ME 218: ENGR COMPUTATIONAL METHODS
Lecture Notes (Set #4)
Secant Method
To overcome the need in the NewtonRaphson iterative scheme to evaluate first
derivative of the function, or the possibility of the derivative going to zero, the Secant
Method can be used. It utilizes the information of a second point to evaluate the
derivative, by assuming that the function is linear in the domain of interest.
From the NewtonRaphson’s method:
x
2
=
x
0
– f(x
0
)/f’(x
0
)
But using another point,
x
1
, between
x
2
and
x
0
to determine the derivative,
f’(x
0
)
:
f’(x
0
) = (f(x
1
) – f(x
0
))/(x
1
– x
0
)
Thus, generalizing the iterative scheme:
x
N
+
1
=
x
N
−
f(x
N
)
x
N
−
x
N
−
1
f(x
N
)
−
f(x
N
−
1
)
Λ
Ν
Μ
Ξ
Π
Ο
The computation of the derivative is avoided by using two starting values,
x
0
and
x
1
!
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Matrix Algebra
A matrix is a rectangular array of numbers in which not only the value of the number
is important, but also its position in the array. It is very easy to solve a system of linear
equations using matrices.
Ω
Ψ
Χ
=
=
=
Υ
Υ
Υ
Υ
Φ
Τ
Σ
Ρ
=
m
j
n
i
for
a
a
a
a
a
a
a
a
a
a
ij
nm
n
n
m
m
,
,
2
,
1
,
,
2
,
1
]
[
2
1
2
22
21
1
12
11
…
…
…
A
.
'
'
'
'
.
'
'
'
'
position
column
the
denotes
j
position
row
the
denotes
i
a
ij
→
i)
Matrix Addition
Two matrices of the same size may be added or subtracted.
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 Fall '08
 Unknown
 Linear Algebra, Invertible matrix, elementary row operations

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