1. ~ Least Squares example
There are 3 mountains u,y,z that from one site have been measured as 2474 ft., 3882 ft., and 4834 ft.. But from u, y looks 1422 ft. taller and the z looks 2354 ft. taller, and from y, z looks 950 ft. taller. Set up the overdetermined system.
2474
3882
4834
1422
2354
950 ~ u y z =b
Ax=
Want to minimize ||Ax-b||2 1

2. Exponential example Assume you are given the following data:1 2 3 4 5 6 7 8 9 10 t
1.3
0.85
0.5
0.4
0.3
0.2
0.15
0.14
0.12
0.11 b
Which you think fits the model
b = ut + w e-.5t
for some values of u and w which you wish to determine so that
The Matrix A would look like:

3. Approaches to solve Ax b
Normal equations-quick and dirty
QR- standard in libraries uses orthogonal decomposition
SVD - decomposition which also gives indication how linear independent columns are
Conjugate gradient- no decompositions, good for large sparse problems

4. Quick and Dirty Approach
Multiply by AT to get the normal equations:
AT A x = AT b
For the mountain example the matrix AT A is 3 x 3.
The matrix AT A is symmetric .
However, sometimes AT A can be nearly singular or singular.
Consider the matrix A =
e 0
0 e
The matrix AT A = 1+ e
1 1+ e2
becomes singular if e is less than the square root of the machine precision.

5. Normal equations for least squares - slide 2
For the hill problem: u
AT A x = y = = ATb
x 8138
Solution is u =2472, y= 3886, z=4816.

6. QR: A more stable approach
Assume the m x n matrix A and the m-vector b can be partitioned into
A = R and b = c d
where R is a nonsingular n x n matrix and c has length n.
Then ||Ax-b|| = ||Rx -c || || d ||22
So pick x such that ||Rx -c ||22
which implies ||Ax-b||22 = || d ||22 .This is the best one can do.

7. QR for least squares-slide 2
But most A matrices do not have this form, but using orthogonal transformations we can transform matrices of full column rank to this form.
A matrix Q is orthogonal if QTQ=I.
If Q is orthogonal then for any x, ||Qx ||22 =||x ||22 ,
that is an orthogonal matrix preserves the 2 norm.
Examples of orthogonal matrices:
cos y sin y
sin y cos y
for some angle y
Givens
rotations 3

8. QR for least squares -slide 3
We wish to pick a sequence of orthogonal matrices such that A might be transformed into upper triangular form:
x x x x x x x x x x x x
x x x 0 x x 0 x x 0 x x = R
x x x 0 x x x 0 0 x
x x x 0 x x 0 0 x
The transformations are then applied to the data vector b
The solution is now found by solving Rx =c, where c
is the first n elements of the transformed b.

9. QR for least squares -slide 4
Householder transformations of the form I -2uuT / uTu can easily eliminate whole columns.
If the A matrix is almost square the QR approach and the normal equations approach require about the same number of operations.
If A is tall and skinny, the QR approach takes about the twice number of operations.
Most good least squares solvers use the QR approach.
In Matlab: x= A \b.
Good project: Investigate structure of R if A is sparse.

10. Singular Value Decomposition
The singular value decomposition(SVD) of a matrix A is given by
A = USVT
where U and V are orthogonal and S is a diagonal matrix of the form.
If any of the s’s are 0, the matrix is singular. Thus one can determine how close A is to a singular matrix by looking at the smallest s’s.
Good project: Investigate an application of SVD

11. SVD for least squares
If A is an m x n matrix of rank n, then
A =USVT = [ U1 U2 ] VT = U1S1VT
where U1 has the first n rows of U and S1 is n x n.
The solution to the least squares problem Ax b is given by
x=VS1-1U1Tb
Requires 4 to 10 times more work than QR but shows dependencies in model.

12. Conjugate gradient Does not require decomposition of matrix
Good for large sparse problem-like PET
Iterative method that requires matrix vector multiplication by A and AT each iteration
In exact arithmetic for n variables guaranteed to converge in n iterations- so 2 iterations for the exponential fit and 3 iterations for the hill problem.
Does not zigzag

13. Conjugate gradient algorithm
Initialization:
Guess x
Set r= b-Ax , p = Atr ( p will be a direction),
Set γ=pTp
Until convergence
Set q= Ap , α = γ/qTq
Reset x to x + α p
Reset r to r - α q
Set s = Atr , γnew =sTs, β = γnew/ γ , γ = γnew
Reset p to s+ β p.

14. Nonlinear example Assume in the exponential example that the model was
b = u + we-kt with w added to the list of unknowns.
The variable k is nonlinear while u and w are linear.
Possible approaches:
Treat all three variables as nonlinear and use a nonlinear solver
Use a 1-dimensional solver for k and each time a function value is requested one solves for u and w using a linear least squares solver, plug the best value for them in and give back the residual.

15. One dimensional minimization of f(x)
Assumption
Given an interval [a,b], where one “knows” there is a minimum, that f is unimodal on [a,b], i.e. there is only one local minimum.
With this assumption we can find the minimum by sampling and discarding portions of the interval that cannot have the solution.
Best algorithms are combinations of golden section search and quadratic interpolation with 3 points and finding minimum of the quadratic-Brent.

16. Golden section on -(2x3-9x2=12x+6)
Originally a =0, x1= .7639, x2= , b=2.0
Subsequent values of a and b
a b
etc.

17. Unconstrained minimization of f(x) where x has n elements
Steepest descent- requires first derivatives (gradient)
might zigzag
good beginning strategy
sped up by conjugate gradient
Newton- requires first and second derivatives(Hessian)
requires solution of linear system with n variables
fast if close to solution
Quasi-Newton(most practical)- requires first derivative
no linear system to solve
builds up approximation to inverse of Hessian

18. Newton’s method for minimization
Let g = gradient, H = matrix of second partials
Taylor’s Theorem:
f(x+s) f(x) + gTs sTHs.
This quadratic function in s is minimized when
s =- H-1g
Algorithm: guess x
Until convergence
Solve H(x) s =- g(x) {needs H and solver}
Reset x to x + s

19. Quasi- Newton Method
Builds up approximation to Hessian in directions that have been searched
Almost as fast as Newton
Initial: pick x, set B = I.
Until convergence:
set s = - Bg (no linear system to solve)
set xnew = x+s
let γ= g(xnew) -g(x); δ = xnew - x; x= xnew
reset B to B + δ δT/ δT γ - B γ (B γ)T/ γT B γ

20. Comparison of steepest descent, Newton, and quasi Newton on f(x) = 0
Comparison of steepest descent, Newton, and quasi Newton on f(x) = 0.5x x22
Iteration Steepest Newton Quasi-Newton

21. Large Scale Problems Conjugate gradient vs. Limited Memory Quasi-Newton
Conjugate gradient- each step is linear combination of
previous step and current gradient
Limited Memory-(Nocedal,Schnabel, Byrd, Kaufman)
Do not multiply B out but keep vectors. Need to keep 2 vectors per iteration
After k steps(k is about 5) reset B to I and start again .
Experimentation favors LMQN over CG
Good project: How should LMQN be done