Numerical Integration

Question 1

What is the main question in this chapter?

Answer 1

How do we calculate integrals numerically?

Question 2

What is a definite integral.

Answer 2

Question 3

What is a quadrature formula?

Answer 3

Question 4

What are x₀, …. x_n and σ₀, … ,σ_n in the following?

Answer 4

• x₀, …. x_n are a set of nodes
• σ₀, … ,σ_n are a set of corresponding weights

Question 5

What are the nodes also known as in the following?

Answer 5

Quadrative points

Question 6

What is the quadrature formula for the trapezium rule?

Answer 6

Question 7

How can you derive quadrative formulas?

Answer 7

By integrating polynomials interpolants at equally spaced points.

Question 8

What is the basic form of the interpolating polynomial in Lagrange form?

Answer 8

Question 9

By integrating the following p_n(x) what is the quadrative formula you get?

Answer 9

Question 10

If the nodes are equidistant what is the following formula called?

Answer 10

Newton-Cotes formula

Question 11

When is the Newton-Cotes formula called closed?

Answer 11

If x₀=a and x_n=b

Question 12

Prove the trapezium with using the closed Newton-Cotes formula.

Answer 12

Question 13

What is the theorem about the Newton-Cotes formula error bound?

Answer 13

Question 14

Prove the following theorem.

Answer 14

Question 15

Why don’t Newton-Cotes formulae converge as n ➝ ∞?

Answer 15

Because they are based on polynomials interpolation at equally-spaced nodes

Question 16

What is the idea with composite Newton-Cotes?

Answer 16

Divide [a,b] into m subintervals [x_i-1,x_i] of equal length h = (b-a)/h and use a N-C formula with small n on each interval.

Question 17

What is the composite trapezium rule?

Answer 17

Question 18

What is the overal error for the composite trapezium rule?

Answer 18

Question 19

What is the trapezium rule exact for?

Answer 19

All lienar polynomials

Question 20

Define degree of exactness.

Answer 20

Question 21

How do you check for degree of exactness?

Answer 21

Check the standard basis 1, x, x², … ,xⁿ

Question 22

What is the Newton Cote’s formula for n=2 called?

Answer 22

Simpson’s rule

Question 23

What is simpsons rule?

Answer 23

Question 24

What is the main idea in Gaussina Quadrative?

Answer 24

Question 25

Finish the following lemma.

Answer 25

Question 26

Prove the following lemma.

Answer 26

Question 27

What is the Gaussian quadrature theorem?

Answer 27

Question 28

What is the difference between Newton-Cotes and Gaussian quadrature?

Answer 28

Both are based on integrating an interpolating polynomial, but for Gaussian the nodes are the roots of an orthogonal polynomial, rather than equally spaced points.

Question 29

Prove the following theorem.

Answer 29

Question 30

What is the Mean value theorem for integrals?

Answer 30

Question 31

Prove the following theorem.

Answer 31

Question 32

What is the Error estimate for Gaussian quadrature theorem?

Answer 32

Question 33

Prove the following theorem.

Answer 33