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7/29/2019 MECH0801 Wk 03 Excel Lec
1/31
MUT/ENGPG-Class
Dr.Thitaphol Huyanan
Preliminary Basics II
MECH-0851FINITE ELEMENTANALYSIS
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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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SystemofLinearEquations
(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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ClassDr.Thitaphol Huyanan
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.
SystemofLinearEquations
(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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MUT/ENG - PG
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
non- singular
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ClassDr.Thitaphol Huyanan
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
|
|
From the given system of equations ,
(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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ClassDr.Thitaphol Huyanan
Gauss-Jordan Elimination17
|
|
From the given system of equations ,
(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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ClassDr.Thitaphol Huyanan
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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ClassDr.Thitaphol Huyanan
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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ClassDr.Thitaphol Huyanan
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