k3math

k3math is a toy math impl

Documentation for the Code

Classes

class k3math.Vector(*args, **kwargs)

A Vector is a list supporting operations:

  • +: vector adds vector

  • -: vector subtracts vector

  • *: vector times scalar

  • **: vector powers scalar

inner_product(b)

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

class k3math.Matrix(vectors)
determinant()

Calculate determinant of this matrix. E.g.:

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

float

minor(i, j)

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

solve(ys)

Solve equations:

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

y (Vector) – a vector of y0, y1, y2.

Returns

Vector

class k3math.Polynomial(iterable=(), /)

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

classmethod fit(xs, ys, degree)

Find a polynomial curve with least squares method.

Parameters

  • 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

classmethod get_fitting_equation(xs, ys, degree)

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])

…

classmethod interpolation(xs, ys, degree, x)

guess value at x with polynomial regression

classmethod plot(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
#                                        .
#                                      ..
#                                    ..
#                                  ..
#                               ...
#                            ...
#                         ...
#                     ....
#                 ....
#            .....
# ...........
Parameters

  • 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

Indices and tables