MODULE 4: CURVE
FITTING
INTERPOLATION
CURVE FITTING
• It is the method of constructing a curve or a derived function that will “fit” to the set of data subject to
constraints to obtain intermediate estimates.
• General approaches for curve fitting:
➢ Least-squares regression
➢ Interpolation
CURVE FITTING
• Interpolation
➢ It is a process of estimating a value between the data points which are known to be precise and
directly pass by the fitted curve or series of curves.
➢ The most common method used is the polynomial interpolation.
➢ Newton’s Divided Difference and Lagrange Interpolating Polynomials.
INTERPOLATION
INTERPOLATION
• Newton’s Divided-Difference Interpolating Polynomials
• Linear Interpolation
• Quadratic Interpolation
• General form of Newton’s divided-difference interpolation
INTERPOLATION
• Linear Interpolation
➢ Simplest form of interpolation that connects two
data points to form a straight line.
•
𝑓1 𝑥 −𝑓 𝑥0
𝑥−𝑥0
=
𝑓 𝑥1 −𝑓 𝑥0
𝑥1 −𝑥0
• 𝒇𝟏 𝒙 = 𝒇 𝒙𝟎 +
𝒇 𝒙𝟏 −𝒇 𝒙𝟎
𝒙𝟏 −𝒙𝟎
𝒙 − 𝒙𝟎
• 𝑓1 𝑥 = 𝑎0 + 𝑎1 𝑥
• 𝑓1 𝑥 = 𝑓 𝑥0 −
𝑓 𝑥1 −𝑓 𝑥0
𝑥0
𝑥1 −𝑥0
+
𝑓 𝑥1 −𝑓 𝑥0
𝑥1 −𝑥0
𝑥
INTERPOLATION
• Quadratic Interpolation
• 𝑓2 𝑥 = 𝑏0 + 𝑏1 𝑥 − 𝑥0 + 𝑏2 𝑥 − 𝑥0 𝑥 − 𝑥1
• 𝑓2 𝑥 = 𝑏0 − 𝑏1 𝑥0 + 𝑏2 𝑥0 𝑥1 + 𝑏1 − 𝑏2 𝑥0 − 𝑏2 𝑥1 𝑥 + 𝑏2 𝑥 2
• 𝑏0 = 𝑓 𝑥0
• 𝑏1 =
𝑓 𝑥1 −𝑓 𝑥0
𝑥1 −𝑥0
• 𝑏2 =
𝑓 𝑥2 −𝑓 𝑥1 𝑓 𝑥1 −𝑓 𝑥0
− 𝑥 −𝑥
𝑥2 −𝑥1
1 0
𝑥2 −𝑥0
INTERPOLATION
• General form of Newton’s Divided-Difference Interpolation
➢ Fitting an 𝑛th order of polynomial to n+1 data points using finite divided difference
❑ 𝑓𝑛 𝑥 = 𝑏0 + 𝑏1 𝑥 − 𝑥0 + 𝑏2 𝑥 − 𝑥0 𝑥 − 𝑥1 + ⋯ + 𝑏𝑛 𝑥 − 𝑥0 𝑥 − 𝑥1 ⋯ 𝑥 − 𝑥𝑛−𝑧
➢ Newton’s divided-difference interpolating polynomial
❑ 𝑓𝑛 𝑥 = 𝑓 𝑥0 + 𝑥 − 𝑥0 𝑓 𝑥1 , 𝑥0 + 𝑥 − 𝑥0 𝑥 − 𝑥1 𝑓 𝑥2 , 𝑥1 , 𝑥0 + ⋯ + 𝑥 − 𝑥0 𝑥 − 𝑥1 ⋯ ሺ𝑥 −
𝑥𝑛−1 ሻ𝑓 𝑥𝑛 , 𝑥𝑛−1 , ⋯ , 𝑥2 , 𝑥1 , 𝑥0
INTERPOLATION
• Finite Divided Difference
❑ 𝑏0 = 𝑓 𝑥0
❑ 𝑏1 = 𝑓 𝑥1 , 𝑥0 =
𝑓 𝑥1 −𝑓 𝑥0
𝑥1 −𝑥0
•
•
•
❑ 𝑏𝑛 = 𝑓 𝑥𝑛 , 𝑥𝑛−1 , ⋯ , 𝑥2 , 𝑥1 , 𝑥0 =
𝑓 𝑥𝑛 ,𝑥𝑛−1 ,⋯,𝑥2 ,𝑥1 −𝑓 𝑥𝑛−1 ,𝑥𝑛−2 ,⋯,𝑥2 ,𝑥1 ,𝑥0
𝑥𝑛 −𝑥0
INTERPOLATION
• Lagrange Interpolating Polynomials
➢ The 𝑛th order polynomial can be represented as
❑ 𝑓𝑛 𝑥 = σ𝑛𝑖=0 𝐿𝑖 𝑥 𝑓 𝑥𝑖
❑ 𝑓𝑛 𝑥 = 𝐿0 𝑥 𝑓 𝑥0 + 𝐿1 𝑥 𝑓 𝑥1 + 𝐿2 𝑥 𝑓 𝑥2 + ⋯ + 𝐿𝑛 𝑥 𝑓 𝑥𝑛
➢ First-order version
𝑥−𝑥
𝑥−𝑥
❑ 𝑓1 𝑥 = 𝑥 −𝑥1 𝑓 𝑥0 + 𝑥 −𝑥0 𝑓 𝑥1
0
1
1
0
➢ Second-order version
𝑥−𝑥
𝑥−𝑥
𝑥−𝑥
𝑥−𝑥
𝑥−𝑥
𝑥−𝑥
❑ 𝑓2 𝑥 = 𝑥 −𝑥1 𝑥 −𝑥2 𝑓 𝑥0 + 𝑥 −𝑥0 𝑥 −𝑥2 𝑓 𝑥1 + 𝑥 −𝑥0 𝑥 −𝑥1 𝑓 𝑥2
0
1
0
2
1
0
1
2
2
0
2
1
INTERPOLATION
• SAMPLE PROBLEM: Find the polynomial that interpolates the points (1, 2) and (4, 5). Evaluate p(1/2)
INTERPOLATION
• SAMPLE PROBLEM: Using Newton’s divided-difference, determine the parabolic equation of the
given data points.
x
0
1
2
y
1
4
9
INTERPOLATION
INTERPOLATION
• SAMPLE PROBLEM: using Newton’s form, show that the interpolating polynomial of the points (-1, 3),
1
1
21
(0, 2), (2, 0), and (4, -1) is 𝑝 𝑥 = 40 𝑥 3 − 40 𝑥 2 − 20 𝑥 + 2
INTERPOLATION
INTERPOLATION
INTERPOLATION
• SAMPLE PROBLEM: Solve problem no. 3 using Lagrange’s Interpolation
INTERPOLATION