Handout CIV 208 Analisis Numerik CIV 208 P7

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Respect, Professionalism, & Entrepreneurship

Interpolasi

Pertemuan - 7 Mata Kuliah : Analisis Numerik

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• _{You will frequently have occasion to estimate}

intermediate values between precise data points.

• _{The most common method used for this purpose}

is polynomial interpolation.

• _{Recall that the general formula for an nth-order}

polynomial is

f(x)=a0+a1x+a2x2+···+anxn

• _{For n+1 data points, there is one and only one}

polynomial of order n that passes through all the points.

• _{For example, there is only one straight line (that}

is, a first-order polynomial) that connects two points (Fig. a).

• _{Similarly, only one parabola connects a set of}

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• _{Polynomial interpolation consists of determining}

the unique nth-order polynomial that fits n+1 data points.

• _{This polynomial then provides a formula to}

compute intermediate values.

• _{Although there is one and only one nth-order}

polynomial that fits n+1 points, there are a variety of mathematical formats in which this polynomial can be expressed.

• _{Two alternatives that are well-suited for}

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Newton Linear Interpolation

• _{The simplest form of}

interpolation is to connect two data points with a straight line. This technique, called linear

interpolation, is depicted

graphically in Fig. Using similar triangles :

• _{which is a linear-interpolation}

formula. The notation f₁(x) designates that this is a first order interpolating polynomial

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Example 1 :

Estimate the natural logarithm of 2 using linear interpolation.

First, perform the computation by interpolating between ln 1=0 and ln6=1.791759.

Then, repeat the procedure, but use a

smaller interval from ln 1 to ln 4 (1.386294). Note that the true value of ln 2 is

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Newton Quadratic Interpolation

• _{If three data points are available,}

this can be accomplished with a

second-order polynomial (also called a quadratic polynomial or a

parabola).

• _{A particularly convenient form for}

this purpose is

f₂(x)=b₀+b₁(x−x₀)+b₂(x−x₀)

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• _{A simple procedure can be used to}

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Example 2 :

Fit a second-order

polynomial to the three points used in Example 1:

x0=1 f(x0)=0

x1=4 f(x1)=1.386294 x2=6 f(x2)=1.791759

• _{Use the polynomial to}

evaluate ln 2.

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General Form of Newton’s Interpolating Polynomials

• _{The preceding analysis can be generalized to fit}

an nth-order polynomial to n+1 data points.

• _{The nth-order polynomial is}

fn(x)=b0+b1(x−x0)+···+bn(x−x0)

(x−x1)···(x−xn−1)(1)

• _{We use these data points and the following}

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• _b_o ₌_f₍_x_o_{) (2)}

• _b₁₌_f_[_x₁_.x_o_{] (3)}

• _b₂₌_f_[_x₂_{, x}₁_{, x}_o_{] (4)} • _…....

• _b_n₌_f_[_x_n_{, x}_n_₁_{, ……, x}₁_{, x}_o_{] (5)} • _{where the bracketed function}

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• _{For example, the first finite divided}

difference is represented generally as

• _{The second finite divided difference,}

which represents the difference of two first divided differences, is

expressed generally as

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• _{Similarly, the n}th finite divided

difference is

• _{These differences can be used to}

evaluate the coefficients in Eqs. (2) through (5), which can then be

substituted into Eq. (1) to yield the interpolating polynomial

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Example 3

In Example 2, data points at x₀=1, x₁=4,

and x₂=6 were used to estimate ln 2 with a parabola.

Now, adding a fourth point [x₃=5; f(x₃)=

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Lagrange Interpolating Polynomials

• _{The Lagrange interpolating}

polynomial is simply a reformulation of the Newton polynomial that

avoids the computation of divided differences.

• _{It can be represented concisely as} 



    (6)



 n

i

i i

n x L x f x

f

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• _Where

• _where__{designates the “product of.”} • _{For example, the linear version}

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• _{and the second-order version is}

• _{the summation of all the products}

designated by Eq. (6) is the unique nth order polynomial that passes

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Example 4

• _{Use a Lagrange interpolating}

polynomial of the first and second

order to evaluate ln 2 on the basis of the data given in Example 2:

x₀=1 f(x₀)=0

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