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Chapter 3

- Solution of Simultaneous Linear Algebraic

Equations Lecture (II)

Note Besides the main textbook, also see Ref

Applied Numerical Methods with MATLAB for

Engineers and Scientists, S. Chapra, Ch. 9.

Outline

- How to solve small (n ? 3) sets of linear

algebraic equations (does not require a

computer)? - The graphical method
- Cramers rule
- How to solve large (n ? 3) sets of linear

algebraic equations (solution algorithms that can

be implemented on a computer)? - Gauss elimination method-an introduction

The graphical method

- Graphical solution of a set of two linear

algebraic equations The solution is represented

by the intersection point of the two lines. - Notes
- Can work if the system is small (n ? 3)

impractical for larger systems. - The resulting solution is not very accurate.
- Useful for visualizing the properties of the

solutions.

Some examples

- Singular systems (a) no solution and (b)

infinite solutions - Ill-conditioned systems (c) the slopes of the

two lines are so close that the point of

intersection is difficult to detect visually.

Cramers rule

- The determinant A of a matrix A
- For small matrices
- Notes For large matrices, the determinants can

be very complicated The determinant A is a

single number.

Minors

Cramers rule (cont.)

- Solving a small set of linear equations using

Cramers rule - An example Solving three linear equations given

by - Ax b, where
- Solution by Cramers rule
- Notes Large computational effort accuracy

limited by round-off errors impractical for

large systems.

What should be x2 and x3?

An example

- Example 3.9 Find the solution of the following

system of equations using Cramers rule - x1 x2 x3 3,
- 2x1 x2 x3 0,
- and 3x1 2x2 2x3 15.

Naïve Gauss Elimination

- Naïve The algorithm does not avoid the problem

of division by zero. - Gauss elimination
- Inspiration Elimination of unknowns
- Two major steps
- (1) Forward elimination
- (2) Back substitution.

An example

- Solving the following set of equations using

elimination of unknowns a11x1 a12x2

a13 x3 b1 - a21x1 a22x2 a23 x3 b2
- a31x1 a32x2 a33 x3 b3
- Can we design a systematic solution algorithm so

that it can be implemented on a computer? - Yes!-Gauss elimination!
- Continue in Lecture (III)

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