Lab 20: Numerical Integration Lab

Type f(x), a and b. Choose the rule and the number of panels; give the exact value if you know it (otherwise a reference value is computed).

Try this

Try this

Q1 Derive the trapezoidal rule from the general quadrature formula.

With

n = 1

:

\int_{x_0}^{x_1} y\,dx = h[y_0 + \frac12\Delta y_0] = \frac h2(y_0 + y_1)

. Adding over all intervals gives

\frac h2[y_0 + 2(y_1 + \cdots + y_{n-1}) + y_n]

.

Q2 Why is Simpson's 1/3 rule exact for cubics, though it uses parabolas?

In the general formula with

n = 2

, the

\Delta^3 y_0

coefficient

n(n-2)^2/24

is zero. Equivalently, its error term involves

f^{(4)}

, which is zero for cubics.

Q3 Evaluate

\int_0^1 \frac{dx}{1+x}

by Simpson's rule with

h = 1/6

.

\frac{1}{18}[1 + 4(0.857143 + 0.666667 + 0.545455) + 2(0.75 + 0.6) + 0.5] = 0.693170

(true

\ln 2 = 0.693147

).

Q4 What is Romberg integration?

Repeated Richardson extrapolation of trapezoidal values

T(h), T(h/2), T(h/4), \dots

:

R_{k,j} = R_{k,j-1} + (R_{k,j-1} - R_{k-1,j-1})/(4^j - 1)

, removing the

h^2, h^4, \dots

error terms in turn.

Q5 State one advantage and one disadvantage of Gauss quadrature.

Advantage:

n

points integrate polynomials of degree

2n - 1

exactly, so very few function values are needed. Disadvantage: the points are irrational, so it cannot be used on equally spaced tabulated data.