Mech300 Numerical Methods, Hong Kong University of Science and Technology.

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Part Three

Linear Algebraic Equations

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

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Motivation

The system of linear equations occur very frequently in engineering applications. For small numbers of equations (n < 4) equations can be solved manually by some simple techniques. However, for four or more equations, solutions become arduous and computers must be utilized.

Historically, the inability to solve all but the smallest sets of equations by hand has limited the scope of problems addressed in many engineering applications.

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

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An Example of Linear Equations in Engineering

xi: mass in reactor i

ai: properties and characteristics

of the system

bi: the forcing functions acting

on the system, e.g., feed rate.Lumped variable system

Distributed variable system

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

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Mathematical Background – Matrix Notation

nc

c

c

C

.

.

.2

1

Row vector: [B] = [b1 b2 . . . bm]

A is a square matrix if n = m

Column vector: {C}

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

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Special Types of Square Matrices

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

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Mathematical Background – Matrix Operating Rules

n

kkjikij bac

1

Matrix addition: [C] = [A] ± [B] (A and B must have the same number of rows n and the same number of columns m)

cij = aij ± bij

[A] ± [B] = [B] ± [A]

([A][B])[C] = [A]([B][C])

([A]([B] + [C]) = [A][B] + [A][C] or ([A] + [B])[C] = [A][C] + [B][C]

[A][B] ≠ [B][A]

Matrix multiplication: [C] = [A][B]

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

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Mathematical Background – Matrix Operating Rules

n

iiia

1

Inverse Matrix: [A][A]-1 = [A]-1[A] = [I]

• a non-square matrix can’t have an inverse

• not every square matrix has an inverse

Transpose of a matrix: [B] = [A]T

bij = cji

Trace of a matrix: tr [A] =

Augmentation of a matrix: addition of column(s) to the original matrix

Example:

100

010

001

333231

232221

131211

333231

232221

131211

aaa

aaa

aaa

aaa

aaa

aaa

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

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Linear Algebraic Equations in Matrix Form

nnnn

n

n

aaa

aaa

aaa

A

...

...

...

...

...

...

21

22221

11211

nbbb ...21

[A]{X} = {B}

{B}T = {X}T = nxxx ...21

[A]-1[A]{X} = [A]-1{B}

{X} = [A]-1{B}

Mech300 Numerical Methods, Hong Kong University of Science and Technology.

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Overall Structure