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CE 341/441 - Review 1 - Fall 2004
p. R1.
REVIEW NO. 1TAYLOR SERIES • Find
away from
given
and the derivatives of
evaluated at
and the derivatives of
evaluated at
are constant and
not
x
-dependent.
- When we use Taylor Series we do not carry all terms!• Our derivations typically carry enough terms to allow us to establish the error in our
f formula.
x ( )
x^
a
f^
a (
)^
f^
x (
)^
x^
a
f^
x (
)^
f a (
)^
x^
a
(^
df ) ----- dx
x^
a =
x^
a
(^
d - 2
f d x
2 --------
x^
a =
x^
a
(^
d - 3
f d x
3 --------
x^
a =
x^
a
(^
d - 4
f d x
4 --------
x^
a =
x^
a
(^
n ) n
d - n^
f d x
n ---------
x^
a =
n
(^
-^
x^
a
(^
n )
1 +^
d^ n^
1 + d x
n^
1 +
---------------
ξ
a^
ξ^
x
f^
a (
)^
f^
x^
a
CE 341/441 - Review 1 - Fall 2004
p. R1.
- Depending on what our purposes are, we may:
- Truncate the series and only carry the
term
- The error term is
- Carry enough terms so that we have a detailed form of the largest portion of the error
term
The
term is carried to ensure that we know that this next term is
where we systematically truncate all terms
O x
a
(^
n )
f^
x ( )
f^
a (
)^
x^
a
(^
df ) ----- dx
x^
a
x^
a
(^
d - 2
f d x
2
x^
a
O x
a
(^
E
O x
a
(^
f^
x ( )
f^
a (
)^
x^
a
(^
df -----) dx
x^
a
x^
a
(^
d 2 f d x
2
x^
a
x^
a
(^
d - 3
f d x
3
x^
a
O x
a
(^
E
x^
a
(^
d
3 f d x
3
x^
a
O x
a
(^
CE 341/441 - Review 1 - Fall 2004
p. R1.
- Carry only the remainder term which represents
all
terms in the truncated series
- Error term is• We note that
therefore depends on where you’re evaluating
- Typically we just estimate
as a starting or a mid point in the interval as a
constant!
- However when you differentiate or integrate the error terms which involve
you must be very careful. It is best to consider a sequence of terms evaluated at
f^
x ( )
f^
a (
)^
x^
a
(^
df -----) dx
x^
a
x^
a
(^
d - 2 f d x
2
x^
a
x^
a
(^
d - 3
f d x
3
x^
ξ
a^
ξ^
x
E
x^
a
(^
d - 3 f d x
3
x^
ξ
ξ^
ξ^
x ( )
=
f^
x ( )
ξ
ξ
x^
a
CE 341/441 - Review 1 - Fall 2004
p. R1.
NUMERICAL SOLUTION TO LINEAR SYSTEMS OF ALGEBRAIC EQUA-TIONS • Solve the system of linear algebraic equations Direct Methods • All direct methods are based on some type of triangulation Gauss elimination • Develop an upper triangular matrix by manipulating
and
→
operations
- Perform backward solution sweep
→
operations
A
X
B
A
B
O n
(^
O n
(^
CE 341/441 - Review 1 - Fall 2004
p. R1.
Matrix conditioning • Ill-conditioned matrices lead to inaccurate solutions for• Diagonally dominant matrices are not ill-conditioned.• We use pivoting to improve structure/conditioning of the matrix• Roundoff effects how badly ill-conditioning effects the solution
- Use larger word size• Use iterative methods
⇒
only
versus
operations
Matrix storage • full• banded• symmetrical• skyline• non-zero locations only (must still store pointers to identify matrix locations)
X
O n
(^
O n
(^
CE 341/441 - Review 1 - Fall 2004
p. R1.
Iterative Methods • Based on using a starting approximation to all terms in every equation except the term
on the diagonal which you solve for
-^
Point Jacobi Method:
Iterate using values from the previous iteration. It is the simplest
method
-^
Gauss-Seidel:
Like Point Jacobi except you update all values with the most recently
computed value
-^
Point Relaxation Methods:
Improve estimates based on previous estimates (either
average or extrapolate out)
- Stability of iterative methods is only guaranteed if the matrix is diagonally dominant.
The solution may or may not be stable if the matrix is diagonal.
- Iterative methods are useful for
- very large sparse systems (benefits include storage and number of operations)• ill-conditioned systems
CE 341/441 - Review 1 - Fall 2004
p. R1.
Method 1 to derive g(x): Power series • Set up generic
N
th^
degree polynomial
-^
Force
at
nodes
(or data or interpolation points)
- Solve for unknown coefficients
,^
from the resulting constraint equations
Method 2 to derive g(x): Lagrange basis functions • Each Lagrange basis function is associated with a data point/node• Results in the same function
as method 1.
g x
(^
)^
a^ o^
a^1
x
a
x 2 2
a^ N^
x^ N
g x
i (^
)^
f^ i =
i^
N ,
a^ i^
i^
N ,
g x
(^
)^ g x
(^
)^
f^ i
V
i^
x (
i^
0
N ∑ =
V
i^
x (
)^
x^
x^ o
(^
)^
x^
x^1
(^
)^
x^
x^2
(^
x
x^ i
1
(^
)^
)^
x^
x^ i
1 +
(^
x
x^ N
(^
x^ i
x^ o
(^
)^
x^ i
x 1
(^
)^
x^ i
x 2
(^
x
i^
x^ i^
1
(^
)^
x^ i
x^ i^
1 +
(^
x^ i
x^ N
(^
V
i^
x^ j (^
)^
i^
j
0
i^
j ≠
CE 341/441 - Review 1 - Fall 2004
p. R1.
- Example for a 3 data point quadratic interpolation function
g x
(^
)^
f^ o
V
o^
x ( )
f^^1
V
^1
x (
)^
f^^2
V
^2
x ( )
1.0 V
0 (x)
x
4.0 V
(x) 2
x^1
= 4
x^0
= 3
x^2
= 5
x^1
= 4
x^0
= 3
x^2
= 5
x^1
= 4
x^0
= 3
x^2
= 5
x x
2.0 V
(x) 1
CE 341/441 - Review 1 - Fall 2004
p. R1.
Method 3 to derive g(x): Newton forward interpolation
where 0 1 2 3 4
g x
(^
)^
f^ o
x^
x^ o
(^
f o h ---------
-^
x^
x^ o
(^
)^
x^
x^1
(^
2
f^ o h
2 -----------
x
x^ o
(^
)^
x^
x 1
(^
)^
x^
x 2
(^
3 f^ o h 3 -----------
1 ------ N!
x^
x^ o
(^
)^
x^
x 1
(^
)^
x^
x 2
(^
x
x^ N
1
(^
N f o h
N -------------
i^
f^ i
f^
i^
2 f^ i
3 f^ i
4 f^ i
f^ o
f^
o^
f^^1
f^ o
2
f^ o
f^
1
f^
o
3 f^ o
2 f^^1
2 f^ o
4 f^ o
3 f^^1
3 f^ o
f^^1
f^
1
f^^2
f^^1
2
f^^1
f^
2
f^
1
3 f^^1
2 f^^2
2 f^^1
f^^2
f^
2
f^^3
f^^2
2
f^^2
f^
3
f^
2
f^^3
f^
3
f^^4
f^^3
f^^4
CE 341/441 - Review 1 - Fall 2004
p. R1.
same
N
th
degree polynomial interpolating function as Lagrange except
using forward difference operators
- However it is much more efficient to implement computationally as compared to the
Power series or Lagrange basis functions methods
- The error is again given by:
and can be approximated as:
e x (
)^
f^
x (
)^
g x
(^
–^
x^
x^ o
(^
)^
x^
x^1
(^
x
x^ N
(^
N
(^
-^
f^
N^
1 + (^
)^
ξ(
x^ o
ξ^
x
e x (
)^
x^
x^ o
(^
)^
x^
x^1
(^
x
x^ N
(^
N
(^
-^
N
1 +^
f^ o
h
N
1 +
--------------------
CE 341/441 - Review 1 - Fall 2004
p. R1.
Extrapolation • Extend the interpolation to range outside of the range of the data points Hermite Interpolation • Develop an interpolating function which passes through
functional values as
well as the 1st, 2nd and subsequent derivatives at the
data points
degree polynomial to match up to the
p
th^
derivative
at
data points.
N
N
f^0
,^ f
0 ,... ,
f^0
xN
x^0
x^1
x
(1)
(P) f^1
,^ f
1 ,... ,
f^1
(1)
(P)
fN
,^ f
N
,... ,
fN
(1)
(P)
p^
(^
)^
N
(^
)^
N
CE 341/441 - Review 1 - Fall 2004
p. R1.
ROOT FINDING ALGORITHMSBisection Method for Finding Roots • Solve for the roots of nonlinear algebraic equations • Based on interval halving and the sign of the function changing on the interval • Problem with double roots, multiple roots in an interval and singularities Newton-Raphson Method • Iterative formula for finding roots derived by developing a Taylor Series expansion for
and using only two terms of the Taylor Series and setting this to zero.
- Since we are trying to find the root such that
⇒
f^
x (
f^
x ( )
f^
x^ ( o
)^
f^
(^1) ( )
x
o (^
)^
x^
x^ o
(^
)^
H.O.T.
f^
x^ ( r
)^
f^
x^ ( o
)^
f^
(^1) ( )
x
o (^
)^
x^ r
x^ o
(^
