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Section ASyllabus
NUMERICAL DIFFERENTIATION AND INTEGRATION:- Approximating the
derivative, Numerical differentiation formulas, Introduction to Numerical quadrature,
Newton-Cotes formula, Gaussion Quadrature.
SOLUTION OF NONLINEAR EQUATIONS:- Bracketing methods for locating a root,
S i B
Q g g ,
Initial approximations and convergence criteria, Newton- Raphson and Secant methods,
Solution of problems through a structural programming language such as C or Pascal.
ERRORS IN NUMERICAL CALCULATIONS:-Introduction, Numbers and their accuracy,
Absolute relative and percentage errors and their analysis General error formula
Section B
Absolute, relative and percentage errors and their analysis, General error formula.
INTERPOLATION AND CURVE FITTING:- Taylor series and calculation of functions,
Introduction to interpolation, Lagrange approximation, Newton Polynomials, Chebyshev
Polynomials, Least squares line, curve fitting, Interpolation by spline functions.
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Section C
SOLUTION OF LINEAR SYSTEMS:- Direct Methods, Gaussian elimination and
pivoting, Matrix inversion, UV factorization, Iterative methods for linear systems,
Solution of problems through a structured programming language such as C or Pascal.
EIGEN VALUE PROBLEMS:- Jacobi, Given’s and Householder’s methods for symmetric
matrices Rutishauser method for general matrices Power and inverse power methods matrices, Rutishauser method for general matrices, Power and inverse power methods.
Section DSOLUTION OF DIFFERENTIAL EQUATIONS:- Introduction to differential
ti I iti l l bl E l ’ th d H ’ th d R K ttequations, Initial value problems, Euler’s methods, Heun’s method, Runge-Kutta
methods, Taylor series method, PredictorCorrector methods, Systems of differential
equations, Boundary valve problems, Finite-difference method, Solution of problems
through a structured programming language such as C or Pascal.
PARTIAL DIFFERENTIAL EQUATIONS, EIGENVALUES AND EIGENVECTORS:- Solution of
h b li b li d lli ti ti Th i l bl Th th d d th J bi’ th dhyperbolic, parabolic and elliptic equations, The eigen value problem, The power method and the Jacobi’s method
for eigen value problems, Solution of problems through a structural programming language such as C or Pascal.
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SECTION ASECTION A
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Approximations and Round-Off Errors
For many engineering problems, we cannot obtain analyticalsolutionssolutions.Numerical methods yield approximate results, results that areclose to the exact analytical solution. We cannot exactly computethe errors associated with numerical methodsthe errors associated with numerical methods.
Only rarely given data are exact, since they originate frommeasurements. Therefore there is probably error in the inputinformationinformation.Algorithm itself usually introduces errors as well, e.g.,unavoidable round-offs, etc …The output information will then contain error from both ofThe output information will then contain error from both ofthese sources.
How confident we are in our approximate result?Th ti i “h h i t i l l tiThe question is “how much error is present in our calculationand is it tolerable?”
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A H l i t d d lAccuracy. How close is a computed or measured value to the true valuePrecision (or reproducibility) How close is a computedPrecision (or reproducibility). How close is a computed or measured value to previously computed or measured values.Inaccuracy (or bias). A systematic deviation from the actual value.Imprecision (or uncertainty). Magnitude of scatter.
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Fi Fig. 1.1
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Significant Figures
Number of significant figures indicates precision. Significant digits of a number are those that can be used with confidence, e.g., the number of f g ,certain digits plus one estimated digit.
53,800 How many significant figures?
5.38 x 104 35.380 x 104 45 3800 x 104 55.3800 x 104 5
Zeros are sometimes used to locate the decimal point not significant figures.g
0.00001753 40.0001753 40.001753 4
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Error DefinitionsError Definitions
True Value = Approximation + ErrorTrue Value Approximation + Error
= True value – Approximation (+/-)E = True value Approximation (+/ )
True error
Et
valuetrueerror true error relative fractional True =valuetrue
%100error true error,relativepercent True t ×=ε %valuetrue
,p t
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For numerical methods, the true value will be known l h d l h f h b l donly when we deal with functions that can be solved
analytically (simple systems). In real world applications, we usually not know the answer a priori. pp , y pThen
%100ionApproximaterror eApproximat a ×=ε
%100ionapproximat Previous -ion approximatCurrent ×=ε
Iterative approach, example Newton’s method
%100ionapproximatCurrent
a ×ε
(+ / -)
Iterative approach, example Newton s method
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Use absolute value.Computations are repeated until stopping criterion is Computations are repeated until stopping criterion is satisfied.
sa εε ⟨ Pre-specified % tolerance based on the knowledge of your sa ⟨ on the knowledge of your solution
If the following criterion is met
)%10(0.5 n)-(2×=ε
you can be sure that the result is correct to at least n significant figures
)%10 (0.5s ×ε
significant figures.
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Round-off Errors
Numbers such as π, e, or cannot be expressed 7, , pby a fixed number of significant figures.Computers use a base-2 representation, they cannot precisely represent certain exact base-10 numbers.Fractional quantities are typically represented in computer using “floating point” form, e.g.,
t
Integer part
em.b exponent
Base of the number system used
mantissaused
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Figure 1.2
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Figure 1.3
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Figure 1.4
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156.78 0.15678x103 in a floating i t b 10 tpoint base-10 system
Suppose only 4 029411765.0
341 =
Suppose only 4 decimal places to be stored1
21100294.0
340 <≤× m
Normalized to remove the leading zeroes. Multiply
2
Normalized to remove the leading zeroes. Multiply the mantissa by 10 and lower the exponent by 1
0.2941 x 10-1Additional significant figure g gis retained
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11<≤
Thereforefor a base-10 system 0.1 ≤m<1
1<≤ mb
for a base-2 system 0.5 ≤m<1Floating point representation allows both fractions and very large numbers to be expressed fractions and very large numbers to be expressed on the computer. However,
Floating point numbers take up more room.T k l t th i t bTake longer to process than integer numbers.Round-off errors are introduced because mantissa holds only a finite number of significant figures.
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17
Interpolation
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Taylor’s Series and Interpolation18
Taylor s Series and Interpolation
Taylor Series interpolates at a specific point:Taylor Series interpolates at a specific point:
The function
Its first derivative
…
It may not interpolate at other points.
We want an interpolant at several f(c)’s.We want an interpolant at several f(c) s.
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Basic Scenario
We are able to prod some function, but do not know what
19
p ,it really is.This gives us a list of data points: [xi,fi]
f(x)
fifi+1
x xxi xi+1
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Interpolation & Curve-fitting 20
Often, we have data sets from experimental/observational measurements
Typically, find that the data/dependent variable/output varies…
As the control parameter/independent variable/input varies. Examples:
Classic gravity drop: location changes with time
Pressure varies with depth
Wind speed varies with timeWind speed varies with time
Temperature varies with location
Scientific method: Given data identify underlying relationshipy y g p
Process known as curve fitting:
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Interpolation & Curve-fitting21
Given a data set of n+1 points (xi,yi) identify a function f(x) (theGiven a data set of n+1 points (xi,yi) identify a function f(x) (the
curve), that is in some (well-defined) sense the best fit to the data
Used for:
Identification of underlying relationship (modelling/prediction)
Interpolation (filling in the gaps)
Extrapolation (predicting outside the range of the data)Extrapolation (predicting outside the range of the data)
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Interpolation Vs Regression 22
Distinctly different approaches depending on the quality of the data
Consider the pictures below:
extrapolate
fid
interpolate extrapolate
h h l i hi iPretty confident:there is a polynomial relationship
Little/no scatterWant to find an expression
h l h h ll h i
Unsure what the relationship isClear scatter
Want to find an expressionthat captures the trend:
i i i f h that passes exactly through all the points minimize some measure of the error Of all the points…
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Interpolation 23
Concentrate first on the case where we believe there is no error in the data (and
round-off is assumed to be negligible).
So we have yi=f(xi) at n+1 points x0,x1…xi,…xn: xj > xj-1
(Often but not always evenly spaced)
In general, we do not know the underlying function f(x)
Conceptually, interpolation consists of two stages:
Develop a simple function g(x) that
Approximates f(x)
Passes through all the points xi
Evaluate f(xt) where x0 < xt < xnf( t) 0 t n
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Interpolation24
Clearly the crucial question is the selection of the Clearly, the crucial question is the selection of the
simple functions g(x)
Types are:
Polynomialsy
Splines
T i t i f tiTrigonometric functions
Spectral functions…Rational functions etc…
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Curve Approximation25
We will look at three possible approximations (time We will look at three possible approximations (time
permitting):
l i l i l iPolynomial interpolation
Spline (polynomial) interpolation
Least-squares (polynomial) approximation
If you know your function is periodic, then trigonometric
functions may work better.
Fourier Transform and representationsFourier Transform and representations
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Polynomial Interpolation 26
Consider our data set of n+1 points yi=f(xi) at n+1 points p yi f( i) px0,x1…xi,…xn: xj > xj-1
In general, given n+1 points, there is a unique polynomial gn(x) of order n:
That passes through all n+1 points2
0 1 2( ) nn ng x a a x a x a x= + + + +K
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Polynomial Interpolation27
There are a variety of ways of expressing the sameThere are a variety of ways of expressing the same
polynomial
Lagrange interpolating polynomials
Newton’s divided difference interpolating polynomialsNewton s divided difference interpolating polynomials
We will look at both forms
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Polynomial Interpolation28
Existence – does there exist a polynomial that exactlyExistence – does there exist a polynomial that exactly
passes through the n data points?
Uniqueness – Is there more than one such polynomial?
We will assume uniqueness for now and prove it latterWe will assume uniqueness for now and prove it latter.
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Lagrange Polynomials29
Summation of terms, such that:,Equal to f() at a data point.
Equal to zero at all nEqual to zero at allother data points.
Each term is a nth-
( )0
( ) ( ) ( )n i ii
p x L x f x=
=∑degree polynomial ( )
( )0,
( )n
ki
k k i i k
x xL x
x x= ≠
−=
−∏ ( ),
1( )
0
i k
i j ij
i jL x
i jδ
=⎧= = ⎨ ≠⎩
Existence!!! 0j j i j≠⎩
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Linear Interpolation30
Summation of two lines:
1
( ) ( )
1
10
1 0
( ) ( ) ( )
( ) ( )
i ii
p x L x f x
x x x xf f
=
=
− −+
∑( )( )
( )( )
1 00 1
0 1 1 0
( ) ( )f x f xx x x x
= +− −
Remember this when we talk about piecewise-
linear splines
x0 x1
p
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Lagrange Polynomials31
2nd Order Case => quadratic polynomialsq p y
The first quadratic has roots at x1
and x2 and a value equal to the function dataat x
The second quadratic has roots at x0 and x2 and a value equal to the function data at x1.
• P(x ) = 0
The third quadratic has roots at x0 and x1 and a value equal to the function data at x2.
• P(x ) = 0
Adding them all together, we get the interpolating quadratic polynomial, such that:
• P(x ) = fat x0.• P(x0) = f0
• P(x1) = 0• P(x2) = 0
• P(x0) = 0• P(x1) = f1
• P(x2) = 0
• P(x0) = 0• P(x1) = 0• P(x2) = f1
• P(x0) = f0
• P(x1) = f1
• P(x2) = f2
x0 x2x1
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Lagrange Polynomials32
Sum must be a unique 2nd order polynomial through all the data
ipoints.
What is an efficient implementation?implementation?
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Newton Interpolation 33
Newton Interpolation
Consider our data set of n+1 points yi=f(xi) at x0,x1…xi,…xn: xn > x0
Since pn(x) is the unique polynomial pn(x) of order n, write it: pn q p y pn
[ ]
0 1 0 2 0 1 0 1 1
0 0
1 01 1 0
( ) ( ) ( )( ) ( )( ) ( )( )
( ) ( ),
n n np x b b x x b x x x x b x x x x x xb f x
f x f xb f x x
−= + − + − − + + − − −=
−= =
K L
[ ]
[ ]
1 1 01 0
2 1 1 02 2 1 0
2 0
,
[ , ] [ , ], ,
b f x xx xf x x f x xb f x x x
x x
−−
= =−
M
[ ] [ ] [ ]1 1 01 0
0
, , , ,, , , n n
n n nn
f x x f x xb f x x x
x x−
−
−= =
−
M
K KK
f[xi,xj] is a first divided differencef[x2,x1,x0] is a second divided difference, etc.
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Invariance Theorem34
Note, that the order of the data points does not , pmatter.
All that is required is that the data points are distinct.
Hence, the divided difference f[x0, x1, …, xk] is i i t d ll t ti f th ‘invariant under all permutations of the xi‘s.
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Linear Interpolation35
Simple linear interpolation results from having only p p g y2 data points.
1 01 0 0
( ) ( )( ) ( ) ( )f x f xp x f x x xx x−
= + −−1 0x x
slope
x0 x1
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Quadratic Interpolation36
Three data points:p
( ) ( )f f
2 1 1 0
1 02 0 0 0 1 2 0 1
1 0
( ) ( ) ( ) ( )
( ) ( )
( ) ( )( ) ( ) ( ) [ , , ]( )( )
f x f x f x f x
f f
f x f xp x f x x x f x x x x x x xx x
− −−
−= + − + − −
−
⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦2 1 1 0
2 0
1 0
1 0
1 0
0 0 0 1
0 0
( ) ( )
( ) ( )
( ) ( ) ( )( )
( ) ( )
x x x x
x x
f x f xx x
f x f x
f x x x x x x x
f x x x
− −
−
−
−
−
⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦= + − + − −
= + −1 0
2 1
2 1
( ) ( ) (
x x
f x f x x xx x
−
−−
−+
1 01 0 0 1
1 0
( ) ( )) ( ) ( ) ( )f x f xx x x x x xx x
⎛ ⎞⎛ ⎞ ⎡ ⎤⎡ ⎤ −− − − −⎜ ⎟⎜ ⎟ ⎢ ⎥⎢ ⎥ −⎣ ⎦ ⎣ ⎦⎝ ⎠ ⎝ ⎠+
2 0x x−
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Newton Interpolation37
Let’s look at the recursion formula:
For the quadratic term:[ ] [ ] [ ]1 1 01 0
0
, , , ,, , , n n
n n nn
f x x f x xb f x x x
x xwhere
−−
−= =
−K K
K
[ ] ( )i if x f x=
[ ]1 02 1
2 1 1 0 2 1 1 0
( ) ( )( ) ( )[ , ] [ , ]
f x f xf x f xf x x f x x x x x xb f
−−−
− − −[ ] 2 1 1 0 2 1 1 02 2 1 0
2 0 2 0
2 11
2 1
[ , ] [ , ], ,
( ) ( )
f fb f x x xx x x x
f x f x bx x
= = =− −
−−
−2 1
2 0x x=
−
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Evaluating for x238
( ) ( )( )
( ) ( )
2 0 1 2 0 2 2 0 2 1
2 1
( )f x b b x x b x x x x
f ff b x x b x x
= + − + − −
⎛ ⎞−= + + ⎜ ⎟( ) ( )0 1 2 0 1 2 1
2 1
f b x x b x xx x
= + − + − −⎜ ⎟−⎝ ⎠
( )0 1 1 0 2 1f b x x f f= + − + −( )
( )
0 1 1 0 2 1
1 00 1 0 2 1
1 0
f b x x f f
f ff x x f fx x
+ +
⎛ ⎞−= + − + −⎜ ⎟−⎝ ⎠
2f=
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Example: ln(x) 39
Interpolation of ln(2): given ln(1); ln(4) and ln(6)Data points: {(1,0), (4,1.3863), (6,1.79176)} Linear Interpolation: 0 + {(1.3863-0)/(4-1)}(x-1) = 0.4621(x-1)Quadratic Interpolation: 0.4621(x-1)+((0.20273-0.4621)/5)(x-1)(x-4)Q p 4 ( ) (( 73 4 )/5)( )( 4)
= 0.4621(x-1) - 0.051874 (x-1)(x-4)
Note the divergence for values outside ofthe data rangethe data range.
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Example: ln(x)40
Quadratic interpolation catches some of Quadratic interpolation catches some of the curvature
I th lt h tImproves the result somewhat
Not always a good idea: see later…y g
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Calculating the Divided-Differences
41
A divided-difference table can easily be constructed ff yincrementally.
Consider the function ln(x).
x ln(x) f[I,I+1] f[I,I+1,…,I+7]x ln(x) f[I,I+1] f[I,I+1,…,I+7]1 0.0000002 0.693147 0.6931473 1.098612 0.405465 -0.1438414 1.386294 0.287682 -0.058892 0.0283175 1.609438 0.223144 -0.032269 0.008874 -0.0048616 1 791759 0 182322 -0 020411 0 003953 -0 001230 0 0007266 1.791759 0.182322 -0.020411 0.003953 -0.001230 0.0007267 1.945910 0.154151 -0.014085 0.002109 -0.000461 0.000154 -0.0000958 2.079442 0.133531 -0.010310 0.001259 -0.000212 0.000050 -0.000017 0.000013x ln(x) b10-b9)/(A10-A9(c10-c9)/(a10-a8)d10-d9)/(a10-a7d10-d9)/(a10-a6e10-e9)/(a10-a5f10-f9)/(a10-a4 (g10-g9)/(a10-a3)
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Calculating the Divided-Differences
42
x ln(x) f[I,I+1] f[I,I+1,…,I+7]x ln(x) f[I,I+1] f[I,I+1,…,I+7]1 0.0000002 0.693147 0.6931473 1.098612 0.405465 -0.1438414 1.386294 0.287682 -0.058892 0.0283175 1.609438 0.223144 -0.032269 0.008874 -0.0048616 1 791759 0 182322 -0 020411 0 003953 -0 001230 0 000726
( )1
1
( ) ( )[ . 1] i i
i i
f x f xf i ix x+
+
−+ =
−6 1.791759 0.182322 -0.020411 0.003953 -0.001230 0.0007267 1.945910 0.154151 -0.014085 0.002109 -0.000461 0.000154 -0.0000958 2.079442 0.133531 -0.010310 0.001259 -0.000212 0.000050 -0.000017 0.000013x ln(x) b10-b9)/(A10-A9(c10-c9)/(a10-a8)d10-d9)/(a10-a7d10-d9)/(a10-a6e10-e9)/(a10-a5f10-f9)/(a10-a4 (g10-g9)/(a10-a3)
( )
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Calculating the Divided-Differences
43
x ln(x) f[I,I+1] f[I,I+1,…,I+7]x ln(x) f[I,I+1] f[I,I+1,…,I+7]1 0.0000002 0.693147 0.6931473 1.098612 0.405465 -0.1438414 1.386294 0.287682 -0.058892 0.0283175 1.609438 0.223144 -0.032269 0.008874 -0.0048616 1 791759 0 182322 -0 020411 0 003953 -0 001230 0 000726
[ ] [ ]( )2
1, 2 , 1[ . 1, 2]
i i
f i i f i if i i i
x x+
+ + − ++ + =
−6 1.791759 0.182322 -0.020411 0.003953 -0.001230 0.0007267 1.945910 0.154151 -0.014085 0.002109 -0.000461 0.000154 -0.0000958 2.079442 0.133531 -0.010310 0.001259 -0.000212 0.000050 -0.000017 0.000013x ln(x) b10-b9)/(A10-A9(c10-c9)/(a10-a8)d10-d9)/(a10-a7d10-d9)/(a10-a6e10-e9)/(a10-a5f10-f9)/(a10-a4 (g10-g9)/(a10-a3)
( )2i i+
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Calculating the Divided-Differences
44
[ ] [ ]1 2 3 1 2f i i i f i i i+ + + + +
x ln(x) f[I I+1] f[I I+1 I+7]
[ ] [ ]( )3
1, 2, 3 , 1, 2[ , , 3]
i i
f i i i f i i if i i
x x+
+ + + − + ++ =
−K
x ln(x) f[I,I+1] f[I,I+1,…,I+7]1 0.0000002 0.693147 0.6931473 1.098612 0.405465 -0.1438414 1.386294 0.287682 -0.058892 0.0283175 1.609438 0.223144 -0.032269 0.008874 -0.0048616 1 791759 0 182322 0 020411 0 003953 0 001230 0 0007266 1.791759 0.182322 -0.020411 0.003953 -0.001230 0.0007267 1.945910 0.154151 -0.014085 0.002109 -0.000461 0.000154 -0.0000958 2.079442 0.133531 -0.010310 0.001259 -0.000212 0.000050 -0.000017 0.000013x ln(x) b10-b9)/(A10-A9(c10-c9)/(a10-a8)d10-d9)/(a10-a7d10-d9)/(a10-a6e10-e9)/(a10-a5f10-f9)/(a10-a4 (g10-g9)/(a10-a3)
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Calculating the Divided-Differences
45
[ ] [ ]1 4 3f i i f i i+ + +
x ln(x) f[I,I+1] f[I,I+1,…,I+7]
[ ] [ ]( )4
1, , 4 , , 3[ , , 4]
i i
f i i f i if i i
x x+
+ + − ++ =
−K K
K
x ln(x) f[I,I+1] f[I,I+1,…,I+7]1 0.0000002 0.693147 0.6931473 1.098612 0.405465 -0.1438414 1.386294 0.287682 -0.058892 0.0283175 1.609438 0.223144 -0.032269 0.008874 -0.0048616 1 791759 0 182322 -0 020411 0 003953 -0 001230 0 0007266 1.791759 0.182322 -0.020411 0.003953 -0.001230 0.0007267 1.945910 0.154151 -0.014085 0.002109 -0.000461 0.000154 -0.0000958 2.079442 0.133531 -0.010310 0.001259 -0.000212 0.000050 -0.000017 0.000013x ln(x) b10-b9)/(A10-A9(c10-c9)/(a10-a8)d10-d9)/(a10-a7d10-d9)/(a10-a6e10-e9)/(a10-a5f10-f9)/(a10-a4 (g10-g9)/(a10-a3)
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Calculating the Divided-Differences
46
[ ] [ ]1 5 4f i i f i i+ + +
x ln(x) f[I,I+1] f[I,I+1,…,I+7]
[ ] [ ]( )5
1, , 5 , , 4[ , , 5]
i i
f i i f i if i i
x x+
+ + − ++ =
−K K
K
x ln(x) f[I,I+1] f[I,I+1,…,I+7]1 0.0000002 0.693147 0.6931473 1.098612 0.405465 -0.1438414 1.386294 0.287682 -0.058892 0.0283175 1.609438 0.223144 -0.032269 0.008874 -0.0048616 1 791759 0 182322 -0 020411 0 003953 -0 001230 0 0007266 1.791759 0.182322 -0.020411 0.003953 -0.001230 0.0007267 1.945910 0.154151 -0.014085 0.002109 -0.000461 0.000154 -0.0000958 2.079442 0.133531 -0.010310 0.001259 -0.000212 0.000050 -0.000017 0.000013x ln(x) b10-b9)/(A10-A9(c10-c9)/(a10-a8)d10-d9)/(a10-a7d10-d9)/(a10-a6e10-e9)/(a10-a5f10-f9)/(a10-a4 (g10-g9)/(a10-a3)
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Calculating the Divided-Differences
47
[ ] [ ]1 6 5f i i f i i+ + +
x ln(x) f[I,I+1] f[I,I+1,…,I+7]
[ ] [ ]( )6
1, , 6 , , 5[ , , 6]
i i
f i i f i if i i
x x+
+ + − ++ =
−K K
K
x ln(x) f[I,I+1] f[I,I+1,…,I+7]1 0.0000002 0.693147 0.6931473 1.098612 0.405465 -0.1438414 1.386294 0.287682 -0.058892 0.0283175 1.609438 0.223144 -0.032269 0.008874 -0.0048616 1 791759 0 182322 -0 020411 0 003953 -0 001230 0 0007266 1.791759 0.182322 -0.020411 0.003953 -0.001230 0.0007267 1.945910 0.154151 -0.014085 0.002109 -0.000461 0.000154 -0.0000958 2.079442 0.133531 -0.010310 0.001259 -0.000212 0.000050 -0.000017 0.000013x ln(x) b10-b9)/(A10-A9(c10-c9)/(a10-a8)d10-d9)/(a10-a7d10-d9)/(a10-a6e10-e9)/(a10-a5f10-f9)/(a10-a4 (g10-g9)/(a10-a3)
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Calculating the Divided-Differences
48
Finally, we can calculate the last coefficient.y,
[ ] [ ]1 7 6f i i f i i+ + +
x ln(x) f[I,I+1] f[I,I+1,…,I+7]
[ ] [ ]( )7
1, , 7 , , 6[ , , 7]
i i
f i i f i if i i
x x+
+ + − ++ =
−K K
K
x ln(x) f[I,I+1] f[I,I+1,…,I+7]1 0.0000002 0.693147 0.6931473 1.098612 0.405465 -0.1438414 1.386294 0.287682 -0.058892 0.0283175 1.609438 0.223144 -0.032269 0.008874 -0.0048616 1 791759 0 182322 -0 020411 0 003953 -0 001230 0 0007266 1.791759 0.182322 -0.020411 0.003953 -0.001230 0.0007267 1.945910 0.154151 -0.014085 0.002109 -0.000461 0.000154 -0.0000958 2.079442 0.133531 -0.010310 0.001259 -0.000212 0.000050 -0.000017 0.000011x ln(x) b10-b9)/(A10-A9(c10-c9)/(a10-a8)d10-d9)/(a10-a7d10-d9)/(a10-a6e10-e9)/(a10-a5f10-f9)/(a10-a4 (g10-g9)/(a10-a3)
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Calculating the Divided-Differences
49
Divided Differences
All of the coefficients All of the coefficients for the resulting polynomial are in bold.
x ln(x) f[I,I+1] f[I,I+1,…,I+7]
b0 b4
bx ln(x) f[I,I+1] f[I,I+1,…,I+7]1 0.0000002 0.693147 0.6931473 1.098612 0.405465 -0.1438414 1.386294 0.287682 -0.058892 0.0283175 1.609438 0.223144 -0.032269 0.008874 -0.0048616 1 791759 0 182322 -0 020411 0 003953 -0 001230 0 000726
b7
6 1.791759 0.182322 -0.020411 0.003953 -0.001230 0.0007267 1.945910 0.154151 -0.014085 0.002109 -0.000461 0.000154 -0.0000958 2.079442 0.133531 -0.010310 0.001259 -0.000212 0.000050 -0.000017 0.000011x ln(x) b10-b9)/(A10-A9(c10-c9)/(a10-a8)d10-d9)/(a10-a7d10-d9)/(a10-a6e10-e9)/(a10-a5f10-f9)/(a10-a4 (g10-g9)/(a10-a3)
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Polynomial Form for Divided-Differences
50
The resulting polynomial comes from the divided-g p ydifferences and the corresponding product terms:
( )7 0p x =
( )( )( )( )( ) ( )
0.693 1
0.144 1 2
0.28 1 2 3
x
x x
x x x
+ −
− − −
+ − − −
( )( )( )( )( )( )( )( ) ( )( )( )( )( ) ( )( )
4
5
0.0049 1 2 3 4
7.26 10 1 2 3 4 5
9.5 10 1 2 3 4 5 6
x x x x
x x x x x
x x x x x x
−
−
− − − − −
+ • − − − − −
− • − − − − − −
( )( )( )( )( )( ) ( )51.1 10 1 2 3 4 5 6 7x x x x x x x−+ • − − − − − − −
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Many polynomials51
Note, that the order of the numbers (xi,yi)’s only , ( i,yi) ymatters when writing the polynomial down.
The first column represents the set of linear splines between two adjacent data pointstwo adjacent data points.
The second column gives us quadratics thru three adjacent points.
Etc.
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Adding an Additional Data Point52
Adding an additional data point, simply adds an g p , p yadditional term to the existing polynomial.
Hence, only n additional divided-differences need to be l l d f h st d icalculated for the n+1st data point.
x ln(x) f[I,I+1] f[I,I+1,…,I+7]x ln(x) f[I,I+1] f[I,I+1,…,I+7]1.0000000 0.00000002.0000000 0.6931472 0.69314723.0000000 1.0986123 0.4054651 -0.14384104.0000000 1.3862944 0.2876821 -0.0588915 0.02831655.0000000 1.6094379 0.2231436 -0.0322693 0.0088741 -0.00486066 0000000 1 7917595 0 1823216 -0 0204110 0 0039528 -0 0012303 0 0007261
b8
6.0000000 1.7917595 0.1823216 -0.0204110 0.0039528 -0.0012303 0.00072617.0000000 1.9459101 0.1541507 -0.0140854 0.0021085 -0.0004611 0.0001539 -0.00009548.0000000 2.0794415 0.1335314 -0.0103096 0.0012586 -0.0002125 0.0000497 -0.0000174 0.00001111.5000000 0.4054651 0.2575348 -0.0225461 0.0027192 -0.0004173 0.0000819 -0.0000215 0.0000082 -0.0000058
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Adding More Data Points53
Quadratic interpolation: does linear interpolationThen add higher-order correction to catch the curvature
C bi Cubic, …Consider the case where the data points are organized such the the first two are the endpoints organized such the the first two are the endpoints, the next point is the mid-point, followed by successive mid-points of the half-intervals.
Worksheet: f(x)=x2 from -1 to 3.
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Uniqueness54
Suppose that two polynomials of degree n (or less) pp p y g ( )existed that interpolated to the n+1 data points.
Subtracting these two polynomials from each other also leads to a polynomial of at most n degree.
( ) ( ) ( )n n nr x p x q x= −
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Uniqueness55
Since p and q both interpolate the n+1 data points,p q p p ,
This polynomial r, has at least n+1 roots!!!
This can not be! A polynomial of degree-n can only p y g yhave at most n roots.
( ) ( )n
∏Therefore, r(x) ≡ 01
1
( ) ( )
( ) ( )
n n ii
n
p x a x r
p x a x r
=
+
= −
= −
∏
∏1 11
( ) ( )n n ii
p x a x r+ +=
= ∏
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Example56
Suppose f was a polynomial of degree m, where m<n.pp f p y g ,
Ex: f(x) = 3x-2
We have evaluations of f(x) at five locations: (-2,-8), f( ) ( , ),(-1,-5), (0,-2), (1,1), (2,4)
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Error57
Define the error term as:
If f(x) is an nth order polynomial p (x) is of course exact
( ) ( ) ( )n nx f x p xε = −
If f(x) is an nth order polynomial pn(x) is of course exact.Otherwise, since there is a perfect match at x0, x1,…,xn
This function has at least n+1 roots at the interpolation points.
0 1( ) ( )( ) ( ) ( )n nx x x x x x x h xε∴ = − − −L
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Interpolation Errors58
( ) ( )( 1)1( ) ( ) ( )n
nn n ix f x p x f x xε ξ+= − = −∏( ) ( )
( )0
( ) ( ) ( )( 1)!
[ , ], ,
n n ii
f p fn
x a b a b
ξ
ξ=+
∈ ∈
∏
Proof is in the book.
Intuitively, the first n+1 terms of the Taylor Series is
( )[ , ], ,ξ
y, yalso an nth degree polynomial.
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Interpolation Errors59
Use the point x, to expand the polynomial.
0 1{ , , }nn
x x x x∉ K
( )0 10
( ) ( ) ( ) [ , , , ]n n n ii
x f x p x f x x x x x xε=
= − = −∏K
Point is, we can take an arbitrary point x, and create an (n+1)th polynomial that goes thru the point xan (n+1) polynomial that goes thru the point x.
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Interpolation Errors60
Combining the last two statements, we can also get a feel for what these divided differences represent.
( )( )0 1
1[ , , ]!
nnf x x x f
nξ=K
Corollary 1 in book – If f(x) is a polynomial of degree m<n then all (m+1)st divided differences and higher m<n, then all (m+1) divided differences and higher are zero.
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Problems with Interpolation61
Is it always a good idea to use higher and higher order polynomials?
Certainly not: 3 4 points usually good: 5 6 ok:Certainly not: 3-4 points usually good: 5-6 ok:
See tendency of polynomial to “wiggle”
Particularly for sharp edges: see figures
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Chebyshev nodes62
Equally distributed points may not be the optimal q y p y psolution.
If you could select the xi’s, what would they be?
Want to minimize the term.
These are the Chebyshev nodes. For x=-1 to 1:
( )0
n
ii
x x=
−∏
cos (0 )ix i nπ⎡ ⎤⎛ ⎞= ≤ ≤⎜ ⎟⎢ ⎥cos , (0 )ix i nn
π= ≤ ≤⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
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Chebyshev nodes63
Let’s look at these for n=4.4
Spreads the points out inthe center.
00cos 14
x π⎡ ⎤⎛ ⎞= =⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎡ ⎤1
1 2cos 0.7074 2
2
x π⎡ ⎤⎛ ⎞= = ≈⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞2cos 0
4
3 2cos 0 707
x
x
π
π
⎡ ⎤⎛ ⎞= =⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
⎡ ⎤⎛ ⎞⎜ ⎟⎢ ⎥3
4
cos 0.7074 2
4cos 14
x
x
π
π
= = − ≈ −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞= = −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
4 4⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
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Polynomial Interpolation in Two-Dimensions
64
Consider the case in higher-dimensions.g
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Finding the Inverse of a Function65
What if I am after the inverse of the function f(x)?f( )For example arccos(x).
Simply reverse the role of the xi and the fi.
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Section B
Solving Non-Linear Equations
(Root Finding)( g)
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Root finding Methods
What are root finding methods?g
Methods for determining a solution of an equation.
Essentially finding a root of a function, that is, a zero y g , ,of the function.
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Root finding Methods
Where are they used?y
Some applications for root finding are: systems of equilibrium, elliptical orbits, the van der Waals equation, and natural frequencies of spring systems.
The problem of solving non-linear equations or sets f li ti i bl i of non-linear equations is a common problem in
science, applied mathematics.
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Cont...
The problem of solving non-linear equations or sets of non-linear equations is a common
bl i i li d h iproblem in science, applied mathematics.
The goal is to solve f(x) = 0 , for the function f( )f(x).
The values of x which make f(x) = 0 are the roots of the equation roots of the equation.
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Cont...
f(x)
a,b are roots of the function f(x)
f(a)=0 f(b)=0
x
( ) ( )
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There are many methods for solving non-linear equations. The methods, which will be highlighted have the following properties:the f nction f( ) is e pected to be contin o s If 1. the function f(x) is expected to be continuous. If not the method may fail.
2 the use of the algorithm requires that the 2. the use of the algorithm requires that the solution be bounded.
3. once the solution is bounded, it is refined to specified tolerance.
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Cont...
Four such methods are:
Interval Halving (Bisection method)
Regula Falsi (False position)g ( p )
Secant method
Fixed point iterationp
Newton’s method
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Bisection Method
It is the simplest root-finding algorithm.
Requires previous knowledge of two initial
d b h initial pt ‘b’
d guesses, a and b, so that f(a) and f(b) have opposite signs root ‘d’
a c b
d
opposite signs.initial pt ‘a’
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Bisection Method
Two estimates are chosen so that the solution is bracketed. ie f(a) and f(b) have opposite signs
initial pt ‘b’
d opposite signs.
In the diagram this f(a) < 0 and f(b) > 0 root ‘d’
a c b
d
f(a) < 0 and f(b) > 0.
The root d lies between a and b!
initial pt ‘a’
between a and b!
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Bisection Method
The root d must always be bracketed (be between a and b)!
i h dinitial pt ‘b’
d Iterative method.
root ‘d’
a c b
d
initial pt ‘a’
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Bisection Method
The interval between a and b is halved by calculating the average of a and b.
h i initial pt ‘b’
d The new point c = (a+b)/2.
Thi d t root ‘d’
a c b
d
This produces are two possible intervals: a < x < c and c < x < b
initial pt ‘a’
< c and c < x < b.
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Bisection Method
This produces are two possible intervals: a < x < c and c < x < b.
If f(c ) > 0, then x = d b h l f f
initial pt ‘b’
d must be to the left of c :interval a < x < c.
If f( ) th droot ‘d’
a c b
d
If f(c ) < 0, then x = dmust be to the right of c :interval c < x < b
initial pt ‘a’
:interval c < x < b.
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Bisection Method
If f(c ) > 0, let anew = a and bnew = c and repeat process.
If f(c ) < 0, let anew = c d b b d
initial pt ‘b’
d and bnew = b and repeat process. This reassignment
root ‘d’
a c b
d
This reassignment ensures the root is always bracketed!!
initial pt ‘a’
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Bisection Method
This produces are two possible intervals: a < x < c and c < x < b.
If f(c ) > 0, then x = d b h l f f
initial pt ‘b’
d must be to the left of c :interval a < x < c.
If f( ) th droot ‘d’
a c b
d
If f(c ) < 0, then x = dmust be to the right of c :interval c < x < b
initial pt ‘a’
:interval c < x < b.
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Bisection Method
ci = (ai + bi )/2i ( i i )/
if f(ci) > 0; ai+1 = a and bi+1 = c
if f(ci) < 0; ai+1 = c and bi+1 = b( i) ; i+1 i+1
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Bisection Method
Bisection is an iterative process, where the initial interval is halved until the size of the interval decreases until it is below decreases until it is below some predefined tolerance ε: |a-b| ≥ ε or f(x) falls
δ
ε: |a b| ≥ ε or f(x) falls below a tolerance δ: |f(c ) – f(c-1)| ≤ δ.
ε
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Bisection Method
Advantagesg
1. Is guaranteed to work if f(x) is continuous and the root is bracketed.
2. The number of iterations to get the root to specified tolerance is known in advance
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Bisection Method
Disadvantagesg
1. Slow convergence.
2. Not applicable when here are multiple roots. Will pp ponly find one root at a time.
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Secant Method
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Overview of Secant MethodOverview of Secant Method
Again to initial guesses are chosen.However there is not However there is not requirement that the root is bracketed!The method proceeds The method proceeds by drawing a line through the points to get a new point closer get a new point closer to the root.This is repeated until h f dthe root is found.
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Secant MethodSecant Method
First we find two points(x0,x1), which are hopefully near the root (we may use the bisection method) bisection method).
A line is then drawn through the two points through the two points and we find where the line intercepts the x-paxis, x2.
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Secant MethodSecant Method
If f(x) were truly linear, the straight line would intercept the x-axis at the root.
i i i However since it is not linear, the intercept is not at the root but it not at the root but it should be close to it.
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Secant MethodSecant Method
From similar triangles From similar triangles we can write that,
( )( )
( )( ) ( )10
10
1
21
xfxfxx
xfxx
−−
=−
( )xf( )0xf
1x 0x2x
( )1xf
1x 0x2x
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Secant Method
From similar triangles From similar triangles we can write that,
Solving for x2 we get:
( )( )
( )( ) ( )10
10
1
21
xfxfxx
xfxx
−−
=−
Solving for x2 we get:
( ) ( )10 xxf −( ) ( )( ) ( )10
10112 xfxf
xfxx−
−=
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Secant Method
Iteratively this is Iteratively this is written as:
( )( ) ( )( ) ( )nn
nnnnn xfxf
xxxfxx−−
−=−
−+
1
11
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Algorithm
Given two guesses x0, x1 near the root,
If then( ) ( )10 xfxf <
Swap x0 and x1.
Repeat
( ) ( )10 ff
p
Set
Set x0 = x1
( ) ( ) ( )10
10112 *
xfxfxxxfxx
−−
−=
Set x0 x1
Set x1 = x2
Until < tolerance value ( )xfUntil < tolerance value. ( )2xf
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Cont...
Because the new point should be closer the root after pthe 2nd iteration we usually choose the last two points.
After the first iteration there is only one new point. However x1 is chosen so that it is closer to root that xx0.
This is not a “hard and fast rule”!
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False Position
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False Position
The method of false position is seen as an pimprovement on the secant method.
The method of false position avoids the problems of the secant method by ensuring that the root is bracketed between the two starting points and remains bracketing between successive pairs remains bracketing between successive pairs.
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False Position
This technique is similar to the bisection method qexcept that the next iterate is taken as the line of interception between the pair of x-values and the x-
i th th t th id i t axis rather than at the midpoint.
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False Position
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Algorithm
Given two guesses x0, x1 that bracket the root,
Repeat
Set
If is of opposite sign to then
( ) ( ) ( )10
10112 *
xfxfxxxfxx
−−
−=
( )2xf ( )0xfpp g
Set x1 = x2
Else Set x0 = x10 1
End If
Until < tolerance value. ( )2xfUntil < tolerance value. ( )2f
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Discussion of False Position Method
This method achieves better convergence but a more gcomplicated algorithm.
May fail if the function is not continuous.
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Newton’s Method
The bisection method is useful up to a point. p p
In order to get a good accuracy a large number of iterations must be carried out.
A second inadequacy occurs when there are multiple roots to be found.
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Newton’s Method
The bisection method is useful up to a point. p p
In order to get a good accuracy a large number of iterations must be carried out.
A second inadequacy occurs when there are multiple roots to be found.
Newton’s method is a much better algorithm.
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Newton’s Method
Newton’s method relies on calculus and uses linear approximation to the function by finding the tangent to the curve.
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Newton’s Method
Algorithm requires an initial guess, x0, which is close to the root. close to the root. The point where the tangent line to the function (f (x)) meets function (f (x)) meets the x-axis is the next approximation, x1. Thi d i This procedure is repeated until the value of ‘x’ is sufficiently close t to zero.
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Newton’s Method
The equation for Newton’s Method can be determined graphically!
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Newton’s Method
The equation for Newton’s Method can be determined graphically!
h di From the diagram tan Ө = ƒ'(x0) = ƒ(x0)/(x0 –x ) x1)
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Newton’s Method
The equation for Newton’s Method can be determined be determined graphically!From the diagram tan From the diagram tan Ө = ƒ'(x0) = ƒ(x0)/(x0 –x1)
Thus, x1=x0 -ƒ(x )/ƒ'(x ) ƒ(x0)/ƒ (x0).
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Newton’s Method
The general form of Newton’s Method is:
xn+1 = xn – f(xn)/ƒ'(xn)
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Newton’s Method
The general form of Newton’s Method is:
xn+1 = xn – f(xn)/ƒ'(xn)
AlgorithmPick a starting value for x
Repeat
x:= x – f(x)/ƒ'(x)
Return x
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Numerical Differentiation and Integration
Standing in the heart of calculus are the mathematical concepts f d ff dof differentiation and integration:
−Δ+=
Δ ii xfxxfy )()(
Δ−Δ+
=
ΔΔ
Δii
x xxfxxf
dxdy
xx)()(lim0
∫=Δ
b
dxxfI
xdx
)(a
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Figure 4 1Figure 4.1
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Figure 4.2
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Noncomputer Methods for Differentiation and Integration
The function to be differentiated or integrated will typically be in one of the following three forms:
A simple continuous function such as polynomial, an exponential, or a trigonometric function.gA complicated continuous function that is difficult or impossible to differentiate or integrate directly.A tabulated function where values of x and f(x) are given at aA tabulated function where values of x and f(x) are given at a number of discrete points, as is often the case with experimental or field data.
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Figure 4.3
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Figure 4.4
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Newton-Cotes Integration Formulas
The Newton-Cotes formulas are the most commonThe Newton Cotes formulas are the most common numerical integration schemes.
They are based on the strategy of replacing a complicated function or tabulated data with an papproximating function that is easy to integrate:
bb
∫∫nn
an
a
xaxaxaaxf
dxxfdxxfI
++++=
≅=
−
∫∫1)(
)()(
L nnn xaxaxaaxf ++++= −110)( L
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Figure 4.5
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Figure 4.6
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The Trapezoidal RuleThe Trapezoidal Rule
The Trapezoidal rule is the first of the Newton-Cotes closed integration formulas, corresponding to the case where the polynomial is first order:
∫∫ ≅=bb
dxxfdxxfI )()( 1
The area under this first order polynomial is an estimate of the integral of f(x) between the limits of a
aa
estimate of the integral of f(x) between the limits of aand b:
)()()( bfafbI +2
)()()( ffabI −= Trapezoidal rule
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Figure 4.7
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Error of the Trapezoidal Rule/Error of the Trapezoidal Rule/
When we employ the integral under a straight lineWhen we employ the integral under a straight line segment to approximate the integral under a curve, error may be substantial:error may be substantial:
3))((1 abfEt −′′−= ξ
where ξ lies somewhere in the interval from a to b
))((12
abfEt ξ
where ξ lies somewhere in the interval from a to b.
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The Multiple Application Trapezoidal RuleThe Multiple Application Trapezoidal Rule
One way to improve the accuracy of the trapezoidal rule is to y p y pdivide the integration interval from a to b into a number of segments and apply the method to each segment.The areas of indi id al segments can then be added to ield theThe areas of individual segments can then be added to yield the integral for the entire interval.
==−
= xbxaabh
∫∫∫ +++=
===
nxxx
n
dxxfdxxfdxxfI
xbxan
h
21
)()()(
0
L
Substituting the trapezoidal rule for each integral yields:
−nxxx 110
)()()()()()( ffffff2
)()(2
)()(2
)()( 12110 nn xfxfhxfxfhxfxfhI +++
++
+= −L
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Figure 4.8
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Simpson’s Rules
More accurate estimate of an integral is obtained if a high-order polynomial is used to connect the points. The formulas that result from taking the integrals under such polynomials are called Simpson’s rulespolynomials are called Simpson s rules.
Simpson’s 1/3 Rule/pResults when a second-order interpolating polynomial is used.
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Figure 4.9
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b b
∫ ∫ )()( 2
b
dxxfdxxfIa a
≅= ∫ ∫
)())((
))(()())((
))(()())((
))((2
1202
101
2101
200
2010
21
20
2
0
dxxfxxxx
xxxxxfxxxx
xxxxxfxxxx
xxxxI
xbxax
x⎥⎦
⎤⎢⎣
⎡−−−−
+−−−−
+−−−−
=
==
∫
[ ]2
)()(4)(3 210
abhxfxfxfhI −=++≅
3
Simpson’s 1/3 Rule
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S ti CSection C
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Solution of linear system of equations126
Circuit analysis (Mesh and node equations)y ( q )
Numerical solution of differential equations (Finite Difference Method)
Numerical solution of integral equations (Finite Element Method, Method of Moments)
nn
bbxaxaxa =+++ L 11212111
⎥⎤
⎢⎡
⎥⎤
⎢⎡⎥⎤
⎢⎡ n
bbxaaa L 1111211
nn
b
bxaxaxa
+++
=+++M
L 22222121
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
n
b
b
x
x
aaa
aaaMMMOMM
L 2222221⇒
nnnnnn bxaxaxa =+++ L2211 ⎦⎣⎦⎣⎦⎣ nnnnnn bxaaa L21
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Consistency (Solvability)127
The linear system of equations Ax=b has a solution, y q ,or said to be consistent IFF
Rank{A}=Rank{A|b}
A system is inconsistent when
Rank{A}<Rank{A|b}
Rank{A} is the maximum number of linearly independent columns or rows of A. Rank can be found by using ERO (Elementary Row Oparations) or ECO (Elementary column operations).
ERO # f ith t l t tERO⇒# of rows with at least one nonzero entryECO⇒# of columns with at least one nonzero entry
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Solution Techniques128
Direct solution methodsDirect solution methodsFinds a solution in a finite number of operations by transforming the system into an equivalent system that is ‘ i ’ t l ‘easier’ to solve. Diagonal, upper or lower triangular systems are easier to solve
b f i i f i f iNumber of operations is a function of system size n.
Iterative solution methodsComputes succesive approximations of the solution vector Computes succesive approximations of the solution vector for a given A and b, starting from an initial point x0. Total number of operations is uncertain, may not converge.
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Direct solution Methods129
Gaussian EliminationBy using ERO, matrix A is transformed into an upper triangular matrix (all elements below diagonal 0)
Back substitution is used to solve the upper triangular systemBack substitution is used to solve the upper-triangular system
n
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡ ni
b
bxaaaMMMMOM
LL 111111
⇒ ⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡ ni
b
bxaaa
~~~0
111111
MMMMOM
LL
stit
uti
on
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ n
i
n
i
nnnin
iniii
b
b
x
x
aaa
aaaMM
LL
MOMM
LL
1
1⇒
ERO⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣ n
i
n
i
nn
inii
b
b
x
x
a
aa
~~00
0MM
LL
MOMM
LL
ack
subs
Ba
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First step of elimination130
⎤⎡⎤⎡⎤⎡ )1()1()1()1()1( bxaaaaPi t l l t
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
=⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
)1(3
)1(2
)(1
3
2
1
)1(3
)1(33
)1(32
)1(31
)1(2
)1(23
)1(22
)1(21
)(1
)(13
)(12
)(11
n
n
n
bbb
xxx
aaaaaaaaaaaa
L
L
LPivotal element
⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣)1()1()1(
3)1(
2)1(
1 nnnnnnn bxaaaaMM
L
MOMMM
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
= )2(2
)1(1
2
1)2(
2)2(
23)2(
22
)1(1
)1(13
)1(12
)1(11
)()(
)1(11
)1(211,2 0/ n
n
bb
xx
aaaaaaa
aam L
L
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣=
=
)2(
)2(33
)2()2()2(
)2(3
)2(33
)2(32
)1()1(
)1(11
)1(311,3
0
0
/
/ n
b
b
x
x
aaa
aaa
aam
aamMM
L
MOMMM
L
M
⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣= 321111, 0/ nnnnnnnn bxaaaaam L
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Second step of elimination131
⎤⎡⎤⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
)2(
)2(2
)1(1
2
1
)2()2()2(
)2(2
)2(23
)2(22
)1(1
)1(13
)1(12
)1(11
00 n
n
bbb
xxx
aaaaaaaaaa
L
L
Pivotal element
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣⎥⎥⎥⎥
⎦⎢⎢⎢⎢
⎣)2(
)(33
)2()2(3
)2(2
)(3
)(33
)(32
0
0
nnnnnn
n
b
b
x
x
aaa
aaaMM
L
MOMMM
L
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
)2(2
)1(1
2
1)2(
2)2(
23)2(
22
)1(1
)1(13
)1(12
)1(11
0n
bb
xx
aaaaaaa
L
L
⎦⎣⎦⎣⎦⎣ 32 nnnnnn
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
=
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
= )3(3
2
3
2)3(
3)3(
33
22322)2(
22)2(
322,3 000
/ n
n
bb
xx
aaaaa
aamMMMOMMM
L
M
⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣= )3()3()3(3
)2(22
)2(22, 00/ nnnnnnn bxaaaam L
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Gaussion elimination algorithm132
/ )()(, = p
ppp
rppr aam
0)( =prpa
)()()1( ppp bmbb ×−=+
For c=p+1 to n
)()()1( ppp+
, pprrr bmbb ×=
)(,
)()1( ppcpr
prc
prc amaa ×−=+
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Back substitution algorithm133
⎤⎡⎤⎡⎤⎡ )1()1()1()1()1( bxaaaa
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎡
)3(3
)2(2
)(1
3
2
1
)3(3
)3(33
)2(2
)2(23
)2(22
)(1
)(13
)(12
)(11
000
n
n
n
bbb
xxx
aaaaaaaaa
L
L
L
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣
=
⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣⎥⎥⎥⎥⎥
⎦⎢⎢⎢⎢⎢
⎣
−−−−−−
)(
)1(1
3
1)(
)(1
)(11
333
0000000
n
nnn
n
nnn
nnn
n
bbxaa
MMMOMMM
⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣)()(0000 n
nnn
nn bxa
[ ]1 1)1()(
−== −− xabxbx nnn
n [ ]
1211 )()(
11)1(11
1)(
−−=⎥⎤
⎢⎡
−=
==
∑
−−−−−
−
nnixabx
xaba
xa
x
nii
nnnnnnn
nnnn
n
1,,2,11
)( K=⎥⎦
⎢⎣
= ∑+=
nnixaba
xik
kikiiii
i
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Operation count134
Number of arithmetic operations required by the algorithm to complete its task.Generally only multiplications and divisions are counted counted Elimination process
65
23
23 nnn−+
Back substitution623
2
2 nn + Dominatesff f
Total 22
3 nnn−+
Not efficient fordifferent RHS vectors
33n+
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LU Decomposition135
A=LU
Ax=b ⇒LUx=b
Define Ux=yy
Ly=b Solve y by forward substitution
ERO’s must be performed on b as well as AERO s must be performed on b as well as A
The information about the ERO’s are stored in L
Indeed y is obtained by applying ERO’s to b vectorIndeed y is obtained by applying ERO s to b vector
Ux=y Solve x by backward substitution
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LU Decomposition by Gaussian elimination136
There are infinitely many different ways to decompose AThere are infinitely many different ways to decompose A.Most popular one: U=Gaussian eliminated matrix
L=Multipliers used for elimination
⎥⎥⎥⎤
⎢⎢⎢⎡
⎥⎥⎥⎤
⎢⎢⎢⎡
)3()3(
)2(2
)2(23
)2(22
)1(1
)1(13
)1(12
)1(11
1,2 0001000100001
n
n
aaaaaaa
m L
L
L
L
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
=
)(1
)(11
)3(3
)3(33
312111
2,31,3
000
00
10001
nn
n
aa
aa
mmm
mmA
MOMMM
L
ML
MOOMM
L
⎥⎥⎦⎢
⎢⎣⎥⎥⎦⎢
⎢⎣
−−−−−−
)(111
4,3,2,1,
3,12,11,1
0000000
11
nnn
nnnn
nnnn
nnn
aaa
mmmmmmm
L
M
Compact storage: The diagonal entries of L matrix are all 1’s, Compact storage: The diagonal entries of L matrix are all 1 s, they don’t need to be stored. LU is stored in a single matrix.
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Operation count137
A LU Decomposition 3 nn Done only once
A=LU Decomposition
Ly=b forward substitution33nn
−2 nn −
y
y
Ux=y backward substitution 2 nn +2
nn
Total 2
332
3 nnn−+
For different RHS vectors, the system can be efficiently solved.
33
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Pivoting138
Computer uses finite-precision arithmeticp f p
A small error is introduced in each arithmetic operation, error propagates
When the pivotal element is very small, the multipliers will be large.
Adding numbers of widely differening magnitude can lead to loss of significance.
T d i h d To reduce error, row interchanges are made to maximise the magnitude of the pivotal element
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Example: Without Pivoting139
⎤⎡⎤⎡⎤⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡− 93.22
414.6210.114.24
281.5133.1
2
1
xx
4-digit arithmetic⎦⎣⎦⎣⎦⎣
⎥⎤
⎢⎡
⎥⎤
⎢⎡⎥⎤
⎢⎡ 414.6281.5133.1 1x
312114.24⎥⎦
⎢⎣−
=⎥⎦
⎢⎣⎥⎦
⎢⎣ − 8.1137.113000.0 2
1
x31.21
133.1.
21 ==m
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡001.19956.0
2
1
xx
Loss of significance⎦⎣⎦⎣ 001.12x
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Example: With Pivoting140
⎤⎡⎤⎡⎤⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡ −414.6
93.22281.5133.1210.114.24
2
1
xx
⎦⎣⎦⎣⎦⎣
⎥⎤
⎢⎡
⎥⎤
⎢⎡⎥⎤
⎢⎡ − 93.22210.114.24 1x
046930133.1⎥⎦
⎢⎣
=⎥⎦
⎢⎣⎥⎦
⎢⎣ 338.5338.5000.0 2
1
x04693.0
14.2433.
21 ==m
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡000.1000.1
2
1
xx
⎦⎣⎦⎣ 000.12x
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Pivoting procedures141
⎤⎡ )1()1()1()1()1()1(
⎥⎥⎥⎤
⎢⎢⎢⎡
)3()3()3()3(
)2(2
)2(2
)2(2
)2(23
)2(22
)1(1
)1(1
)1(1
)1(13
)1(12
)1(11
000 nji
nji
aaaaaaaaaaa
LLL
LLL
⎥⎥⎥⎥
⎢⎢⎢⎢
)()()(
)3(3
)3(3
)3(3
)3(33
000
00
iii
nji
aaa
aaaaMOMOMOMMM
LLLEliminated
part Pivotal
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
)()()(
)()()(
000
000
iii
inijii
aaa
aaaMOMOMOMMM
LLL Pivotal row
⎥⎥⎥
⎦⎢⎢⎢
⎣)()()(
)()()(
000
000
iii
jnjjji
aaa
aaa
LLL
MOMOMOMMM
LLL
⎦⎣ 000 nnnjni aaa LLL
Pivotal column
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Row pivoting142
Most commonly used partial pivoting procedureMost commonly used partial pivoting procedure
Search the pivotal column
Find the largest element in magnitude
Th i h hi i h h i l Then switch this row with the pivotal row
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Row pivoting143
⎤⎡ )1()1()1()1()1()1(
⎥⎥⎥⎤
⎢⎢⎢⎡
)3()3()3()3(
)2(2
)2(2
)2(2
)2(23
)2(22
)1(1
)1(1
)1(1
)1(13
)1(12
)1(11
000 nji
nji
aaaaaaaaaaa
LLL
LLL
Interchange
⎥⎥⎥⎥
⎢⎢⎢⎢
)()()(
)3(3
)3(3
)3(3
)3(33
000
00
iii
nji
aaa
aaaaMOMOMOMMM
LLL gthese rows
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
)()()(
)()()(
000
000
iii
inijii
aaa
aaaMOMOMOMMM
LLL
⎥⎥⎥
⎦⎢⎢⎢
⎣)()()(
)()()(
000
000
iii
jnjjji
aaa
aaa
LLL
MOMOMOMMM
LLL
⎦⎣ 000 nnnjni aaa LLL
Largest in magnitude
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Column pivoting144
⎤⎡ )1()1()1()1()1()1(
⎥⎥⎥⎤
⎢⎢⎢⎡
)3()3()3()3(
)2(2
)2(2
)2(2
)2(23
)2(22
)1(1
)1(1
)1(1
)1(13
)1(12
)1(11
000 nji
nji
aaaaaaaaaaa
LLL
LLL
⎥⎥⎥⎥
⎢⎢⎢⎢
)()()(
)3(3
)3(3
)3(3
)3(33
000
00
iii
nji
aaa
aaaaMOMOMOMMM
LLL
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
)()()(
)()()(
000
000
iii
inijii
aaa
aaaMOMOMOMMM
LLL
Largest in magnitude
⎥⎥⎥
⎦⎢⎢⎢
⎣)()()(
)()()(
000
000
iii
jnjjji
aaa
aaa
LLL
MOMOMOMMM
LLL
⎦⎣ 000 nnnjni aaa LLL
Interchange these columns
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Complete pivoting145
⎤⎡ )1()1()1()1()1()1(
⎥⎥⎥⎤
⎢⎢⎢⎡
)3()3()3()3(
)2(2
)2(2
)2(2
)2(23
)2(22
)1(1
)1(1
)1(1
)1(13
)1(12
)1(11
000 nji
nji
aaaaaaaaaaa
LLL
LLL
Interchange
⎥⎥⎥⎥
⎢⎢⎢⎢
)()()(
)3(3
)3(3
)3(3
)3(33
000
00
iii
nji
aaa
aaaaMOMOMOMMM
LLL gthese rows
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
)()()(
)()()(
000
000
iii
inijii
aaa
aaaMOMOMOMMM
LLL
⎥⎥⎥
⎦⎢⎢⎢
⎣)()()(
)()()(
000
000
iii
jnjjji
aaa
aaa
LLL
MOMOMOMMM
LLL
Largest in magnitude
⎦⎣ 000 nnnjni aaa LLL g
Interchange these columns
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Row Pivoting in LU Decomposition 146
When two rows of A are When two rows of A are interchanged, those rows of b should also be
⎥⎥⎥⎤
⎢⎢⎢⎡
321
⎥⎥⎥⎤
⎢⎢⎢⎡21
interchanged.
Use a pivot vector. Initial pivot vector is integers ⎥
⎥⎥⎥⎥
⎢⎢⎢⎢⎢
= jpM
3
⎥⎥⎥⎥
⎢⎢⎢⎢
= ipM
3
pivot vector is integers from 1 to n.
When two rows (i and j) ⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
i
jp
M
M
⎥⎥⎥⎥⎥
⎢⎢⎢⎢⎢
j
ipM
of A are interchanged, apply that to pivot vector. ⎥
⎥
⎦⎢⎢
⎣nM
⎥⎥⎥
⎦⎢⎢⎢
⎣nM
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Example147
⎤⎡⎤⎡⎤⎡ 112230
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−=
321
35
12
241124230
pbA
⎦⎣⎦⎣⎦⎣
Column search: Maximum magnitude second rowInterchange 1st and 2nd rows
⎥⎤
⎢⎡
⎥⎤
⎢⎡ −− 2124
Interchange 1st and 2nd rows
⎥⎥⎥
⎦⎢⎢⎢
⎣
=⎥⎥⎥
⎦⎢⎢⎢
⎣ −=′
31
241230 pA
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Example continued...148
⎥⎥⎥⎤
⎢⎢⎢⎡
=⎥⎥⎥⎤
⎢⎢⎢⎡ −−
=′ 12
230124
pA⎥⎦⎢⎣⎥⎦⎢⎣ − 3241
Eliminate a21 and a31 by using a11 as pivotal element
A LU i f (i i l i )A=LU in compact form (in a single matrix)
⎥⎤
⎢⎡
⎥⎤
⎢⎡ −− 2124
⎥⎥⎥
⎦⎢⎢⎢
⎣
=⎥⎥⎥
⎦⎢⎢⎢
⎣ −−=′
31
75.15.325.0230 pA
Multipliers (L matrix)
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Example continued...149
⎤⎡⎤⎡ 2124
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −−=′
312
75153250230124
pA⎥⎦⎢⎣⎥⎦⎢⎣ −− 375.15.325.0
Column search: Maximum magnitude at the third rowI t h nd d rd
⎤⎡⎤⎡ −− 2124
Interchange 2nd and 3rd rows
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−=′
132
23075.15.325.0
124pA
⎦⎣⎦⎣
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Example continued...150
⎤⎡⎤⎡ 2124
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−
−−=′
132
23075.15.325.0
124pA
⎥⎦⎢⎣⎥⎦⎢⎣ 1230
Eliminate a32 by using a22 as pivotal element
⎥⎤
⎢⎡
⎥⎤
⎢⎡ −− 2124
⎥⎥⎥
⎦⎢⎢⎢
⎣
=⎥⎥⎥
⎦⎢⎢⎢
⎣
−−=′
13
5.35.3/3075.15.325.0 pA
Multipliers (L matrix)
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Example continued...151
⎤⎡⎤⎡⎤⎡ 2124001
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=′
132
530075.15.30
124
153/300125.0001
pA⎥⎦⎢⎣⎥⎦⎢⎣⎥⎦⎢⎣ 15.30015.3/30
⎥⎤
⎢⎡−
′⎥⎤
⎢⎡
⎥⎤
⎢⎡ 5122
⎥⎥⎥
⎦⎢⎢⎢
⎣
=′⇒⎥⎥⎥
⎦⎢⎢⎢
⎣
−=⎥⎥⎥
⎦⎢⎢⎢
⎣
=123
35
13 bbp
A’x=b’ LUx=b’Ux=y
b’Ly=b’
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Example continued...152
Ly=b’
⎥⎥⎤
⎢⎢⎡ −
⎥⎥⎤
⎢⎢⎡
75151
yy
⎥⎥⎤
⎢⎢⎡−
⎥⎥⎤
⎢⎢⎡
⎥⎥⎤
⎢⎢⎡
35
01250001 1
yy
Ly=b
⎥⎥⎥
⎦⎢⎢⎢
⎣
=⎥⎥⎥
⎦⎢⎢⎢
⎣ 5.1075.1
3
2
yy
Forwardsubstitution⎥
⎥⎥
⎦⎢⎢⎢
⎣
=⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣
−123
15.3/300125.0
3
2
yy
⎥⎤
⎢⎡
⎥⎤
⎢⎡ 11x
⎥⎤
⎢⎡ −
⎥⎤
⎢⎡⎥⎤
⎢⎡ −− 5124 1x
Ux=y
⎥⎥⎥
⎦⎢⎢⎢
⎣
=⎥⎥⎥
⎦⎢⎢⎢
⎣ 32
3
2
xxBackward
substitution⎥⎥⎥
⎦⎢⎢⎢
⎣
=⎥⎥⎥
⎦⎢⎢⎢
⎣⎥⎥⎥
⎦⎢⎢⎢
⎣
−5.10
75.15.30075.15.30
3
2
xx
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Gauss-Jordan elimination153
The elements above the diagonal are made zero at the same time that zeros are created below the diagonal
⎤⎡ )1()1()1()1( baaa ⎤⎡ )1()1()1()1( baaa
⎥⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎢⎡
)1()1()1()1(
)1(2
)1(2
)1(22
)1(21
)(1
)(1
)(12
)(11
n
n
baaabaaa
MMOMM
L
L
⎥⎥⎥⎥⎥⎤
⎢⎢⎢⎢⎢⎡
)2()2()2(
)2(2
)2(2
)2(22
)(1
)(1
)(12
)(11
0 n
n
baabaaa
MMOMM
L
L
⎥⎥⎦⎢
⎢⎣
)1()1()1(2
)1(1 nnnnn baaa L ⎥
⎥⎦⎢
⎢⎣
)2()2()2(20 nnnn baa L
⎤⎡ − )1(1
)1(11 00 nba L⎤⎡ )2(
1)2()1(
11 0 baa L
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡−
)()(
)1(2
)2(22
111
00
0000
nn
n
b
baba
MMOMM
L
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
)3()3(
)2(2
)2()2(22
111
00
00
nn
nn
b
baabaa
MMOMM
L
⎥⎦⎢⎣)()(00 n
nn
nn baL⎥⎦⎢⎣)3()3(00 nnn baL
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Gauss-Jordan Elimination154
Almost 50% more arithmetic operations than 5 pGaussian elimination
Gauss-Jordan (GJ) Elimination is prefered when the inverse of a matrix is required.
[ ]IAApply GJ elimination to convert A into an identity matrix.
[ ]IA
[ ]1−AI[ ]AI
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Eigen values and Eigenvectors
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Suppose we have some vector A, in the equation Ax=b and we want to find the which vectors x are pointing the same direction after the transformation. These vectors are called Eigenvectors vectors are called Eigenvectors.
The vector b must be a scalar multiple of x. The scalar that multiplies x is called the Eigen valuethat multiplies x is called the Eigen value
The main equation for this section is
A λAx = λx
Any vector x that satisfies this equation is an Eigenvector the corresponding λ is the Eigen valueEigenvector, the corresponding λ is the Eigen value
Note: for this section we are only considering square matricesmatrices.
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Example A
Let’s examine some vectors that we are already familiar with and determine the Eigenvectors
d i land Eigenvalues.
id j i i i h jConsider a Projection matrix P in R3, that projects vectors on to a plane. What are the Eigenvectors and Eigenvalues?and Eigenvalues?
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Answer to Example A
Some Eigenvectors are the vectors that are already in the plane that is being projected on. In that case the vector does not change so the Eigenvalue for these vectors is 1Eigenvalue for these vectors is 1
Other Eigenvectors are those orthogonal to the g gplane that is being projected on. Those vectors become the zero vector (which is considered parallel to all vectors) The Eigen value for these parallel to all vectors). The Eigen value for these vectors is zero.
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Look at the case λ = 0
If A is a singular matrix, then we can solveg ,
Ax = λx
What did we previously call these values?
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Answer
If λ= 0 then we are solving Ax=0 which is the null gspace (Kernel)
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The following statements are equivalentA is in ertibleA is invertibleThe linear system Ax=b has a unique solution x for all b
f(A) Irref(A) = In
Rank(A) = n
I (A) RIm (A) = Rn
ker(A) = 0
The column vectors of A form a basis of Rn
The column vectors of A span Rn
The column vectors of A are linearly independent
detA ≠0
0 fails to be an eigenvalue of A
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How can I solve Ax = λxBring everything on one sideBring everything on one sideAx – λx = 0(A- λI)x = 0( )If this can be solved then the matrix(A- λI) must be singularWhich means that det (A- λI) =0 This equation is called the characteristic equation.There should be n values to this equation (although some
could be repeated)Once we find λ find the nullspace of(A λI)x = 0Once we find λ find the nullspace of(A- λI)x = 0to find the x’s (Eigenvectors)
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Find the Eigenvalues
3 1
1 3
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Find the Eigenvalues3 1
1 3 1 1
1 1 Note: this equation is
Find det (A- λI) =0 Plug in
(3- λ)2 – 1 λ=2 and find
Note: this equation is called the characteristic equation
λ2 - 6 λ + 8 = 0 a basis for
(λ-4)(λ-2)= 0 kernel
3- λ 1
1 3- λ
λ=4 λ=2
Plug in λ=4 to find the Eigenvectorsg g
find a basis for the null space (kernel)-1 1
1 -1
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Eigenvalues of triangular matrices
Find the Eigenvalues ofg
3 1
0 3 0 3
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Triangular matrices slide 1 of solutions
Find the Eigenvalues 3 1
0 3 A=
A- λ I= 3- λ 1
0 3 - λ
Det(A)= (3 – λ)2 = 0 λ =3
This matrix has a repeated Eigenvalue. p g
Note: for triangular matrices, the values on the diagonal of the matrix are the Eigenvalues
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Triangular matricesTriangular matrices
Find the Eigenvectors A- λ I=
Replace λ by 3
3- λ 1
0 3 - λ
Find the null space0 1
0 0
This matrix has only 1 Eigenvector!
A repeated λ gives the possibility of a lack of Eigenvectors
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Section D
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Ordinary Differential Equationsy q
Equations which are composed of an unknownEquations which are composed of an unknown function and its derivatives are called differential equations.
Differential equations play a fundamental role in q p yengineering because many physical phenomena are best formulated mathematically in terms of their rate of change.
vcgdv−=
v- dependent variable
t independent variablemg
dt t- independent variable
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When a function involves one dependent variable, the equation is called an ordinary differential equation (or ODE). A partial y ff q ( ) pdifferential equation (or PDE) involves two or more independent variables.
Differential equations are also classified as to their order.A first order equation includes a first derivative as its highest d i iderivative.A second order equation includes a second derivative.
Higher order equations can be reduced to a system of first order equations, by redefining a variable.
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ODEs and Engineering PracticeODEs and Engineering Practice
Figure 7.1
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Figure 7.2
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Runga-Kutta Methods
This chapter is devoted to solving ordinaryThis chapter is devoted to solving ordinary differential equations of the form
)(fdy
Euler’s Method
),( yxfdxy=
Euler s Method
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Figure 7.3
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The first derivative provides a direct estimate of the slope at xip i
),( ii yxf=φ
where f(xi,yi) is the differential equation evaluated at xiand yi. This estimate can be substituted into the
tiequation:hyxfyy iiii ),(1 +=+
A new value of y is predicted using the slope to extrapolate linearly over the step size hextrapolate linearly over the step size h.
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10
5.820122),( 23 +−+−==
i tSt ti
xxxyxfdxdy
25550*581)( =+=+= hyxfyy
1,0 00 == yxpointStarting
25.55.0*5.81),(1 =+=+=+ hyxfyy iiii
Not good
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Error Analysis for Euler’s Method/Error Analysis for Euler s Method/Numerical solutions of ODEs involves two types of error:error:
Truncation errorLocal truncation error
!2),( 2hyxfE ii
a′
=
P t d t ti
)(!2
2hOEa =Propagated truncation error
The sum of the two is the total or global truncation errorRound-off errorsRound off errors
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The Taylor series provides a means of quantifying the error in Euler’s method. However;
The Taylor series provides only an estimate of the local truncation error-that is, the error created during a single step of the method.In actual problems, the functions are more complicated than simple polynomials. Consequently, the derivatives needed to evaluate the Taylor series expansion would not always beevaluate the Taylor series expansion would not always be easy to obtain.
In conclusion,h b d d b d i h ithe error can be reduced by reducing the step size
If the solution to the differential equation is linear, the method will provide error free predictions as for a straight p p gline the 2nd derivative would be zero.
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Figure 7.4
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Improvements of Euler’s methodp
A fundamental source of error in Euler’s method is that theA fundamental source of error in Euler s method is that the
derivative at the beginning of the interval is assumed to apply
th ti i t lacross the entire interval.
Two simple modifications are available to circumvent this
shortcoming:
Heun’s Method
The Midpoint (or Improved Polygon) Method
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Heun’s Method/Heun s Method/One method to improve the estimate of the slope involves the determination of two derivatives for theinvolves the determination of two derivatives for the interval:
At the initial pointpAt the end point
The two derivatives are then averaged to obtain an improved estimate of the slope for the entire interval.
hyxfyy )(:Predictor 0 +=
hyxfyxfyy
hyxfyy
iiiiii
iiii
2),(),(:Corrector
),( :Predictor0
111
1
+++
+
++=
+=
ii 21+
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Figure 7.5
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The Midpoint (or Improved Polygon) Method/p ( p yg )Uses Euler’s method t predict a value of y at the midpoint of the interval:
hyxfyy iiii ),( 2/12/11 +++ +=
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Figure 7.6
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Runge-Kutta Methods (RK)g ( )
Runge-Kutta methods achieve the accuracy of aRunge Kutta methods achieve the accuracy of a Taylor series approach without requiring the calculation of higher derivatives.
),,(
2211
1
kakakahhyxyy
nn
iiii+
+++=+=
Lφφ
Increment function
)(),(
constants'
1
hkqyhpxfkyxfk
sa
ii
++===
p’s and q’s are
),(),(
22212133
11112
hkqhkqyhpxfkhkqyhpxfk
ii
ii
+++=++=
M
p s and q s are constants
),( 11,122,1111 hkqhkqhkqyhpxfk nnnnninin −−−−−− +++++= L
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k’s are recurrence functions. Because each k is a functional l i hi k RK h d ffi i fevaluation, this recurrence makes RK methods efficient for
computer calculations.Various types of RK methods can be devised by employingVarious types of RK methods can be devised by employing different number of terms in the increment function as specified by n.Fi d RK h d i h 1 i i f E l ’ h dFirst order RK method with n=1 is in fact Euler’s method.Once n is chosen, values of a’s, p’s, and q’s are evaluated by setting general equation equal to terms in a Taylor seriessetting general equation equal to terms in a Taylor series expansion.
hkakayy ii )( 22111 ++=+
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Values of a1, a2, p1, and q11 are evaluated by setting the second order equation to Taylor series expansionthe second order equation to Taylor series expansion to the second order term. Three equations to evaluate four unknowns constants are derived.
hyxfhyxfyyHowever
hkakayyhaveWe
iiiiii
ii
21
22111
!2),('),(
)(:
++=
++=
+
+
hdyyxfyxf
dxdy
yyxf
xyxfyxfBut iiii
ii
2)()(
),(),(),('
!2
⎤⎡ ∂∂
∂∂
+∂
∂=
yxfk
hdxdy
yyxf
xyxfhyxfyyThen
ii
iiiiiiii
1
1
),(!2
),(),(),(
=
⎥⎦
⎤⎢⎣
⎡∂
∂+
∂∂
++=+
hkyxfhyxffk
hkqyhpxfkexpandnowWehkqyhpxfk
iiii
ii
ii
11112
11112
),(),()(
),(),(
∂∂++=
++=
hkqy
yxfhpx
yxfyxfk iiiiii 11112
),(),(),(∂
∂+
∂∂
+=
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hkakayy ii )( 22111 ++=+
hhkqy
yxfhpx
yxfyxfayxfayy iiiiiiiiii
⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+
∂∂
+++=+ 1111211),(),(),(),(
yxfx
yxfhpayxfhayxfhayy iiiiiiii
∂∂
∂+++=+
)(
),(),(),(
2
212211
)()( 2hyxfyxf ⎤⎡ ∂∂
yyxfhyxfqa ii
ii ∂∂
+),(),( 2
112
!2),(),(),(),(1
hyxfy
yxfx
yxfhyxfyy iiiiii
iiii ⎥⎦
⎤⎢⎣
⎡∂
∂+
∂∂
++=+
121 =+ aa
121
12
21
=pa
2112 =qa
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Because we can choose an infinite number of values for a2, there are an infinite number of second-order RK methods.E i ld i ld l h l ifEvery version would yield exactly the same results if the solution to ODE were quadratic, linear, or a constant.constant.However, they yield different results if the solution is more complicated (typically the case).Three of the most commonly used methods are:
Huen Method with a Single Corrector (a2=1/2)The Midpoint Method (a2=1)Raltson’s Method (a =2/3)Raltson s Method (a2=2/3)
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Figure 7.7
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191Partial Differential Equations
Figure 8.1
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Fi i Diff Elli i E i192Finite Difference: Elliptic EquationsSolution TechniqueSolution Technique
Elliptic equations in engineering are typically used to characterize steady-state, boundary value problems.
For numerical solution of elliptic PDEs, the PDE is f d i l b i diff itransformed into an algebraic difference equation.
B f it i li it d l l t tBecause of its simplicity and general relevance to most areas of engineering, we will use a heated plate as an example for solving elliptic PDEs. p g p
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Figure 8.2
193
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The Laplacian Difference Equations/The Laplacian Difference Equations/
02
2
2
2
=∂∂
+∂∂
yT
xT
Laplace Equation
194
22
,1,,12
2
Δ
+−=
∂∂
∂∂
−+ jijiji
xTTT
xT
yx Laplace Equation
O[Δ(x)2]
22
1,,1,2
2
Δ
+−=
∂∂ −+ jijiji
yTTT
yT
O[Δ(y)2]
022
21,,1,
2,1,,1
ΔΔ
=Δ
+−+
Δ
+− −+−+ jijijijijiji
yxy
TTTx
TTT
04 ,1,1,,1,1 =−+++Δ=Δ
−+−+ jijijijiji TTTTTyx
Laplacian difference equation.q
Holds for all interior points
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Figure 8.3
195
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In addition boundary conditions along the edges must be196
In addition, boundary conditions along the edges must be specified to obtain a unique solution.The simplest case is where the temperature at the boundary is set at a fixed value, Dirichlet boundary condition.A balance for node (1,1) is:
04TTTTT
075
04
01
1110120121
==−+++
TT
TTTTT
040
211211
10
=++−=
TTTT
Similar equations can be developed for other interior points to result a set of simultaneous equations.
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• The result is a set of nine simultaneous equations with nine 197unknowns:
04754
22132111
122111
=−−+−=−−
TTTTTTT
754504
13221211
323121
22132111
=−−+−=−+−
TTTTTTT
50404
33322231
2332221221
=−+−−=−−+−−
TTTTTTTTT
150410041754
33231322
231312
+=−+−−=−+−
TTTTTTT
TTT
1504 332332 =+−− TTT