1. Math 175: Numerical Analysis II
Lecturer: Jomar Fajardo Rabajante
IMSP, UPLB
2nd Sem AY

2. SOLVING AND SOLVING AGAIN…
Solutions of Linear Equations [high-school/ Math 17]
Solutions of Nonlinear Equations (finding roots of nonlinear equations) [Math 17/Math 175]
Solutions of Linear Systems (Systems of Linear Equations) [high school/Math 17/Math 120/ then later in Math 175]
Solutions of Nonlinear Systems (Systems of Nonlinear Equations) [Math 17/Math175] 
Equations – one dimension; Systems of equations – n-dimension

3. SOLVING AND SOLVING AGAIN…
We will DELAY the discussion of Numerical Solutions to LINEAR SYSTEMS.
Your LABORATORY instructors will discuss Numerical Solutions to NONLINEAR SYSTEMS, specifically applying Multivariate Newton’s Method (similar to our Newton-Raphson but here we will use the concept of Jacobian).

4. NOW,
We will discuss an optional topic which is NUMERICAL OPTIMIZATION (unconstrained)
We will consider two methods:
Golden-section Search (univariate) (lecture class)
Newton’s Method (laboratory class)
Univariate (Math 36)
Multivariate (Math 38)
Of course Exhaustive Search is still applicable but not encouraged. Another famous method is the Steepest Descent.

5. GOLDEN SECTION SEARCH We will only consider MINIMIZATION. Why?
Similar to bisection method. It uses a bracket. It is also linearly but globally convergent.
The function should be UNIMODAL on the interval in consideration (i.e. there is only ONE dip ).

6. GOLDEN SECTION SEARCH
It uses the golden ratio or the divine proportion φ: Theorem: After k steps of Golden Section Search with starting interval [a,b], the midpoint of the final interval is within of the minimum, where
The theorem says that (b-a) is cut approx (38.2%)^k times, then after that get the midpoint.

8. GOLDEN SECTION SEARCH
Algorithm: Given f unimodal with minimum in [a,b]
while (0.5*(g^k)*(b-a)>tol)
if f(a+(1-g)*(b-a))<f(a+g*(b-a))
b=a+g*(b-a)
else
a=a+(1-g)*(b-a)
end
approxmin=(a+b)/2