Writing programs

• Choosing appropriate data types → min memory, can store all values required
• Global/static/local variables → should select one in a justified manner → pro/cons for taking a decision
• Functions to write your programs, main() should be very small → type of arguments → our choice from C++. Value, reference, pointer? Justify
  → overloading
  → template functions

Numerical algorithms

• Linear equations
• Polynomial roots
• Numerical integration
• Solving ODE/PDE

Use numerical algorithms when we can’t solve analytically. Don’t use it for roots of quadratic/cubic/quartic. Same when we can actually integrate or solve PDEs.
Use numerical algorithms when no closed form.

Interval halving

We need to first find two points between which the function changes its sign.
Suppose we have a polynomial, can we use the coefficients to find a suitable interval?

Newton Raphson

\(x_{n+1} = x_n - f(x_n)/f'(x_n)\)

Polynomial can be evaluated at any point.
Construct \(p'(x)\) given \(p(x)\).

\(e_{n+1} = ce_n^2\), \(e_n\) is the error at \(n\)-th step.
Using this, justify how many times we should run a while loop.

NR shows different behaviour if we have repeated roots.

Polynomial: Suppose we have distinct roots which are reasonably separated.
\(P(x)\) : roots are 1, 2, 3, 4, 5
Suppose we can create a polynomial with square roots
\(P_1(x)\) : roots are 1, 4, 9, 16, 25
And then, \(P_2(x)\) - 1, 16, 81, 256, 625

coeff of \(x^4\) in the last polynomial: 625 + 256 + 81 + 16 + 1 ~ 625 → take fourth roots (this is the spirit) In reality, the sum is 979 → 5.59…
Then find the next by looking at \(x^3\)

Programming part - find the coefficients of \(P_1\), \(P_2\)
Need a method to do that without knowing the roots.

Hint: Consider \(P(x)P(-x)\)

Prev | Next

CS 213