Romberg's method is used to estimate definite integrals by applying Richardson extrapolation repeatedly to the trapezoidal rule or rectangular rule. This generates a triangular array that increases in accuracy. The method is an extension of trapezoidal and rectangular rules. It works by recursively calculating the integral using smaller step sizes to generate values in the triangular array. Convergence is reached when two successive values are very close. An example calculates a definite integral using Romberg's method in three cases with decreasing step sizes to populate the triangular array.

1. Romberg’s Integration
Dr. Varun Kumar
Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-3 1 / 8

2. Outlines
1 Introduction to Romberg’s Rule
2 Mathematical Formulation
3 Example
Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-3 2 / 8

3. Introduction to Romberg’s Rule
Important points
⇒ In numerical analysis, Romberg’s method (Romberg 1955) is used to
estimate the definite integral
F(x) =
Z b
a
f (x)dx
⇒ By applying Richardson extrapolation repeatedly on the trapezium
rule or the rectangle rule.
⇒ The estimates generate a triangular array.
⇒ It increases the accuracy with greater extent.
⇒ It is the extension of trapezoidal and rectangular rule.
Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-3 3 / 8

4. Other Integration Methods
Rectangular rule
Z b
a
f (x)dx = h
h
f (a) +
f (x1) + f (x2) + ....f (xn−1)
+ f (b)
i
where
h = step size → (b−a)
n
Total numbers of sample = n + 1 (Including point a and b )
x1 = a + h, x2 = a + 2h,.......
Trapezoidal rule
Z b
a
f (x)dx =
h
2
h
f (a) + 2
f (x1) + f (x2) + ....f (xn−1)
+ f (b)
i
Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-3 4 / 8

5. Romberg’s Integration
Steps for solving Romberg’s Integration
hn = (b−a)
2n → Variable step size
The method can be inductively defined by
R(0, 0) = h1 f (a) + f (b)
R(n, 0) =
1
2
R(n − 1, 0) + hn
2n−1
X
k=1
f (a + (2k − 1)hn)
Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-3 5 / 8

6. More on Romberg’s Integration
Triangular array
Rectangular or Trapezoidal Method Recursively
Step size Step-1 Step-2 Step-3 Step-4
h I1
I∗
1 = I2 + 1
3
(I2 − I1)
I∗
2 = I3 + 1
3
(I3 − I2)
I∗
3 = I4 + 1
3
(I4 − I3)
I∗∗
1 = I2 ∗ + 1
3
(I2 ∗ −I1∗)
I∗∗
2 = I3 ∗ + 1
3
(I3 ∗ −I2∗)
I∗∗∗
1 = I∗∗
1 + 1
3
(I∗∗
2 − I∗∗
1 )
h/2 I2
h/4 I3
h/8 I4
This method can be stopped when two successive values are very
close to each other.
Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-3 6 / 8

7. Example–
Example
Q Evaluate the following definite integral J using Romberg’s integration
rule, where
J =
Z 1
0
1
1 + x
dx
Ans Solution: According to question, a = 0, b = 1. We solve this by
trapezoidal rule
Case 1: Taking h = 0.5, the value of x and f (x) is
At x = 0, f (x) = 1
At x = 0.5, f (x) = 0.66667
At x = 1, f (x) = 0.5
At I = 1
4[1 + 2 × 0.66667 + 0.5] = 0.70835
Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-3 7 / 8

8. Continued–
Case 2: Taking h = 0.25, the value of x and f (x) is
x 0 0.25 0.5 0.75 1
f(x) 1 0.8 0.667 0.5714 0.5
By trapezoidal rule I = 0.25
2 [1 + 2(0.8 + 0.667 + 0.5714) + 0.5] = 0.6970
Case 3: Taking h=0.125, x and f (x) value is
x 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1
f(x) 1 0.8889 0.8 0.7273 0.667 0.6154 0.5714 0.5333 0.5
By trapezoidal rule
I = 0.125
2
[1+2(0.8889+0.8+0.7273+0.667+0.6154+0.5714+0.5333)+0.5] = 0.6914
I(h) = 0.7084 I(h/2) = 0.6970 I(h/4) = 0.6914
I∗
1 = 0.6932, I∗
2 = 0.6931 and I∗∗
1 = 0.6931
Dr. Varun Kumar (IIIT Surat) Unit 5 / Lecture-3 8 / 8