numsolve 0.0.3

Numerical Methods to solve various equation

NumSolve

Package to solve various equations using Numerical Methods

List of Methods Implemented

• Euler Forward method

• Newton Raphson Method 1D

• Euler Backward method

• Modified Euler method

• Euler Cauchy Method/Hune Method

• Ranga Kutta method of thita

• General 2nd Order method

• Nystrom 3rd order method

• Hune 3rd order method

• Classical 3rd Order method

• Nearly optimal 3rd Order method

• Kutta Method 4th order method

• Classical 4th Order method

• General Explicit Ranga Kutta method

• Newton Raphson Method for solving for y for a given x in F(x,y) = 0

• Implicit RK Method of order 2

• Implicit RK Method of order 4

• Forward Difference operator

• Backward Difference operator

• Adam Bashford method

• Adam Moulton method

• Newton Raphson Method for solving for y for a given x in y-coeff* F(x,y) = intercept

• Milne Simpson method

• Adams Bashforth Moulton PC method

• Milne PC method

• Euler PC method

• Thomas Algorithm for solving tridiagonal system

• Finite Difference Method to solve general 2nd order BVP

Further scope

• Methods to solve PDEs

Examples

import the package

>>> import numsolve as ns

define the functions needed for the equation

>>> # function

>>> def h(x,y):

...     return -2*x*y*y

...

>>> # partial derivative wrt y

>>> def dhy(x,y):

...     return -4*x*y

...

>>> # partial derivative wrt x

>>> def dhx(x,y):

...     return -2*y*y

check the doc strings and run the methods

>>> help(ns.ECM)

Help on function ECM in module numsolve:



ECM(n, F, y_init, limit=[0, 1])

    Euler Cauchy Method or Hune Method for solution of

    y' = F(x,y)



    n : number of steps

    F : F(x,y) takes 2 input as x and y then outputs the value for this equation

    y_init = initial value at point x=a

    limit : [a,b], region on x to solve the equation for



>>> ns.ECM(5,F=h,y_init=1)

[1, 0.96, 0.86029775536128, 0.7350425008089037, 0.6115716703493418]

>>> help(ns.General_Explicit)

Help on function General_Explicit in module numsolve:



General_Explicit(order, W, A, n, F, y_init, limit=[0, 1], C=None)

    General Explicit Ranga Kutta method for solution of

    y' = F(x,y)



    order : order of method say m

    W = weight numpy Array for, y_n+1 = y_n + ( w1*k1 + w2*k2.....+ wm*km ), here W = [w1,w2..,wm]

    A = weight numpy matrix Matrix for, ki = h*F(x_n + ci*h,y_n + ( a_{i,i}*k1 + a_{i,2}*k2 + ...+ a_{i,i-1}*k_i-1 )) for i = 0...m, h is step size = (b-a/n)

        note values beyound i-1 are all supposed to be zero for Explicit method

    n : number of steps

    F : F(x,y) takes 2 input as x and y then outputs the value for this equation

    y_init = initial value at point x=a

    limit : [a,b], region on x to solve the equation for

    C : weight numpy Array [c1,c2...c], usually ci = a_{i,1}+ a_{i,2}+..+  a_{i,i-1}

        this can be changed with help of this input



>>>

>>> ns.General_Explicit(3,[0.5,0.3,0.2],[[0,0,0],[1,0,0],[0.5,0.2,0]],5,F=h,y_init=1)

[1, 0.9651555328, 0.8676644565853738, 0.7409120390270509, 0.6143029355673265]