MATH4141/MATH6651/PHYS5070

Formula Sheet

**Fixed-Point Theorem in R ^{n} :**

Let . Suppose is a continuous function with
continuous partial derivatives fromR^{n} into R^{n} 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 *J _{g} *is the Jacobian of

converges to the fixed point provided .

Newton's method in R^{n} :

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 = .

Conjugate Gradient Method

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

P_{0} = 1; P_{1} = x; P_{2
}= (3x^{2} – 1);
P_{3} = (5x^{3} – 3x)