MATH4141/MATH6651/PHYS5070

Formula Sheet

Fixed-Point Theorem in Rn :

Let . Suppose is a continuous function with continuous partial derivatives fromRn into Rn with the property that whenever . Then has a fixed point in D.

Furthermore, if a constant k exists with 0 < k < 1 such that

for

where Jg is the Jacobian of then has a unique fixed point in D and the sequence defined by

converges to the fixed point provided .

Newton's method in Rn :

Given the system of equations Newton's method for finding a root of this system is to construct the iterative scheme

where J is the Jacobian matrix of the system.

Golden Search method

Given the interval [a, b] the formula for the points c and d required for this method are

c = (R–1)a + (2 –R)b

d = (2–R)a + (R–1)b

with R = .

To find a minimum of the function :

1) set ;

2) do steps 3) to 5) for k = 0, 1, 2 ...

3) search along the direction to minimum point ;

4) set

5) set where

Conjugate Directions Method

To find a minimum of the function :

1) initialize the search directions , i = 1,n to the co-ordinate directions;

2) do steps 3) to 6) n times;

3) starting from search in each of the directions , i = 1, n in turn to reach ;

4) put , i = 1, n-1;

5) put ;

6) search in this new direction to reach new ;

7) after n iterations of steps 3) to 6) reinitialize the search directions , i = 1,n to the co-ordinate directions and return to step 2).

Symmetric Power Method:

To find the dominant eigenvalue of the symmetric matrix A:

1) Choose a vector with unit Euclidean norm.

2) Do the following steps for k = 1, 2, 3, ....

3) Calculate

4) Calculate an approximation to the eigenvalue

5) Normalize

6) Stop if and/or has converged.

Elementary reflectors

The elementary reflector has the property that where and . Here and is the elementary vector with unit y in the first position and zeros everywhere else.

Legendre polynomials

P0 = 1; P1 = x; P2 = (3x2 – 1); P3 = (5x3 – 3x)