solution of linear equations using gaussian elimination

FAQ Reference s Summary Info Resources Learning Objectives Introduction Gaussian elimination method Gaussian elimination procedure Programming exercise Home HOME Introduction Gaussian E

Solution of linear equations using Gaussian elimination

FAQ Reference s Summary Info

Resources

Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

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HOME

Introduction

Gaussian Elimination

Method

Gaussian Elimination

Procedure Programming Exercise

Resources

Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

Learning Objectives

Learning objectives in this module

1 Develop problem solution skills using computers and numerical methods

2 Review the Gaussian elimination method for solving simultaneous linear equations

3 Develop programming skills using FORTRAN New FORTRAN elements in this module

-use of NAG-library

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Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

Introduction

Solution of sets of linear equations is required in many petroleum applications Many methods exist for this purpose, direct methods as well as iterative methods The reference mentioned in the end may

be consulted for a review of such methods

A direct method frequently used in petroleum applications is the Gaussian elimination method, and the simplest form of this method will be discussed below First, let’s review the concept of

simultaneous linear equations A set of linear simultaneous equations may be written as:

a11x1 + a12x2 + a13x3 + + a1N xN = d1

a21x1 + a22x2 + a23x3 + + a2N xN = d2

a31x1 + a32x2 + a33x3 + + a3N xN = d3

aM 1x1 + aM 2x2 + aM 3x3 + + aMN xN = dM

Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

Introduction

 Here we have a total of N unknowns (xj, j=1, 2… N), related through M equations The coefficients in the left sides of the equations (aij, i=1, 2… N ; j=1, 2… M ) are known parameters, and so are also the coefficients on the right side (bi, i=1, 2… M)

a11x1 + a12x2 + a13x3 + + a1N xN = d1

a21x1 + a22x2 + a23x3 + + a2N xN = d2

a31x1 + a32x2 + a33x3 + + a3N xN = d3

aM 1x1 + aM 2x2 + aM 3x3 + + aMN xN = dM

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Introduction Gaussian elimination method

Gaussian elimination procedure

Programming exercise

Introduction

The equations may alternatively be written in a compact form:

where A is the coefficient matrix, and b is the right hand side vector:

A ⋅ x = b

A =

a11 a12 a 1N

a21 a22 a 2N

a M 1 a M 2 a MN





























b =

b1

b2

b M





























x =

x1

x2

x M





























If the number of unknowns

is equal to the number of equations, N=M, we may be able to solve the set of

equations, provided that the equations are unique

Gaussian Elimination Method

Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

Gaussian elimination method

For simplicity, let’s use the following set of 3 equations and 3 unknowns, ie N=3 and M=3, in order to illustrate the Gaussian elimination method:

The method starts by multiplying Eq (4) by –a21/a11 and then

add it to Eq (5) The resulting equation becomes:

We then multiply Eq.(4) by and add it to Eq.(6), resulting in:

a 11 x 1+ a 12 x 2 + a 13 x 3 = d 1

a 21 x 1 +a 22 x 2 +a 23 x 3 = d 2

a 31 x 1 +a 32 x 2 +a 33 x 3 = d 3

(4) (5) (6)

a ′ 22x2 + ′ a 23x3 = ′ d 2

a ′ 32 x 2 + ′ a 33 x 3 = ′ d 3

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Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

Gaussian elimination method

The set of equation has now become

Next, we multiply Eq (8) by –a’ 32 /a’ 22 and add it to Eq (9), so that the set of equations become:

This completes the first part of the Gaussian elimination method, called the forward elimination process

a11x1 + a12x2 + a13x3 = d1

a ′ 22x2 + ′ a 23x3 = ′ d 2

a ′ 32x2 + ′ a 33x3 = ′ d 3

(7) (8) (9)

a11x1+ a12x2 + a13x3 = d1

a ′ 22x2 + ′ a 23x3 = ′ d 2

a ′ ′ 33x3 = ′ ′ d 3

(10) (11) (12)

Continue to the Second Part

Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

Gaussian elimination method

Eq (12) may now be used to solve directly for x3:

After completion of the forward elimination process, determined the last unknown of the vector (x3) by Eq (13), we will perform a

back substitution process This simply means that as the unknowns are calculated, in our simple example from x3 and downwards, they are substituted into the equations above, and the next unknown may be computed For Eqs (12) and (11) this process is carried out as follows:

Based on the example above, we may formulate a general procedure for the forward elimination, solution for the last unknown, and back substitution to get the rest of the unknowns

x3 = ′ ′ d 3 / a ′ ′ 33

x 2 = (d ′ 2− ′ a 23 x 3) / a ′ 22

x 1 = (d 1 − a 12 x 2 − a 13 x 3 ) / a 11

(13)

(14) (15)

Gaussian Elimination Procedure

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Learning Objectives

Introduction Gaussian elimination method

Gaussian elimination procedure

Programming exercise

Gaussian elimination procedure

1) Forward elimination:

2) Solving for xN:

3) Back substitution:

a i, j = a i, j + a k, j − a i,k

a k,k



 



d i = d i + d k − a i,k

a k ,k



 



xN = dN

aN,N

x i = 1

a i,i d i − a i, j

j=i+ 1

n

∑ x j



 



  , i = n −1, ,1

Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

Programming Exercise

Make a FORTRAN program consisting of a main program, that reads the coefficients of the system of equations (n, a1,1…an,n ,

d1…dn) from an input file (IN.DAT) and writes the results (x1…xn)

to an out-file (OUT.DAT), and a subroutine, SUBROUTINE GAUSS(X,A,D,N), that uses the Gaussian elimination method in order to solve the set of equations and returns n values of x to the main program

Test the program on the following set of equations:

+27

= +6t

+4s -2z

+5y 3x

-19

= -9t

+3s +8z

-7y 5x

-33

= -2t

-3s +6z

4y

+53

= +3t

+9s -3z

-2y 8x

-35

= +7t

-5s +4z

+3y 2x

Continue Push

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Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

Programming Exercise

The Petra-server includes a NAG-library of scientific subroutines Find a subroutine in this library (use naghelp) that may be used for solution of simultaneous equations (Gaussian elimination method or some other method) Modify the program above so that is asks you on the screen if you want to use the

programmed Gaussian-routine or the NAG-routine for the solution Check that the solutions from the two methods are identical (Note that all real variables should be declared as

REAL*8 (double precision) since the NAG-routines require this) The NAG-routine may be linked in by the command

xlf –o prog fil.f –L/localiptibm3/lib –l nag

Resources

Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

Resources

Introduction to Fortran Fortran Template here The whole exercise in a printable format here Web sites

 Numerical Recipes In Fortran

 Fortran Tutorial

 Professional Programmer's Guide to Fortran77

 Programming in Fortran77

Fortran Tem plate

Solution of lin ear equation using Gaussi

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Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

General information

Title: Solution of linear equations using Gaussian elimination Teacher(s): Professor Jon Kleppe

Assistant(s): Per Jørgen Dahl Svendsen Abstract: Provide a good background for solving problems within

petroleum related topics using numerical methods

4 keywords: Linear equations, Gaussian elimination, matrices,

subroutines Topic discipline:

Learning goals: Develop problem solution skills using computers and

numerical methods Size in megabytes: 0.6 MB

Software requirements: MS Power Point 2002 or later, Flash Player 6.0 Estimated time to complete:

Copyright information: The author has copyright to the module and use of the

content must be in agreement with the responsible author or in agreement with

http://www.learningjournals.net.

About the author

Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

FAQ

No questions have been posted yet However, when questions are asked they will be posted here.

Remember, if something is unclear to you, it is a good chance that there are more people that have the same question

For more general questions and definitions try these

Dataleksikon Webopedia Schlumberger Oilfield Glossary

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Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

References

W H Preuss, et al., “Numerical Recipes in Fortran”, 2nd edition Cambridge University Press, 1992

References to the textbook :

 Gauss Jordan elimination: page 27

 Gaussian Elimination with backsubstitution: page 33

Learning Objectives

Introduction

Gaussian elimination

method

Gaussian elimination

procedure

Programming

exercise

Summary

Subsequent to this module you should

in Fortran