MECH0801 Wk 03 Excel Lec

7/29/2019 MECH0801 Wk 03 Excel Lec

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Dr.Thitaphol Huyanan

Preliminary Basics II

MECH-0851FINITE ELEMENTANALYSIS


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ClassDr.Thitaphol Huyanan

Agenda1

Solving of Linear Equation System

Introduction to MS Excel


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SolvingofLinearEquationSystem

2


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Class

Overviews3


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4

A linear equation is an algebraic equation in

which each term is either a constant or a product of

a constant times the first power of a variable. The

linear equation is equivalent to equating the 1st-order polynomial to zero.

0 1 1 2 2 2 2 1 1 0 1 2 0 20 ( / ) ( / )

linear equation1 -order polynomialst

m b

a a x a x a x a x a y a a x a a

y x

This type of equation is called a linear equationsince it represents a straight line in the Cartesian

coordinates and it can be written in a general form

as 1 1 2 2a x a x b

SystemofLinearEquations


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(continued)5

The linear algebraic equation can involve morethan two variables. Thus a general form of any n-variable linear equation can be expressed as

where each a is coefficient constants, b is constant,and each x is an variable.

A collection ofn linear equations that

simultaneously involves the same set ofnvariables is called a system of linear equations orso-called a linear system.

1 1 2 2 n na x a x a x b

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

n n

n n

n n nn n n

a x a x a x ba x a x a x b

a x a x a x b


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6

The linear system ofn simultaneous equations

with n variables can also be rewritten in a matrix

form.11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

11 11 1 1 1

21 22 2 2 2

1 2

n n

n n

nn n nn n

n

n

n nn n nn

a x a x a x b

a x a x a x b

ba x a x a x

a a a x b

a a a x b

x ba a a

x bA

A solver that directly uses the above system of

linear equations to determine values of a series of

variables simultaneously is called a direct solver.


(continued)


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Augmented Matrix7

A augmented matrixfor a system of equations is

the matrix of numbers in which each row represents

those constants from one equation (both the

coefficients and the constant on the other side ofthe equal sign) and each column represents all the

coefficients for a same variable.

The augmented matrix is constructed by

appending the vector of the constants next to the

coefficients matrix for the purpose of performing the

same elementary row operations on each of the

given matrices.

augmenting A x b A b


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Echelon Form8

In linear algebra, a matrix is in row echelon form if

All nonzero rows the rows with at least onenonzero element are above any rows of all

zeroes, The leading coefficient the first nonzero number

from the left, also called the pivot of thenonzero row is always strictly to the right of theleading coefficient of the rowabove it,

All entries in a columnbelow a leading entry are zeroes(implied by the first two criteria).

11 1 1 1

1

0

0

0 0

i n

ii i i

nn n

u u u b

u u b

u b

U b


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Echelon Form (continued)9

A matrix is in reduced row echelon form (alsocalled row canonical form) if it satisfies theadditional condition which is every leadingcoefficient is one and is the only nonzero entry in itscolumn.

1

1 0 0

0 1 0

0 0 1

i

n

b

b

b

I b


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ElementaryRowOperations10

Consider 3 linear equations with 3 unknown below,

Switching Rows : any two equations can be

interchanged,

1 2 3

1 2 3

1 2 3

augmenting

4

2 3

3 2

2 3

3 4

4 5 1 2 3 42 3 4 33 4 5 2

x

x

x

x x

x x

x x

A b

A bA x b

1 3

3 1

1 2 3

1 2 3

1 2 3

3 2

2 3

4

, 4 5 3 4 5 2

3 4 2 3 4 3

2 3 1 2 3 4,

R R

R R

x

x

x

x x

x x

x x


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ElementaryRowOperations

(continued)11

Multiplying A Row with One Scalar: any equationcan be multiplied by a nonzero constant,

Adding Rows : any equation can be replaced by

the sum of itself and any multiple of another

equation.

2 2

1 2 3

1 2 3

1 2 3

2

3 2

4 64

2

4 5 3 4 5 2

6 8 4 6 8 62 3 1 2

,3 4

R R

x

xx

x x

x xx x

2 2 1

1 2 3

1 2 3

1 2 3

3 2

8

4

4 5 3 4 5 2

10 13 1 10 13 8

2 3 1 2 4

,

3

R R R

x

x

x

x x

x x

x x


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Class

Direct Solvers12


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Inverse of Matrix13

Matrix cannot be divided by another matrix,but analogous to division can be performed by aprocess known as inversion of matrices.

The inverse of a square matrix is anothersquare matrix of same order, such thatA1A1 1 A A A A I

The inverse of matrix is formed by dividing

each member of the adjoint of that matrix by thedeterminant of the same matrix. Therefore

1 adjoint adj

det

A A

AA A


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Direct Matrix Inversion14

Consider the matrix equation of linear system

, if the matrix is nonsingular ( ) then its

inverse is existed. Therefore

A x b

A det 0A -1A

, where

-1 -1 -1

-1 -1A A x A b I x A b

x A b A C AT

1

1

T T

From the given system of equations

(1) Compute to check as if is

(2) Compute then obtain from its definition

(3) Obtain a value of each fromix

A x b

A A

C A C A

x A b

nonsingular


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11 12 1

-

th

i i

i

i

a a b a

From the given system of equations ,

(1) Compute to check as if is

(2) Compute , where is that is replaced its column

by the constants vector

A x b

A A

A A A

b

A

non singular

1

21 22 2 2

1 2

1 2

n

n

n n n nn

i

i i

i n

a a b a

a a b a

x x (3) Obtain a value of each fromA

A

Cramer's rule15


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Gauss Elimination16

|

|


(1) Compose the

(2) Forward eliminate the augmented matrix into a

via a series of row operations(3) Obtain solut

A x b

A b

U b

augmented matrix

row

echelon form

1

1( 1)..1

nn

nn

n

i i ij j

j iii

bx

u

x b u x i nu

ion by backward substitution with the

formula

, then

for


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Gauss-Jordan Elimination17

|

|


(1) Compose the

(2) Forward eliminate the augmented matrix into a

via a series of row operations(3) Obtai

A x b

A b

I b

augmented matrix

reduced

row echelon form1

i ix b i n =n solution from for to , or x b


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3

3

n

n

(a) Direct solvers that suits A whose dimension is less or equal

to three are,

>

>(b) Direct solvers that suits A whose dimension is greater than

three are,

>

Matrix Inversion

Cramer's Rule

Gauss Eli>

minationGauss- Jordan Elimination

Remarks on Direct Solution18


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1 2 3

1 2 3 4

2 3 4

3 4

2 3 21

4 2 11

2 3 7

5 2 11

x x x

x x x x

x x x

x x

12

3

4

2 3 1 0 211 4 2 1 110 2 1 3 70 0 5 2 3

x

x

x

x

A bx

1 1 2 14 2 1 3 1 0

2 1 2 1 3 1 1 2 1 3 0 00 5 2 0 5 2

2 1 34 1 1 35 33 0

A

non - singular

30.272714.54554.09098.7273

x

Matrix InversionCramer's Rule

Gauss EliminationGauss - Jordan Elimination

Direct

Solver

Remarks on Direct Solution

(continued)19


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Introduction to MS Excel20


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Class

Overviews21


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22

Table VS WorkSheet

Text Number Formula

Row

Index

Column

Index


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MS Excel in Big Picture23

workbook

worksheet

range of cells

cell

status bar

name box formula bar

command bar

cursor


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WorkBook24

bookname

navigation bar selection barscrollbars

WorkbookCollection

Workbook

WorksheetHandling Bars

WorksheetCollection

Reference

name

stored folder

navigation bar

selection bar

scroll bars


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WorkSheet25

selection tab

columnindex

column

row

rowindex

sheetname

WorksheetCollection

Worksheet

Format

Rows

Range & Cell

Collection

Columns

Reference

indexes

widths

indexes

heights

name

workbook

background

selection tab color


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Class

Cell & Range in MS Excel26


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Overview27


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Cell Referring28


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Types of Recorded Data29

Constant

Number :

Text :

Formula

Formulas are equations that perform

calculations on values in a worksheet. A formula

starts with an equal sign (=). A formula can alsocontain any or all of the following: functions,

references, operators, and constants.


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Questions and Discussion30

MECH0801 Wk 03 Excel Lec

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