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k3math

Action-CI Documentation Status Package

A toy math implementation with Vector, Matrix, and Polynomial classes for basic linear algebra operations.

k3math is a component of pykit3 project: a python3 toolkit set.

Installation

pip install k3math

Quick Start

from k3math import Vector, Matrix, Polynomial

# Vector operations
v1 = Vector([1, 2, 3])
v2 = Vector([4, 5, 6])
print(v1 + v2)        # [5.0, 7.0, 9.0]
print(v1 * 2)         # [2.0, 4.0, 6.0]
print(v1.inner_product(v2))  # 32.0

# Matrix operations
m = Matrix([
    [1, 2],
    [3, 4]
])
print(m.determinant())  # -2.0
print(m.solve([5, 11])) # [1.0, 2.0] (solves x + 2y = 5, 3x + 4y = 11)

# Polynomial curve fitting
xs = [1, 2, 3, 4, 5]
ys = [2.1, 4.0, 5.9, 8.1, 10.0]
poly = Polynomial.fit(xs, ys, degree=1)
print(poly)  # Displays the fitted polynomial

API Reference

k3math

Matrix

Bases: list

Source code in k3math/mth.py 
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class Matrix(list):
    def __init__(self, vectors):
        vectors = [Vector(x) for x in vectors]
        super(Matrix, self).__init__(vectors)

    def minor(self, i, j):
        """
        Make a new matrix without i-th row and j-th column.
        """
        vectors = [Vector(x) for x in self[:i]]
        vectors += [Vector(x) for x in self[i + 1 :]]
        for v in vectors:
            v.pop(j)

        return Matrix(vectors)

    def determinant(self):
        """
        Calculate determinant of this matrix. E.g.::

            | a b | = a*d - b*c
            | c d |

        Returns:
            float

        """
        if len(self) == 1:
            return self[0][0]

        if len(self) == 2:
            return self[0][0] * self[1][1] - self[0][1] * self[1][0]

        rst = 0
        for i in range(len(self)):
            rst += (-1) ** i * self[0][i] * self.minor(0, i).determinant()
        return rst

    def replace_row(self, i, vec):
        self[i] = Vector(vec)

    def replace_col(self, j, vec):
        for i in range(len(self)):
            self[i][j] = vec[i]

    def solve(self, ys):
        """
        Solve equations::

            |a00 a01 a02|   |x0|   |y0|
            |a10 a11 a12| * |x1| = |y1|
            |a20 a21 a22|   |x2|   |y2|

        Args:
            y(Vector): a vector of `y0, y1, y2`.

        Returns:
            Vector

        """
        # Sovle linear equation M x [x] = [y]
        # with Cramer's rule
        xs = []
        det = self.determinant()
        for i in range(len(self)):
            m = Matrix(self)
            m.replace_col(i, ys)
            x = m.determinant() / det
            xs.append(x)

        return Vector(xs)

    def invert(self):
        # TODO test
        rst = []
        for i in range(len(self)):
            vi = []
            for j in range(len(self)):
                vi.append(self.minor(j, i).determinant())

            vi = Vector(vi)
            vi = vi * (1.0 / self.determinant())

            rst.append(vi)

        return Matrix(rst)

determinant()

Calculate determinant of this matrix. E.g.::

| a b | = a*d - b*c
| c d |

Returns:

Type Description

float
Source code in k3math/mth.py 
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def determinant(self):
    """
    Calculate determinant of this matrix. E.g.::

        | a b | = a*d - b*c
        | c d |

    Returns:
        float

    """
    if len(self) == 1:
        return self[0][0]

    if len(self) == 2:
        return self[0][0] * self[1][1] - self[0][1] * self[1][0]

    rst = 0
    for i in range(len(self)):
        rst += (-1) ** i * self[0][i] * self.minor(0, i).determinant()
    return rst

minor(i, j)

Make a new matrix without i-th row and j-th column.

Source code in k3math/mth.py 
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def minor(self, i, j):
    """
    Make a new matrix without i-th row and j-th column.
    """
    vectors = [Vector(x) for x in self[:i]]
    vectors += [Vector(x) for x in self[i + 1 :]]
    for v in vectors:
        v.pop(j)

    return Matrix(vectors)

solve(ys)

Solve equations::

|a00 a01 a02|   |x0|   |y0|
|a10 a11 a12| * |x1| = |y1|
|a20 a21 a22|   |x2|   |y2|

Parameters:

Name Type Description Default
y Vector

a vector of y0, y1, y2.

 required

Returns:

Type Description

Vector
Source code in k3math/mth.py 
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def solve(self, ys):
    """
    Solve equations::

        |a00 a01 a02|   |x0|   |y0|
        |a10 a11 a12| * |x1| = |y1|
        |a20 a21 a22|   |x2|   |y2|

    Args:
        y(Vector): a vector of `y0, y1, y2`.

    Returns:
        Vector

    """
    # Sovle linear equation M x [x] = [y]
    # with Cramer's rule
    xs = []
    det = self.determinant()
    for i in range(len(self)):
        m = Matrix(self)
        m.replace_col(i, ys)
        x = m.determinant() / det
        xs.append(x)

    return Vector(xs)

Polynomial

Bases: list

It represents a polynomial: y = a₀ + a₁ * x¹ + a₂ * x² ... Where coefficients = [a₀, a₁, a₂ .. ].

xs and ys is array of x-coordinate value and y-coordinate value. They are all real numbers.

xs = [1, 2, 3, 4, 5..] ys = [1, 2, 4, 7, 11..]

With xs and ys to calc the coefficients of a polinomial

degree is the highest power of polinomial: degree=2: y = a0 + a1x + a2x^2

Source code in k3math/mth.py 
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class Polynomial(list):
    """
    It represents a polynomial: ``y = a₀ + a₁ * x¹ + a₂ * x² ..``.
    Where `coefficients = [a₀, a₁, a₂ .. ]`.

    xs and ys is array of x-coordinate value and y-coordinate value.
    They are all real numbers.

    xs = [1, 2, 3, 4, 5..]
    ys = [1, 2, 4, 7, 11..]

    With xs and ys to calc the coefficients of a polinomial

    degree is the highest power of polinomial:
    degree=2: y = a0 + a1*x + a2*x^2

    """

    def __str__(self):
        # TODO test
        super_num = "⁰¹²³⁴⁵⁶⁷⁸⁹"

        rst = []
        for i, coef in enumerate(self):
            if coef == 0:
                continue

            if coef == 1:
                c = ""
            elif int(coef) == coef:
                c = str(int(coef))
            else:
                c1 = "{:>4f}".format(coef)
                c1 = c1.rstrip("0")
                c2 = "{:>4e}".format(coef)
                if len(c1) > len(c2):
                    c = c2
                else:
                    c = c1

            if i == 0:
                rst.append(c)
            elif i == 1:
                rst.append(c + "x")
            else:
                pw = str(i)
                pw = "".join([super_num[int(x)] for x in pw])
                rst.append(c + "x" + pw)

        rst = " + ".join(rst)
        return rst.replace(" + -", " - ")

    @classmethod
    def get_fitting_equation(clz, xs, ys, degree):
        # TODO test
        """
        Curve fit with least squres

        We looking for a curve:

            Y = a0 + a1*x + a2*x^2

        that minimize variance:

            E = sum((Y[i]-ys[i])^2)

        Partial derivatives about a0..an are:

            E'a0 = sum(2 * (a0 + a1*xs[i] + a2*xs[i]^2 - ys[i]) * 1)
            E'a1 = sum(2 * (a0 + a1*xs[i] + a2*xs[i]^2 - ys[i]) * xs[i])
            E'a2 = sum(2 * (a0 + a1*xs[i] + a2*xs[i]^2 - ys[i]) * xs[i]^2)

        The best fit is a curve that minimizes E:
        or all partial derivatives are 0:

            | c00 c01 c02 |   | a0 |   | Y0 |
            | c10 c11 c12 | * | a1 | = | Y1 |
            | c20 c21 c22 |   | a2 |   | Y2 |

            c00 = 2 * n
            c01 = 2 * sum(xs[i])
            c02 = 2 * sum(xs[i]^2)
            Y0  = 2 * sum(ys[i])

            c10 = 2 * sum(xs[i])
            c11 = 2 * sum(xs[i]^2)
            c12 = 2 * sum(xs[i]^3)
            Y1  = 2 * sum(ys[i]*xs[i])

            ...

        """

        xs, ys = Vector(xs), Vector(ys)
        coef = []

        yys = Vector()

        for deg in range(0, degree + 1):
            # o[i] is coefficient of a[i] of E'a[i]
            o = Vector([0] * (degree + 1))
            y = 0
            for i in range(len(xs)):
                # a0 + a1*xs[i] + a2*xs[i]^2
                v = Vector([0] * (degree + 1))
                for ai in range(degree + 1):
                    v[ai] = xs[i] ** ai
                v = v * 2
                v = v * (xs[i] ** deg)
                o = o + v

                y += ys[i] * 2 * xs[i] ** deg

            coef.append(o)
            yys.append(y)

        coef = Matrix(coef)

        return coef, yys

    @classmethod
    def fit(clz, xs, ys, degree):
        """
        Find a polynomial curve with least squares method.

        Args:

            x(Vector): Vector of x positions

            y(Vector): Vector of y positions

            degree(int): the highest power of variable `x` in the polynomial.

        Returns:
            Polynomial

        """
        xs, ys = Vector(xs), Vector(ys)

        m, yys = clz.get_fitting_equation(xs, ys, degree)

        coef = m.solve(yys)

        return clz(coef)

        # return coef

    @classmethod
    def evaluate(clz, coefficients, x):
        # TODO test
        r = 0
        for i, ee in enumerate(coefficients):
            r += ee * (x**i)
        return r

    @classmethod
    def variance(clz, coefficients, xs, ys):
        # TODO test
        yys = [clz.evaluate(coefficients, x) for x in xs]
        pairs = list(zip(yys, ys))
        sm = sum([(a - b) ** 2 for a, b in pairs]) / len(xs)
        return sm

    @classmethod
    def interpolation(clz, xs, ys, degree, x):
        # TODO test
        """
        guess value at x with polynomial regression
        """

        # curve fit
        coef = clz.fit(xs, ys, degree)
        return clz.evaluate(coef, x)

    @classmethod
    def plot(clz, polynomials, rangex, rangey=None, width=120, height=20, points=()):
        """
        Plot a polynomial with text::

            poly = [3.5, 3.4, 1]

            for l in Polynomial.plot([(poly, '.')],
                                     rangex=[-1, 6],
                                     width=40, height=10):
                print l
            #                                        .
            #                                      ..
            #                                    ..
            #                                  ..
            #                               ...
            #                            ...
            #                         ...
            #                     ....
            #                 ....
            #            .....
            # ...........

        Args:

            polynomials: list of a vector of polynomial coefficients and symbol::

                [ ([1, 6], 'x'),    # y = 1 + 6x, plot with "x"
                  ([2, 2, 2], '.'), # y = 2 + 2x + 2x^2, plot with "."
                ]

            rangex(float): is a tuple of two floats that specifies range of x.

            rangey(float): is a tuple of two floats that specifies range of y.

            width(int): specifies plot graph width.

            height(int): specifies plot graph height.

            points: other points to add to the plot.
                It is a vector of ``(x, y[, char])``.
                ``char`` is optional to specify point mark.
                By default it is ``X``.

        Returns:
            list of strings
        """

        # polynomials: is list of coefficients and point symbol
        #
        # [
        #     ([0, 1, 2], '.'),
        #     ([0, 2], 'x'),
        # ]
        # TODO test
        rangex = (float(rangex[0]), float(rangex[1]))
        rng_width = rangex[1] - rangex[0]
        ys = []
        jys = []
        for i in range(width):
            x = float(i) / width * rng_width + rangex[0]
            for poly, sym in polynomials:
                y = clz.evaluate(poly, x)
                ys.append(y)
                jys.append((i, y, sym))

        if rangey is None:
            y_range = ys + [xx[1] for xx in points]
        else:
            y_range = [rangey[0], rangey[1]]

        bot, top = min(y_range), max(y_range)

        lines = []
        for ii in range(height + 1):
            lines.append([" "] * (width + 1))

        for j, y, sym in jys:
            h = y - bot
            h = h * height / (top - bot)
            h = int(h)

            if height - h < 0:
                continue
            try:
                lines[height - h][j] = sym
            except IndexError:
                # point out of range. ignore
                pass

        for ii, xyv in enumerate(points):
            x, y = xyv[:2]
            if len(xyv) > 2:
                v = str(xyv[2])
            else:
                v = "(%d,%d)" % (x, y)
            h = y - bot
            i = h * height / (top - bot)
            i = int(i)
            i = height - i

            j = (float(x) - rangex[0]) * width / rng_width
            j = int(j)

            if 0 <= i <= height and 0 <= j <= width:
                txt = v
                m = j
                while m <= width and m - j < len(txt):
                    lines[i][m] = txt[m - j]
                    m += 1

        lines = ["".join(xx) for xx in lines]
        return lines

fit(clz, xs, ys, degree) classmethod

Find a polynomial curve with least squares method.

Args:

x(Vector): Vector of x positions

y(Vector): Vector of y positions

degree(int): the highest power of variable `x` in the polynomial.

Returns:

Type Description

Polynomial
Source code in k3math/mth.py 
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@classmethod
def fit(clz, xs, ys, degree):
    """
    Find a polynomial curve with least squares method.

    Args:

        x(Vector): Vector of x positions

        y(Vector): Vector of y positions

        degree(int): the highest power of variable `x` in the polynomial.

    Returns:
        Polynomial

    """
    xs, ys = Vector(xs), Vector(ys)

    m, yys = clz.get_fitting_equation(xs, ys, degree)

    coef = m.solve(yys)

    return clz(coef)

get_fitting_equation(clz, xs, ys, degree) classmethod

Curve fit with least squres

We looking for a curve:

Y = a0 + a1*x + a2*x^2

that minimize variance:

E = sum((Y[i]-ys[i])^2)

Partial derivatives about a0..an are:

E'a0 = sum(2 * (a0 + a1*xs[i] + a2*xs[i]^2 - ys[i]) * 1)
E'a1 = sum(2 * (a0 + a1*xs[i] + a2*xs[i]^2 - ys[i]) * xs[i])
E'a2 = sum(2 * (a0 + a1*xs[i] + a2*xs[i]^2 - ys[i]) * xs[i]^2)

The best fit is a curve that minimizes E: or all partial derivatives are 0:

| c00 c01 c02 |   | a0 |   | Y0 |
| c10 c11 c12 | * | a1 | = | Y1 |
| c20 c21 c22 |   | a2 |   | Y2 |

c00 = 2 * n
c01 = 2 * sum(xs[i])
c02 = 2 * sum(xs[i]^2)
Y0  = 2 * sum(ys[i])

c10 = 2 * sum(xs[i])
c11 = 2 * sum(xs[i]^2)
c12 = 2 * sum(xs[i]^3)
Y1  = 2 * sum(ys[i]*xs[i])

...
Source code in k3math/mth.py 
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@classmethod
def get_fitting_equation(clz, xs, ys, degree):
    # TODO test
    """
    Curve fit with least squres

    We looking for a curve:

        Y = a0 + a1*x + a2*x^2

    that minimize variance:

        E = sum((Y[i]-ys[i])^2)

    Partial derivatives about a0..an are:

        E'a0 = sum(2 * (a0 + a1*xs[i] + a2*xs[i]^2 - ys[i]) * 1)
        E'a1 = sum(2 * (a0 + a1*xs[i] + a2*xs[i]^2 - ys[i]) * xs[i])
        E'a2 = sum(2 * (a0 + a1*xs[i] + a2*xs[i]^2 - ys[i]) * xs[i]^2)

    The best fit is a curve that minimizes E:
    or all partial derivatives are 0:

        | c00 c01 c02 |   | a0 |   | Y0 |
        | c10 c11 c12 | * | a1 | = | Y1 |
        | c20 c21 c22 |   | a2 |   | Y2 |

        c00 = 2 * n
        c01 = 2 * sum(xs[i])
        c02 = 2 * sum(xs[i]^2)
        Y0  = 2 * sum(ys[i])

        c10 = 2 * sum(xs[i])
        c11 = 2 * sum(xs[i]^2)
        c12 = 2 * sum(xs[i]^3)
        Y1  = 2 * sum(ys[i]*xs[i])

        ...

    """

    xs, ys = Vector(xs), Vector(ys)
    coef = []

    yys = Vector()

    for deg in range(0, degree + 1):
        # o[i] is coefficient of a[i] of E'a[i]
        o = Vector([0] * (degree + 1))
        y = 0
        for i in range(len(xs)):
            # a0 + a1*xs[i] + a2*xs[i]^2
            v = Vector([0] * (degree + 1))
            for ai in range(degree + 1):
                v[ai] = xs[i] ** ai
            v = v * 2
            v = v * (xs[i] ** deg)
            o = o + v

            y += ys[i] * 2 * xs[i] ** deg

        coef.append(o)
        yys.append(y)

    coef = Matrix(coef)

    return coef, yys

interpolation(clz, xs, ys, degree, x) classmethod

guess value at x with polynomial regression

Source code in k3math/mth.py 
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@classmethod
def interpolation(clz, xs, ys, degree, x):
    # TODO test
    """
    guess value at x with polynomial regression
    """

    # curve fit
    coef = clz.fit(xs, ys, degree)
    return clz.evaluate(coef, x)

plot(clz, polynomials, rangex, rangey=None, width=120, height=20, points=()) classmethod

Plot a polynomial with text::

poly = [3.5, 3.4, 1]

for l in Polynomial.plot([(poly, '.')],
                         rangex=[-1, 6],
                         width=40, height=10):
    print l
#                                        .
#                                      ..
#                                    ..
#                                  ..
#                               ...
#                            ...
#                         ...
#                     ....
#                 ....
#            .....
# ...........

Args:

polynomials: list of a vector of polynomial coefficients and symbol::

    [ ([1, 6], 'x'),    # y = 1 + 6x, plot with "x"
      ([2, 2, 2], '.'), # y = 2 + 2x + 2x^2, plot with "."
    ]

rangex(float): is a tuple of two floats that specifies range of x.

rangey(float): is a tuple of two floats that specifies range of y.

width(int): specifies plot graph width.

height(int): specifies plot graph height.

points: other points to add to the plot.
    It is a vector of ``(x, y[, char])``.
    ``char`` is optional to specify point mark.
    By default it is ``X``.

Returns:

Type Description

list of strings
Source code in k3math/mth.py 
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@classmethod
def plot(clz, polynomials, rangex, rangey=None, width=120, height=20, points=()):
    """
    Plot a polynomial with text::

        poly = [3.5, 3.4, 1]

        for l in Polynomial.plot([(poly, '.')],
                                 rangex=[-1, 6],
                                 width=40, height=10):
            print l
        #                                        .
        #                                      ..
        #                                    ..
        #                                  ..
        #                               ...
        #                            ...
        #                         ...
        #                     ....
        #                 ....
        #            .....
        # ...........

    Args:

        polynomials: list of a vector of polynomial coefficients and symbol::

            [ ([1, 6], 'x'),    # y = 1 + 6x, plot with "x"
              ([2, 2, 2], '.'), # y = 2 + 2x + 2x^2, plot with "."
            ]

        rangex(float): is a tuple of two floats that specifies range of x.

        rangey(float): is a tuple of two floats that specifies range of y.

        width(int): specifies plot graph width.

        height(int): specifies plot graph height.

        points: other points to add to the plot.
            It is a vector of ``(x, y[, char])``.
            ``char`` is optional to specify point mark.
            By default it is ``X``.

    Returns:
        list of strings
    """

    # polynomials: is list of coefficients and point symbol
    #
    # [
    #     ([0, 1, 2], '.'),
    #     ([0, 2], 'x'),
    # ]
    # TODO test
    rangex = (float(rangex[0]), float(rangex[1]))
    rng_width = rangex[1] - rangex[0]
    ys = []
    jys = []
    for i in range(width):
        x = float(i) / width * rng_width + rangex[0]
        for poly, sym in polynomials:
            y = clz.evaluate(poly, x)
            ys.append(y)
            jys.append((i, y, sym))

    if rangey is None:
        y_range = ys + [xx[1] for xx in points]
    else:
        y_range = [rangey[0], rangey[1]]

    bot, top = min(y_range), max(y_range)

    lines = []
    for ii in range(height + 1):
        lines.append([" "] * (width + 1))

    for j, y, sym in jys:
        h = y - bot
        h = h * height / (top - bot)
        h = int(h)

        if height - h < 0:
            continue
        try:
            lines[height - h][j] = sym
        except IndexError:
            # point out of range. ignore
            pass

    for ii, xyv in enumerate(points):
        x, y = xyv[:2]
        if len(xyv) > 2:
            v = str(xyv[2])
        else:
            v = "(%d,%d)" % (x, y)
        h = y - bot
        i = h * height / (top - bot)
        i = int(i)
        i = height - i

        j = (float(x) - rangex[0]) * width / rng_width
        j = int(j)

        if 0 <= i <= height and 0 <= j <= width:
            txt = v
            m = j
            while m <= width and m - j < len(txt):
                lines[i][m] = txt[m - j]
                m += 1

    lines = ["".join(xx) for xx in lines]
    return lines

Vector

Bases: list

A Vector is a list supporting operations:

  • +: vector adds vector
  • -: vector subtracts vector
  • *: vector times scalar
  • **: vector powers scalar
Source code in k3math/mth.py 
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class Vector(list):
    """
    A ``Vector`` is a ``list`` supporting operations:

    -   ``+``: vector adds vector
    -   ``-``: vector subtracts vector
    -   ``*``: vector times scalar
    -   ``**``: vector powers scalar
    """

    def __init__(self, *args, **kwargs):
        super(Vector, self).__init__(*args, **kwargs)
        for i in range(len(self)):
            self[i] = float(self[i])

    def __add__(self, b):
        return Vector([self[i] + b[i] for i in range(len(self))])

    def __sub__(self, b):
        return Vector([self[i] - b[i] for i in range(len(self))])

    def __mul__(self, v):
        return Vector([self[i] * v for i in range(len(self))])

    def __pow__(self, v):
        return Vector([self[i] ** v for i in range(len(self))])

    def inner_product(self, b):
        """
        Calculate inner product of two vector and returns a new Vector.
        """
        return sum([self[i] * b[i] for i in range(len(self))])

inner_product(b)

Calculate inner product of two vector and returns a new Vector.

Source code in k3math/mth.py 
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def inner_product(self, b):
    """
    Calculate inner product of two vector and returns a new Vector.
    """
    return sum([self[i] * b[i] for i in range(len(self))])

License

The MIT License (MIT) - Copyright (c) 2015 Zhang Yanpo (张炎泼)