12-267/Numerical Methods - Revision history

Drorbn: Numerical Methods moved to 12-267/Numerical Methods

2012-10-26T21:59:01Z

Numerical Methods moved to 12-267/Numerical Methods

← Older revision

Revision as of 17:59, 26 October 2012

Twine: Added derivation of local error of Improved Euler method

2012-10-26T01:55:32Z

Added derivation of local error of Improved Euler method

← Older revision		Revision as of 21:55, 25 October 2012
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	<math>y_{n+1} = y_n + \frac{f_n + f(t_n + h, y_n + hf_n)}{2} h</math>		<math>y_{n+1} = y_n + \frac{f_n + f(t_n + h, y_n + hf_n)}{2} h</math>

			To determine local error we took the taylor expansions of <math>y</math> and <math>\Phi</math> and compared them.

			<math>\Phi(x_1) = \Phi(x_0 + h) = \Phi(x_0) + h \Phi'(x_0) + \frac{h^2}{2} \Phi''(x_0) + O(h^3)</math>

			<math>\Phi'(x) = f(x, \Phi(x)) \quad \Phi''(x) = \Phi'(x) = f_x(x, \Phi(x)) + f_y(x, \Phi(x))\Phi'(x)</math>

			<math>\Phi(x_0 + h) = y_0 + h f(x_0, y_0) + \frac{h^2}{2}(f_x + f_y f) + O(h^3)</math>

			We compare this to the computed value <math>y_1</math>

			<math>y_1 = y_0 +\frac{h}{2}(f(x_0, y_0) + f(x_0 + h, y_0 + hf(x_0, y_0)))</math>

			Taking the taylor expansion we get

			<math>y_1 = y_0 + h f(x_0, y_0) + \frac{h^2}{2}(f_x + f_y f) + O(h^3)</math>

			So we have shown that the Improved Euler Formula is accurate up to an error term proportional to <math>h^3</math>

	Local truncation error is proportional to <math>h^3</math>		Local truncation error is proportional to <math>h^3</math>

Twine at 01:00, 26 October 2012

2012-10-26T01:00:44Z

← Older revision		Revision as of 21:00, 25 October 2012
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	==Summary of Numerical Methods==		==Summary of Numerical Methods==

	Based largely off of a note available [http://imgur.com/a/uLSlM here] [[User:Simon1\|Simon1]] --[[User:Twine\|Twine]] 20:55, 25 October 2012 (EDT)		Based largely off of a note available [http://imgur.com/a/uLSlM here] posted by [[User:Simon1\|Simon1]] --[[User:Twine\|Twine]] 20:55, 25 October 2012 (EDT)

	Numerical methods: <math>\frac{dy}{dt} = f(t, y)</math> and <math>y(t_0) = y_0</math>, <math>y = \Phi(t)</math> is a solution.		Numerical methods: <math>\frac{dy}{dt} = f(t, y)</math> and <math>y(t_0) = y_0</math>, <math>y = \Phi(t)</math> is a solution.

Twine: Added python example of Euler's method, headings

2012-10-26T00:59:41Z

Added python example of Euler's method, headings

← Older revision		Revision as of 20:59, 25 October 2012
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			==Summary of Numerical Methods==

		⚫	Based largely off of a note available [http://imgur.com/a/uLSlM here] [[User:Simon1\|Simon1]] --[[User:Twine\|Twine]] 20:55, 25 October 2012 (EDT)

	Numerical methods: <math>\frac{dy}{dt} = f(t, y)</math> and <math>y(t_0) = y_0</math>, <math>y = \Phi(t)</math> is a solution.		Numerical methods: <math>\frac{dy}{dt} = f(t, y)</math> and <math>y(t_0) = y_0</math>, <math>y = \Phi(t)</math> is a solution.

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	Global truncation error is proportional to <math>h^4</math>.		Global truncation error is proportional to <math>h^4</math>.

			==Python Example of Euler's Method==
			In class on October 15th we discussed Euler's Method to numerically compute a solution to a differential equation. <math>x_0</math> and <math>y_0</math> are given as well as an increment amount <math>h</math>, <math>x_{n+1} = x_n + h</math>, and we use the guess <math>y_{n+1} = y_n + f(x_n, y_n)*h</math> where f computes the derivative as a function of x and y.

			Here is an example of code (written in Python) which carries out Euler's Method for the example we discussed in class, <math>y' = -y</math>:

			def f(x, y):
⚫	Based largely off of a note available [http://imgur.com/a/uLSlM here] [[User:Simon1\|Simon1]] --[[User:Twine\|Twine]] 20:55, 25 October 2012 (EDT)
			return -y

			def euler(x, y, f, h, x_max):
			"""Take in coordinates x and y, a function f(x, y) which calculates
			dy/dx at (x, y), an increment h, and a maximum value of x.

			Return a list containing coordinates in the Euler's Method computation
			of the solution to Phi' = f(x, Phi(x)), Phi(x) = y, with the x
			values of those coordinates separated by h, and not exceeding x_max.
			"""
			if x > x_max: # we have already calculated all our values
			return []
			x_next, y_next = (x + h, y + f(x, y)*h) # calculate the next x, y values
			# return the current coordinates, and every coordinates following it, in a list
			return [(x_next, y_next)] + euler(x_next, y_next, f, h, x_max)

			if __name__ == '__main__':
			print euler(0, 1, f, 0.01, 1)[-1]

Twine: Created page, based largely off of http://imgur.com/a/uLSlM posted by Simon1

2012-10-26T00:55:19Z

Created page, based largely off of http://imgur.com/a/uLSlM posted by Simon1

New page

Numerical methods: <math>\frac{dy}{dt} = f(t, y)</math> and <math>y(t_0) = y_0</math>, <math>y = \Phi(t)</math> is a solution.

'''1. Using the proof of Picard's Theorem:'''

<math>\Phi_0(x) = y_0</math>

<math>\Phi_n(x) = y_0 + \int_{x_0}^x f(x, \Phi_{n-1}(x)) dx</math>

<math>\Phi_n(x) \rightarrow \Phi(x)</math>

'''2. The Euler Method:'''

<math>y_{n+1} = y_n + f(t_n, y_n)(t_{n+1} - t_n) = y_n + f_n h</math> if h is constant

Backward Euler formula: <math>y_{n+1} = y_n + h f(t_{n+1}, y_{n+1})</math>

Local truncation error: <math>e_{n+1} = \frac{1}{2} \Phi''(t_n)h^2 \leq \frac{Mh^2}{2}</math> where <math>m = \mathrm{max} |\Phi''(t)|</math>

Local error is proportional to <math>h^2</math>.

Global error is proportional to h.

'''3. Improved Euler Formula (or Heun Formula):'''

<math>y_{n+1} = y_n + \frac{f_n + f(t_n + h, y_n + hf_n)}{2} h</math>

Local truncation error is proportional to <math>h^3</math>

Global truncation error is proportional to <math>h^2</math>

'''4. The Runge-Kutta Method:'''

<math>y_{n+1} = y_n + \frac{k_{n1} + 2k_{n2} + 2k_{n3} + k_{n4}}{6} h</math>

where

<math>k_{n1} = f(t_n, y_n) \quad k_{n2} = f(t_n + \frac{1}{2} h, y_n + \frac{1}{2}hk_{n1})</math>

<math>k_{n3} = f(t_n + \frac{1}{2} h, y_n + \frac{1}{2}hk_{n2}) \quad k_{n4} = f(t_n + h, y_n + hk_{n3})</math>

Local truncation error is proportional to <math>h^5</math>.

Global truncation error is proportional to <math>h^4</math>.

Based largely off of a note available [http://imgur.com/a/uLSlM here] [[User:Simon1|Simon1]] --[[User:Twine|Twine]] 20:55, 25 October 2012 (EDT)