['[-0.989992, 0.540302]', '[-0.989992, 1]', '[-1, 1]'])

Test interval systems

To check the performance of each implemented algorithm, it is tested on well-studied test systems. This subsection describes some of these systems, for which the properties of the solution sets are known, and their analytical characteristics and the complexity of numerical procedures have been previously studied.

The Shary system

def Shary(n, N=None, alpha=0.23, beta=0.35)

One of the popular test systems is the Shary system. Due to its symmetry, it is quite simple to determine the structure of its united solution set as well as other solution sets. Changing the values of the system parameters, you can get an extensive family of interval linear systems for testing the numerical algorithms. As the parameter beta decreases, the matrix of the system becomes more and more singular, and the united solution set enlarges indefinitely.

Parameters:

  • nint

    Dimension of the interval system. It may be greater than or equal to two.

  • Nfloat, optional

    A real number not less than (n − 1). By default, N = n.

  • alphafloat, optional

    A parameter used for specifying the lower endpoints of the elements in the interval matrix. The parameter is limited to 0 < alpha <= beta <= 1. By default, alpha = 0.23.

  • betafloat, optional

    A parameter used for specifying the upper endpoints of the elements in the interval matrix. The parameter is limited to 0 < alpha <= beta <= 1. By default, beta = 0.35.

Returns:

  • out: Interval, tuple

    The interval matrix and interval vector of the right side are returned, respectively.

Examples:

>>> A, b = ip.Shary(3)
>>> print('A: ', A)
>>> print('b: ', b)
A:  Interval([['[2, 3]', '[-0.77, 0.65]', '[-0.77, 0.65]'],
          ['[-0.77, 0.65]', '[2, 3]', '[-0.77, 0.65]'],
          ['[-0.77, 0.65]', '[-0.77, 0.65]', '[2, 3]']])
b:  Interval(['[-2, 2]', '[-2, 2]', '[-2, 2]'])

The Neumaier-Reichmann system

def Neumaier(n, theta, infb=None, supb=None)

This system is a parametric interval linear system, first proposed by K. Reichmann [2], and then slightly modified by A. Neumaier. The matrix of the system can be regular, but not strongly regular for some values of the diagonal parameter. It is shown that n × n matrices are non-singular for theta > n provided that n is even, and, for odd order n, the matrices are non-singular for theta > sqrt(n^2 - 1).

Parameters:

  • nint

    Dimension of the interval system. It may be greater than or equal to two.

  • thetafloat, optional

    Nonnegative real parameter, which is the number that stands on the main diagonal of the matrix А.

  • infbfloat, optional

    A real parameter that specifies the lower endpoints of the components of the right-hand side vector. By default, infb = -1.

  • supbfloat, optional

    A real parameter that specifies the upper endpoints of the components of the right-hand side vector. By default, supb = 1.

Returns:

  • out: Interval, tuple

    The interval matrix and interval vector of the right side are returned, respectively.

Examples:

>>> A, b = ip.Neumaier(2, 3.5)
>>> print('A: ', A)
>>> print('b: ', b)
A:  Interval([['[3.5, 3.5]', '[0, 2]'],
          ['[0, 2]', '[3.5, 3.5]']])
b:  Interval(['[-1, 1]', '[-1, 1]'])

References

[1] S.P. Shary - On optimal solution of interval linear equations // SIAM Journal on Numerical Analysis. – 1995. – Vol. 32, No. 2. – P. 68–630.

[2] Reichmann K. Abbruch beim Intervall-Gauß-Algorithmus // Computing. – 1979. – Vol. 22, Issue 4. – P. 355–361.

[3] С.П. Шарый - Конечномерный интервальный анализ.

Sergey P. Shary, `Finite-Dimensional Interval Analysis`_.