Examination projects

Consider your examination project like an extra homework. Make the project in a separate folder in your repository with an indicative name (like "exam") which I should easily find. Supply a README.TXT (or similar) file with a short description of what you did and (optionally) a self-evaluation of the project on the scale [0,10].

Remember to check that your repository is publicly browseable and publicly cloneable, and that everything is built.

The list of the projects below will be reshuffled randomly the day when the examination period starts.

The index of the project assigned to you (from the indexed list below) is given by your number (index) in our wiki-list, modulo the number N of the projects (N=31). For example,

if your index is 1, your project number is 1,
if your index is 2, your project number is 2,
... ,
if your index is 31, your project number is 31,
if your index is 32, your project number is 1,
if your index is 33, your project number is 2,
... .

| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 |

— 1 —

Symmetric rank-1 update of a size-n symmetric eigenvalue problem

The matrix A to diagonalize is given in the form

A = D +uu^T,

where D is a diagonal matrix and u is a column-vector.

Given the diagonal elements of the matrix D and the elements of the vector u find the eigenvalues of the matrix A using only O(n²) operations (see section "Eigenvalues of updated matrix" in the book).

— 2 —

Implicit Heun's stepper for Ordinary Differential Equations

The step is given by the formula,

y_x+h = y_x + h/2(f(x,y_x)+f(x+h,y_x+h))

where the vector y_x+h has to be found by solving numerically the above equation using your own root-finder.

The start point for your rootfinding can be the Euler's step,

y_x+h ≈ y_x + h f(x,y_x)

and the error estimate could be the difference between the Euler's step and the implicit Heun's step.

— 3 —

Eigenvalues with Rayleigh quotient and locally optimized gradient descent

Introduction

In mathematics the Rayleigh quotient for a given real symmetric matrix H and a non-zero vector v is defined as

R(H,v)=(v^THv)/(v^Tv) .

It can be shown that the Rayleigh quotient reaches its minimum value λ_min—the smallest eigenvalue of H—when v equals v_min (the corresponding eigenvector),

∀v≠0 R(H,v)≥λ_min, R(H,v_min)=λ_min if Hv_min=λ_minv_min.

Analogously,

∀v≠0 R(H,v)≤λ_max, R(H,v_max)=λ_max if Hv_max=λ_maxv_max.

In quantum mechanics this is called Variational Method.

Project

Implement a function that calculates the lowest eigenvalue and the corresponding eigenvector of a given symmetric matrix H by minimizing its Rayleigh quotient R(H,v) using the components of the vector v as free parameters. The function should use newton's method with analytic gradient and optimized linesearch.

The gradient ∂R/∂v_k is given as

∂R/∂v^T=2[Hv-Rv]/(v^Tv) ∝ (H-R)v .

Therefore one can search for the optimal Newton's step as -αΔv, where

Δv = (H-R)v ,

and where α is to be found by minimizing R(v-αΔv) which is given as

R(v-αΔv) = (v-αΔv)^TH(v-αΔv)/(v-αΔv)^T(v-αΔv)
         = (v^THv-2αΔv^THv+α²Δv^THΔv)/(v^Tv+α²Δv^TΔv) .

This is a one-dimensional minimization that can be done numerically precisely with you own numerical minimizer.

Hint: you might need to keep the norm of your v-vector close to unity.

Investigate the scaling of this method (time to find the eigenvalue as function of the size of the marix) and find out how far in the size of the matrix you can realistically go (that is, within few seconds of calculation time).

Calculate the ground state of the Hydrogen atom.

Calculate also the second smallest eigenvalue.

— 4 —

Cubic (sub-)spline for data with derivatives

Introduction

In practice a function is often tabulated together with its derivative (for example, after numerical integration of a second order ordinary differential equation). The table is then given as {x_i, y_i, y'_i}_i=1,..,n , where y_i is the value of the tabulated function, and y'_i is the value of the derivative of the tabulated function at the point x_i.

Project

Given a data-set, {x_i, y_i, y'_i}_i=1,..,n , build a cubic sub-spline,

S(x) _{x∈[x_i,x_i+1]} =S_i(x)

where

S_i(x)= y_i +b_i(x-x_i) +c_i(x-x_i)² +d_i(x-x_i)³.

For each interval the three coefficients b_i, c_i, d_i are determined by the three conditions,

S_i(x_i+1)=y_i+1,
S'_i(x_i)=y'_i,
S'_i(x_i+1)=y'_i+1.

See the subsection "Akima sub-spline interpolation" for the inspiration.

Extra

Derivative and integral of the spline.
Make the second derivative of the spline continuous by increasing the order of the spline to four, for example, in the form
S_i(x)= y_i +b_i(x-x_i) +c_i(x-x_i)² +d_i(x-x_i)³ +e_i(x-x_i)²(x-x_i+1)² ,
and choosing the coefficients e_i such that the spline has continuous second derivative.

— 5 —

Adaptive recursive integrator with subdivision into three subintervals.

Implement a (one-dimensional) adaptive recursive integrator which at each iteration subdivides the interval not into two, but into three sub-intervals. Reuse points. Compare its effectiveness with the adaptive integrator from your homework.

For example, a rule like this (check that the numbers are correct),

x_i={1/6,3/6,5/6} reusable points for division by 3;
w_i={3/8,2/8,3/8} trapezium rule; (check this)
v_i={1/3,1/3,1/3} rectangle rule; (check this)

— 6 —

Particle swarm optimization

Implement the particle swarm optimization algorithm.

— 7 —

Least-squares signal declipping

Implement the least-squares signal declipping method.

— 8 —

Accelerated Particle Swarm Optimization

Implement the Accelerated Particle Swarm Optimization algorithm (APSO).

In this variant of Particle Swarm Optimization (PSO) algorithm one dispenses with the particle's velocity and the particle's best position and instead updates the position of the particle according to the following rule,

x_i ← (1-β)x_i+βg+αv ,

where x_i is the particle's position, g is the global best, β~0.1-0.7 and α~0.1-0.5 are the parameters of the algorithm, and v is a random vector given as v_k=U(-(b-a)_k,(b-a)_k) where a and b are the "lower-left" and "upper-right" corners of the search volume and U(c,d) is a uniform distribution from c to d.

— 9 —

Adaptive two-dimensional integrator

Implement a two-dimensional integrator for integrals in the form

∫_{_a}^{^b}dx ∫_{_d(x)}^{^u(x)}dy f(x,y)

which applies your favourite adaptive one-dimensional integrator along each of the two dimensions. The signature might be something like

static double integ2D(
	Func<double,double,double> f,
	double a, double b,
	Func<double,double> d,
	Func<double,double> u,
	double acc, double eps)

— 10 —

Inverse iteration algorithm for eigenvalues (and eigenvectors)

Implement the inverse iteration algorithm that calculates the eigenvalue that is closest to the given shift-value s (and the corresponding eigenvector) of the given symmetric matrix A. Use your own linear algebra routines.

Inverse iteration algorithm allows calculation of the eigenvalue (and the corresponding eigenvector) that is closest to a given value. Read the book and/or the [inverse iteration] article in Wikipedia.

The interface could be like this (check everything),

(double,vector) inverse_iteration (matrix A,double s,vector x0=null) :
	if x0==null : x0 = random_vector
	Q,R = qrdecomp(A-s*1)
	x1 = qrsolve(Q,R,x0)
	λ1 = s + (x1•x0)/(x1•x1)
	do :
		x0 = x1
		x1 = qrsolve(Q,R,x0)
		λ0 = λ1
		λ1 = s + (x1•x0)/(x1•x1)
		if abs(λ1-λ0) < acc : break
	while true
	return (λ1,x1)

— 11 —

LU decomposition of a real square matrix

Implement the LU-decomposition of a real square matrix.

Implement also the corresponding functions to solve linear equations, calculate the determinant, and calculate the inverse matrix.

Is your LU faster than your QR?

— 12 —

Lanczos tridiagonalization algorithm

Lanczos tridiagonalization is the transformation of a real symmetric N×N matrix A into the form V^TAV=T (where V is an N×n orthonormal matrix and T is an n×n tridiagonal matrix with n≤N) using power iteration and Gram-Schmidt orthogonalization. Read the book and/or the Wikipedia article. The algorithm is often used as the first step in matrix diagonalization. Particularly with lossy (with n<N) compression that preserves extreme eigenvalues.

Implement the Lanczos tridiagonalization algorithm for real symmetric matrices.

Consider the hydrogen atom using our discretization method with the given Δr and r_max=NΔr. Use the Lanczos iteration algorithm and build a size-n (where n≤N) tridiagonal representation T of the Hamiltonian. Calculate the ground state energy of the matrix T. Investigate the convergence of the ground state energy as function of n.

Check whether Jacobi eigenvalue algorithm can be tuned to take advantage of the tridiagonal form of a matrix.

— 13 —

Least-squares signal smoothing

Implement least-squares signal smoothing as described in the Book.

— 14 —

Hessenberg factorization of a real square matrix using Jacobi transformations

Definitions

A lower Hessenberg matrix H is a square matrix that has zero elements above the first sup-diagonal, {H_ij=0}_j>i+1.

An upper Hessenberg matrix H is a square matrix that has zero elements below the first sub-diagonal, {H_ij=0}_i>j+1.

Hessenberg factorization is a representation of a square real matrix A in the form A=QHQ^T where Q is an orthogonal matrix and H is a Hessenberg matrix.

If A is a symmetrix matrix its Hessenberg factorization produces a tridiagonal matrix H.

Why Hessenberg matrix?

If the constraints of a linear algebra problem do not allow a general matrix to be conveniently reduced to a triangular form, reduction to Hessenberg form is often the next best thing. Reduction of any matrix to a Hessenberg form can be achieved in a finite number of steps. Many linear algebra algorithms require significantly less computational effort when applied to Hessenberg matrices.

Hessenberg factorization by Jacobi transformations

Consider a Jacobi transformation with the matrix J(p,q),

A→J(p,q)^TAJ(p,q)

where the rotation angle is chosen not to zero the element A_p,q but rather to zero the element A_p-1,q. Argue that the following sequence of such Jacobi transformations,

J(2,3), J(2,4), ..., J(2,n); J(3,4), ..., J(3,n); ...; J(n-1,n)

reduces the matrix to the lower Hessenberg form.

Project

Implement Hessenberg factorization of a square matrix using Jacobi transformations. Remember to accumulate the total transformation matrix. When multiplying with the J-matrices you should use your Jtimes and timesJ routines from Jacobi-eigenvalue homework that take O(n) operations.

Extra

Find out (by time measurements) which factorization is faster, Hessenberg or QR.

Check if QR-factorization of an upper Hessenberg matrix using Givens rotations can be done in O(n²) operations.

Check if Jacobi eigenvalue algorithm applied cleverly (eigenvalue-by-eigenvalue, and don't zero the elements that are already zero) to a tridiagonal matrix takes O(n) operations.

— 15 —

Bi-linear interpolation on a rectilinear grid in two dimensions

Introduction

A rectilinear grid (note that rectilinear is not necessarily cartesian nor regular) in two dimensions is a set of n_x×n_y points where each point can be adressed by a double index (i,j) where 1 ≤ i ≤ n_x, 1 ≤ j ≤ n_y and the coordinates of the point (i,j) are given as (x_i,y_j), where x and y are vectors with sizes n_x and n_y correspondingly. The values of the tabulated function F at the grid points can then be arranged as a matrix {F_i,j=F(x_i,y_j)}.

Problem

Build an interpolating routine which takes as the input the vectors {x_i} and {y_j}, and the matrix {F_i,j} and returns the bi-linear interpolated value of the function at a given 2D-point p=(p_x,p_y).

Hints

See the chapter "Bi-linear interpolation" in the book.

The signature of the interpolating subroutine can be

static double bilinear(double[] x, double[] y, matrix F, double px, double py)

— 16 —

One-sided Jacobi algorithm for Singular Value Decomposition

Introduction

The singular value decomposition (SVD) of a (real square, for simplicity) matrix A is a representation of the matrix in the form

A = U D V^T ,

where matrix D is diagonal with non-negative elements and matrices U and V are orghogonal. The diagonal elements of matrix D can always be chosen non-negative by multiplying the relevant columns of matrix U with (-1).

SVD can be used to solve a number of problems in linear algebra.

Problem

Implement the one-sided Jacobi SVD algorithm.

Algorithm

In this method the elementary iteration is given as

A → A J(θ,p,q)

where the indices (p,q) are swept cyclicly (p=1..n, q=p+1..n) and where the angle θ is chosen such that the columns number p and q of the matrix AJ(θ,p,q) are orthogonal. One can show that the angle should be taken from the following equation (you should use atan2 function),

tan(2θ)=2a_p^Ta_q /(a_q^Ta_q - a_p^Ta_p)

where a_i is the i-th column of matrix A (check this).

After the iterations converge and the matrix A'=AJ (where J is the accumulation of the individual rotations) has orthogonal columns, the SVD is simply given as

A=UDV^T

where

V=J, D_ii=||a'_i||, u_i=a'_i/||a'_i||,

where a'_i is the i-th column of matrix A' and u_i us the i-th column of matrix U.

— 17 —

Gauss-Newton algorithm

Implement the Gauss-Newton algorithm.

— 18 —

Fast Fourier Transform (FFT)

Implement the (naïve, with complex numbers) Cooley-Tukey algorithm for FFT. Check that the scaling is indeed O(Nlog(N)).
Implement an application of FFT at your choice.
Get rid of the array copying in the naïve implementation.

— 19 —

Bare Bones Particle Swarm Optimization

Implement the Bare Bones Particle Swarm Optimization (BBPSO) algorithm (see the book).

In this variant of the Particle Swarm Optimization (PSO) algorithm one dispences with the particle's velocity and instead updates the position of the particle according to the following rule,

x_i ← G(^(p_i+g)/₂ , ‖p_i-g‖) ,

where x_i is the particle's position, p_i is the particle's best position, g is the global best, ‖…‖ signify the norm of the vector, and G(m,σ) is the multi-dimensional Gaussian (normal) distribution with the mean m and variance σ.

Hint: The normal distribution can be approximated as n=12 Irwin-Hall distribution (read the book),

G(0,1)≈Σ_{_i=1…12}u(0,1) - 6

where u(0,1) is a rangom number from the uniform [0,1] distribution.

— 20 —

Yet another cubic sub-spline

Introduction

The ordinary cubic spline simetimes makes unpleasant wiggles, for example, around an outlier, or around a step-like feature of the tabulated function (read the Akima sub-spline chapter in the lecture notes). Here is yet another attempt to reduce the wiggles by building a sub-spline.

Project

Consider a data set {x_i, y_i}_i=1,..,n which represents a tabulated function.

Implement a sub-spline of this data using the following algorithm:

For each inner point x_i=2,..,n-1, build a quadratic interpolating polynomial through the points x_i-1, x_i, x_i+1 and estimate, using this polynomial, the first derivative, p_i, of the function at the point x_i;
For the the first and the last points estimate p₁ and p_n from the the same polynomial that you used to estimate correspondingly p₂ and p_n-1.
Now you have a data set {x_i, y_i, p_i}_i=1,..,n where the function is tabulated together with its derivative: build a cubic sub-spline for it,
S_i(x)= y_i +b_i(x-x_i) +c_i(x-x_i)² +d_i(x-x_i)³,
where for each interval the three coefficients b_i, c_i, d_i are determined by the three conditions,
S_i(x_i+1)=y_i+1,
S'_i(x_i)=p_i,
S'_i(x_i+1)=p_i+1.

Extra

Derivative and integral of the spline.

— 21 —

Berrut interpolation

Implement the Berrut B₁ rational function interpolation algorithm.

— 22 —

Cholesky decomposition of a real symmetric positive-definite matrix

For a real symmetric positive-definite matrix, A, the Cholesky decomposition, A=LL^T (where L is lower-triangular), is more efficient than the LU-decomposition. Read the book and/or Wikipedia article Cholesky-decomposition.

Implement Cholesky decomposition of a real symmetric positive-definite matrix using Cholesky-Banachiewicz or Cholesky-Crout algorithms.

Try implement also the corresponding linear equation solver, calculation of determinant, and calculation of the inverse matrix.

— 23 —

Akima sub-spline

Implement the Akima (sub-)spline interpolation. See the section "Akima sub-spline interpolation" in the book for the inspiration.

— 24 —

Eigenvalues via rootfinding: Variational method with Lagrange multipliers

Introduction

One can find an eigenvalue and the corresponding eigenvector of a real symmetric matrix A by searching for the stationary point of the following constrained form

E(v)=v^TAv , where v^Tv=1 .

This can be done using the method of Lagrange multipliers: the original problem is reformulated into unconstrained search for the stationary point (the point where all partial derivatives are zero) of the Lagrangian function defined as

L(v,λ) = v^TAv - λ(v^Tv-1) .

in the space {v,λ}. At the stationary point: λ is the eigenvalue and v is the corresponding eigenvector normalized to unity. Indeed, at the stationary point

∂L(v,λ)/∂v = 2(Av - λv) = 0 ,
∂L(v,λ)/∂λ = v^Tv-1 = 0 .

Of course this is equivalent to finding the root in the n+1 dimensional space x={v,λ} of the n+1 dimensional vector-function

f(x) = {Av-λv, v^Tv-1}

where n is the size of v.

Note that the Jacobian of this function is analytic,

∂f_i(x)/∂v_j = A_ij-λδ_ij , where i,j≤n ,
∂f_i(x)/∂λ = -v_i , where i≤n ,
∂f_n+1(x)/∂v_j = 2v_j , where j≤n 
∂f_n+1(x)/∂λ = 0 .

The analytic Jacobian greatly facilitates the search for the root.

Project

Implement a subroutine that calculates the eigenvalue (close to a given value λ_start) and the corresponding eigenvector of a given symmetric matrix A by searching for the root of the function

f(v,λ) = {Av-λv, v^Tv-1}

using the components of the vector v and the (Lagrangian multiplier) λ as free parameters. Use your own Newton's rootfinding routine and customize it to use analytic Jacobian.

Extra

Calculate several lowest state of the Hydrogen atom.

It seems that the line-search here can be done analytically, you might want to try to investigate this possibility.

— 25 —

Least-squares signal extrapolation (linear prediction)

Implement the least-squares signal extrapolation method (linear prediction).

— 26 —

Two-sided Jacobi algorithm for Singular Value Decomposition (SVD)

Introduction

The SVD of a (real square, for simplicity) matrix A is a representation of the matrix in the form

A = U D V^T ,

SVD can be used to solve a number of problems in linear algebra.

Problem

Implement the two-sided Jacobi SVD algorithm.

Algorithm (as described in the "Eigenvalues" chapter of the book)

In the cyclic Jacobi eigenvalue algorithm for symmetric matrices we applied the elementary Jacobi transformation

A → J^T A J

where J ≐ J(θ,p,q) is the Jacobi matrix (where the angle θ is chosen to eliminate the off-diagonal elements A_pq=A_qp) to all upper off-diagonal matrix elements in cyclic sweeps until the matrix becomes diagonal.

The two-sided Jacobi SVD algorithm for general real square matrices is the same as the Jacobi eigenvalue algorithm, except that the elementary transformation is slightly different,

A → J^T G^T A J ,

where

G ≐ G(θ,p,q) is the Givens (Jacobi) matrix where the angle is chosen such that the matrix A' ≐ G^T(θ,p,q) A has identical off-diagonal elements, A'_pq=A'_qp:
tan(θ) = (A_pq - A_qp)/(A_qq + A_pp)
(check that this is correct).
J ≐ J(θ,p,q) is the Jacobi matrix where the angle is chosen to eliminate the off-diagonal elements A'_pq=A'_qp.

The matrices U and V are updated after each transformation as

U ← U G J , V ← V J .

Of course you should not actually multiply Jacobi matrices—which costs O(n³) operations—but rather perform updates which only cost O(n) operations.

— 27 —

ODE: a two-step method

Implement a two-step stepper for solving ODE (as in the book). Use your own adaptive stepsize driver.

— 28 —

Symmetric row/column update of a size-n symmetric eigenvalue problem

The matrix to diagonalize is given in the form

A = D + e(p) u^T + u e(p)^T

where D is a diagonal matrix with diagonal elements {d_k, k=1,...,n}, u is a given column-vector, and the vector e(p) with components e(p)_{_i}=δ_ip is a unit vector in the direction p where 1≤p≤n.

Given the diagonal elements of the matrix D, the vector u, and the integer p, calculate the eigenvalues of the matrix A using O(n²) operations (see section "Eigenvalues of updated matrix" in the book).

— 29 —

Least-squares missing samples recovery (error concealment)

Implement the least-squares missing samples recovery method.

— 30 —

Yet another stochastic global optimizer

Implement a stochastic global optimizer using the following algorithm,

Use the given number of seconds (or the number of samples) to search for the global minimum of the given cost-function in the given volume by sampling the function using a low-discrepancy sequence.
From the best point found at the previous step run your favourite local minimizer.

— 31 —

Generalized eigenvalue problem

Implement a function to solve a generalized eigenvalue problem AV=BVE (where A is a real symmetric matrix, B is a real symmetric positive definite matrix, V is the matrix of the generalized eigenvectors and E is the diagonal matrix with the generalized eigenvalues) via diagonalization of the matrix B. Specifically, the diagonalization of matrix B, given as B=QSQ^T, where S is a diagnoal matrix with positive elements and Q is an orthogonal matrix, turns the eigenproblem
```
AV=BVE
```
into
```
AV=QSQ^TVE .
```
This can be identically rewritten as
```
S^-½Q^TAQS^-½ S^½Q^TV = S^½Q^TV E ,
```
which is an ordinary real symmetric eigenvalue problem
```
Ã X = X E ,
```
where
```
Ã = S^-½Q^TAQS^-½ ,
```
and
```
X = S^½Q^TV .
```
Diagonalization of matrix Ã gives directly the eigenvalues E while the original eigenvectors can be calculated as
```
V = QS^-½X .
```
(Note that since S is diagonal so is S^½ and S^-½ and therefore you don't need to build this matrices explicitly and waste O(n³) operations for matrix multiplication.)
Find the ground state of the hydrogen atom with the Schrödinger equation
```
(-¹/₂ ^d²/_dr² - ¹/_r)u(r) = εu(r),
```
using the variational method,
```
u(r) = ∑_{_i=1..n}c_iφ_i(r)
```
with the basis functions φ_i(r) in the form (satisfying the boundary conditions u(0)=u(∞)=0)
```
φ_i(r) = r e^-α_ir²
```
where α_i are the variational parameters (to be found by your own minimization routine) and where the coefficients c_i should be found by solving the generalized eigenvalue problem
```
H c = ε N c
```
where H and N are matrices with the elements
```
H_ij = ∫_₀^{^∞}dr φ_i(r) (-¹/₂ ^d²/_dr² - ¹/_r) φ_j(r) ,
N_ij = ∫_₀^{^∞}dr φ_i(r) φ_j(r) .
```
The matrix elements are actually analytic (check the formulae before using),
```
∫_₀^{^∞}dr r e^-αr² r e^-βr² = 1/4 √π (α+β)^-3/2 ,
```
```
∫_₀^{^∞}dr r e^-αr² (^d²/_dr²) r e^-βr² = -3/2 √π αβ (α+β)^-5/2 ,
```
```
∫_₀^{^∞}dr r e^-αr² (¹/_r) r e^-βr² = 1/2 (α+β)^-1 .
```
Try n=1,2,3,... gaussians in your basis and investigate the convergence. The method will probably break down at about 5 Gaussians due to their excessive overlap.
Implement a function to solve the generalized eigenvalue problem via Cholesky decomposition of the matrix B.