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Mata Kuliah : Metode Numerik�Minggu ke 11

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Mahasiswa dapat melakukan komputasi untuk menentukan koefisien fungsi polynomial dengan Lagrange Interpolating

Tujuan perkuliahan

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Lagrange’s Method

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Example

The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Lagrangian method for linear interpolation.

Table Velocity as a

function of time

Figure. Velocity vs. time data

for the rocket example

(s)	(m/s)
0	0
10	227.04
15	362.78
20	517.35
22.5	602.97
30	901.67

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Linear Interpolation

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Linear Interpolation (contd)

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Quadratic Interpolation

http://numericalmethods.eng.usf.edu

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Example

http://numericalmethods.eng.usf.edu

The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Lagrangian method for quadratic interpolation.

Table Velocity as a

function of time

Figure. Velocity vs. time data

for the rocket example

(s)	(m/s)
0	0
10	227.04
15	362.78
20	517.35
22.5	602.97
30	901.67

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Quadratic Interpolation (contd)

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Quadratic Interpolation (contd)

The absolute relative approximate error obtained between the results from the first and second order polynomial is

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Cubic Interpolation

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Example

http://numericalmethods.eng.usf.edu

The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Lagrangian method for cubic interpolation.

Table Velocity as a

function of time

Figure. Velocity vs. time data

for the rocket example

(s)	(m/s)
0	0
10	227.04
15	362.78
20	517.35
22.5	602.97
30	901.67

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Cubic Interpolation (contd)

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Cubic Interpolation (contd)

The absolute relative approximate error obtained between the results from the first and second order polynomial is

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Comparison Table

Order of Polynomial	1	2	3
v(t=16) m/s	393.69	392.19	392.06
Absolute Relative Approximate Error	--------	0.38410%	0.033269%

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Why Splines ?

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Why Splines ?

Figure : Higher order polynomial interpolation is a bad idea

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Linear Interpolation

http://numericalmethods.eng.usf.edu

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Linear Interpolation (contd)

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Example

The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using linear splines.

Table Velocity as a

function of time

Figure. Velocity vs. time data

for the rocket example

(s)	(m/s)
0	0
10	227.04
15	362.78
20	517.35
22.5	602.97
30	901.67

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Linear Interpolation

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Quadratic Interpolation

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Quadratic Interpolation (contd)

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Quadratic Splines (contd)

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Quadratic Splines (contd)

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Quadratic Splines (contd)

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Quadratic Spline Example

The upward velocity of a rocket is given as a function of time. Using quadratic splines

Find the velocity at t=16 seconds
Find the acceleration at t=16 seconds
Find the distance covered between t=11 and t=16 seconds

Table Velocity as a

function of time

Figure. Velocity vs. time data

for the rocket example

(s)	(m/s)
0	0
10	227.04
15	362.78
20	517.35
22.5	602.97
30	901.67

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Solution

Let us set up the equations

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Each Spline Goes Through Two Consecutive Data Points

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t	v(t)
s	m/s
0	0
10	227.04
15	362.78
20	517.35
22.5	602.97
30	901.67

Each Spline Goes Through Two Consecutive Data Points

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Derivatives are Continuous at Interior Data Points

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Derivatives are continuous at Interior Data Points

At t=10

At t=15

At t=20

At t=22.5

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Last Equation

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Final Set of Equations

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Coefficients of Spline

i	a_i	b_i	c_i
1	0	22.704	0
2	0.8888	4.928	88.88
3	−0.1356	35.66	−141.61
4	1.6048	−33.956	554.55
5	0.20889	28.86	−152.13

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