Python for Science and Engg: Solving Equations & ODEs

Fossee

Description

Python for Science and Engg:Solving Equations & ODEsFOSSEEDepartment of Aerospace EngineeringIIT BombaySciPy 2010, Introductory tutorialsDay 1, Session 6FOSSEE (IIT Bombay) Solving Equations & ODEs 1 / 34Solving linear equationsOutline1 Solving linear equations2 Finding Roots3 ODEs4 FFTsFOSSEE (IIT Bombay) Solving Equations & ODEs 2 / 34Solving linear equationsSolution of equationsConsider,3x+ 2y-z = 12x- 2y+ 4z =-21-x+ y-z = 02Solution:x = 1y =-2z =-2FOSSEE (IIT Bombay) Solving Equations & ODEs 3 / 34Solving linear equationsSolving using MatricesLet us now look at how to solve this using matricesIn []: A = array([[3,2,-1],[2,-2,4],[-1, 0.5, -1]])In []: b = array([1, -2, 0])In []: x = solve(A, b)FOSSEE (IIT Bombay) Solving Equations & ODEs 4 / 34Solving linear equationsSolution:In []: xOut[]: array([ 1., -2., -2.])FOSSEE (IIT Bombay) Solving Equations & ODEs 5 / 34Solving linear equationsLet’s check!In []: Ax = dot(A, x)In []: AxOut[]: array([ 1.00000000e+00, -2.00000000e+00,-1.11022302e-16])The last term in the matrix is actually 0!We can use allclose() to check.In []: allclose(Ax, b)Out[]: True10 mFOSSEE (IIT Bombay) Solving Equations & ODEs 6 / 34Solving linear equationsProblemSolve the set of equations:x+y+ 2z-w = 32x+ 5y-z- 9w =-32x+y-z+ 3w =-11x- 3y+ 2z+ 7w =-515 mFOSSEE (IIT Bombay) Solving Equations & ODEs 7 / 34Solving linear equationsSolutionUse solve()x =-5y = 2z = 3w = 0FOSSEE (IIT Bombay) ...

Sujets

• Python
• Tutoriel
• Calcul (mathématiques)

Informations

• Publié par: mtoledan
• Publié le: 27 juin 2011
• Nombre de lectures: 191
• Langue: English

Extrait

FOSSE

Python for Science and Engg: Solving Equations & ODEs

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FOSSEE

Department of Aerospace Engineering IIT Bombay

SciPy 2010, Introductory tutorials Day 1, Session 6

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FFTs

4

ODEs

3

Roots

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2

Finding

Solving

linear equations

1

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ODEsons&

Consider,

Solution:

x=1 y= −2 z= −2

3/34

− = 2

3x+2y−z=1 2x−2y+4z= −2 −x+ 1 y−z=0

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Let us now look at how to solve this usingsecmirta

array([[3,2,-1], [2,-2,4], [-1, 0.5, -1]]) array([1, -2, 0]) solve(A, b)

In []: b = In []: x =

In []: A =

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Solution:

In []: Out[]:

x array([

1.,

-2.,

-2.])

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In []: Ax = dot(A, x) In []: Ax Out[]: array([ 1.00000000e+00, -1 11022302e-16]) .

The last term in the matrix is actually0! We can uselaes)(clolto check.

In []: allclose(Ax, b) Out[]: True

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Solve the set of equations:

x+y+2z−w=3 2x+5y−z−9w= −3 2x+y−z+3w= −11 x−3y+2z+7w= −5

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Use

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x y z w

= = = =

−5 2 3 0

solve()

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Solving

linear equations

Outline

1

4/3

FFTs

ODEs

4

Roots

3

2

Finding

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3/4

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Calculates the roots of polynomials To calculate the roots ofx 2 −5x+6

In []: In []: Out[]:

coeffs = [1, -5, roots(coeffs) arra ([3., 2.])

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Finds the roots of a system of non-linear equations Input arguments - Function and initial estimate Returns the solution

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In []:fromscipy.optimizeimportfsolve

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