Introduction to Regular Expressions

Introduction to Regular Expressions

• A regular expression, also called a pattern, is an expression used to specify a set of strings required for a particular purpose.

• Check this: https://regexone.com.

1See https://en.wikipedia.org/wiki/Regular_expression; also https://www.mathworks.com/help/matlab/matlab_prog/

regular-expressions.html.

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Example

1 >> text = 'bat cat can car coat court CUT ct ...

CAT-scan';

2 >> pattern = 'c[aeiou]+t';

3 >> start idx = regexp(text, pattern)

4

5 start idx =

6

7 5 17

• The pattern ’c[aeiou]+t’ indicates a set of strings:

• c must be the first character;

• c must be followed by one of the characters in the brackets [aeiou], followed by t as the last character;

• in particular, [aeiou] must occur one or more times, as indicated by the + operator.

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Metacharacters

Operator Definition

| Boolean OR.

* 0 or more times consecutively.

? 0 times or 1 time.

+ 1 or more times consecutively.

{n} exactly n times consecutively.

{m, } at least m times consecutively.

{, n} at most n times consecutively.

{m, n} at least m times, but no more than n times consecutively.

2See https://www.mathworks.com/help/matlab/ref/regexp.html.

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Operator Definition

. any single character, including white space.

[c1c2c3] any character contained within the brackets.

[∧c₁c₂c₃] any character not contained within the brackets.

[c₁-c₂] any character in the range of c₁ through c₂.

\s any white-space character.

\w a word; any alphabetic, numeric, or underscore character.

\W not a word.

\d any numeric digit; equivalent to [0-9].

\D no numeric digit; equivalent to [∧0-9].

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Output Keywords

Keyword Output

’start’ starting indices of all matches, by default

’end’ ending indices of all matches

’match’ text of each substring that matches the pattern

’tokens’ text of each captured token

’split’ text of nonmatching substrings

’names’ name and text of each named token

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Examples

1 clear; clc;

2

3 text1 = {'Madrid, Spain', 'Romeo and Juliet', ...

'MATLAB is great'};

4 tokens = regexp(text1, '\s', 'split')

5

6 text2 = 'EXTRA! The regexp function helps you ...

relax.';

7 matches = regexp(text2, '\w*x\w*', 'match')

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Exercise: Listing Filtered Files

1 clear; clc;

2

3 file list = dir;

4 filenames = {file list(:).name};

5 A = regexp(filenames, '.+\.m', 'match');

6 mask = cellfun(@(x) ~isempty(x), A);

7 cellfun(@(f) fprintf('%s\\%s\n', pwd, f{:}), A(mask))

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Example: By Names

• You can associate names with tokens so that they are more easily identifiable.

• For example,

1 >> str = 'Here is a date: 01-Apr-2020';

2 >> expr = '(?<day>\d+)-(?<month>\w+)-(?<year>\d+)';

3 >> mydate = regexp(str, expr, 'names')

4

5 mydate =

6

7 day: '01'

8 month: 'Apr'

9 year: '2020'

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Exercise: Web Crawler

• Write a script which collects the names of html tags by defining a token within a regular expression.

• For example,

1 >> str = '<title>My Title</title><p>Here is some ...

text.</p>';

2 >> pattern = '<(\w+).*>.*</\1>';

3 >> [tokens, matches] = regexp(str, pattern, ...

'tokens', 'match')

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More Regexp Functions

• See regexpi, regexprep, and regexptranslate.

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1 >> Lecture 6

2 >>

3 >> -- Special Topic: File Operations & other I/O

4 >>

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Spreadsheets: Excel/CSV Files (Revisited)

• The command xlsread(filename) reads excel files, for example,

1 [~, ~, raw] = xlsread("2330.xlsx");

• By default, it returns a numeric matrix.

• The text part is the 2nd output, separated from the numeric part.

• You may consider the whole spreadsheet by using the 3rd output (stored in a cell array).

• Note that you can use ∼ to drop the output value.

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More Tips for Excel Files

• You can specify the range.

• For example, the string argument"B:B"is used to import column B.

• If you need a single value, say the cell B1, just use"B1:B1".³

• You could specify the worksheet by the sheet name⁴ or the sheet number.

• You could refer to the document for more details.⁵

3Contribution by Mr. Tsung-Yu Hsieh (MAT24409) on August 27, 2014.

4The default sheet name is “工作表”.

5See https://www.mathworks.com/help/matlab/ref/xlsread.html.

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Mat Files

• Recall that I/O is costly.

• To save time, you may consider save matrices to the disk; for example,

1 data1 = rand(1, 10);

2 data2 = ones(10);

3 save('trial.mat', 'data1', 'data2');

• You can use load to fetch the data from mat files.

1 load('trial.mat');

6See https://www.mathworks.com/help/matlab/ref/save.html.

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Selected Read/Write Functions

• For text data, see https:

//www.mathworks.com/help/matlab/text-files.html.

• Try dlmread, dlmwrite, csvread, csvwrite, textread/textscan.

• For images, see https://www.mathworks.com/help/

matlab/images_images.html.

• For video and audio, see https://www.mathworks.com/

help/matlab/audio-and-video.html.

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Selected File Operations

cd Change current folder.

pwd Identify current folder.

ls List folder contents by chars.

dir List folder contents by structures.

exist Check existence of variable, script, function, folder, or class.

mkdir Make new folder.

visdiff Compare two files or folders.

7See

https://www.mathworks.com/help/matlab/file-operations.html.

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Example: Pooling Data from Multiple Files

1 clear; clc;

2

3 cd('./stocks'); % enter the folder

4 files = dir; % get all files in the current folder

5 files = files(3 : end); % drop the first two

6 names = {files(:).name}; % get all file names

7 filter = endsWith(names, '.xlsx'); % filter by .xlsx

8 names = names(filter);

9

10 pool = cell(length(names), 2);

11 for i = 1 : length(names)

12 [~, ~, raw] = xlsread(names{i});

13 pool(i, :) = {names{i}(1 : 4), raw};

14 end

15 save('data pool', 'pool');

8Downloadstocks.zip.

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1 >> Lecture 7

2 >>

3 >> -- Matrix Computation

4 >>

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Vectors

• Let R be the set of all real numbers.

• Rⁿ denotes the vector space of all m-by-1 column vectors:

u = (u_i) =





 u1

... u_m





. (1)

• You can simply use the colon (:) operator to reshape any array in a column major, say u(:).

• Similarly, the row vector v is v = (v_i) =

v1· · · v_n  . (2)

• We consider column vectors unless stated.

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Matrices

• Mm×n(R) denotes the vector space of all m-by-n real matrices, for example,

A = (aij) =







a₁₁ · · · a_1n ... . .. ... a_m1 · · · a_mn





.

• Complex vectors/matrices⁹ follow similar definitions and operations introduced later, simply with some care.

9Matlab treats a complex number as a single value.

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Transposition

1 >> A = [1 i];

2 >> A' % Hermitian operator; see any textbook for ...

linear algebra

3

4 ans =

5

6 1.0000 + 0.0000i

7 0.0000 - 1.0000i

8

9 >> A.' % transposition of A

10

11 ans =

12

13 1.0000 + 0.0000i

14 0.0000 + 1.0000i

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Arithmetic Operations

• Let aij and bij be the elements of the matrices A and B ∈ M^m×n(R) for 1 ≤ i ≤ m and 1 ≤ j ≤ n.

• Then C = A ± B can be calculated by cij = aij ± b_ij. (Try.)

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Inner Product

• Let u, v ∈ R^m.

• Then the inner product, denoted by u · v , is calculated by

u · v = u⁰v = [u1· · · u_m]





 v1

... v_m





.

1 clear; clc;

2

3 u = [1; 2; 3];

4 v = [4; 5; 6];

5 u' * v % normal way; orientation is important

6 dot(u, v) % using the built-in function

10Akaa dot product and scalar product.

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• Inner product is also calledprojection for emphasizing its geometric significance.

• Recall that we know

u · v = 0

if and only if these two are orthogonal to each other, denoted by

u ⊥ v .

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Generalization of Inner Product

• Let x ∈ R, f (x) and g (x) be real-valued functions.

• In particular, assume that g (x ) is a basis function.¹¹

• Then we can define the inner product of f and g on [a, b] by

hf , g i = Z b

a

f (x )g (x )dx .

11See https://en.wikipedia.org/wiki/Basis_function, https://en.wikipedia.org/wiki/Eigenfunction, and https://en.wikipedia.org/wiki/Approximation_theory.

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• For example,Fourier transformis widely used in engineering and science.

• Fourier integral¹² is defined as

F (ω) = Z ∞

−∞

f (t)e^{−i ωt}dt

where f (t) is a square-integrable function.

• The Fast Fourier transform (FFT) algorithm computes the discrete Fourier transform (DFT) in O(n log n) time.^13,14

12See https://en.wikipedia.org/wiki/Fourier_transform.

13Cooley and Tukey (1965).

14See https://en.wikipedia.org/wiki/Fast_Fourier_transform.

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Matrix Multiplication

• Let A ∈ M_m×q(R) and B ∈ Mq×n(R).

• Then C = AB is given by

c_ij =

q

X

k=1

a_ik× b_kj. (3)

• For example,

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Example

1 clear; clc;

2

3 A = randi(10, 5, 4); % 5-by-4

4 B = randi(10, 4, 3); % 4-by-3

5 C = zeros(size(A, 1), size(B, 2));

6 for i = 1 : size(A, 1)

7 for j = 1 : size(B, 2)

8 for k = 1 : size(A, 2)

9 C(i, j) = C(i, j) + A(i, k) * B(k, j);

10 end

11 end

12 end

13 C % display C

• Time complexity: O(n³).

• Strassen (1969): O(n^log27).

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Matrix Exponentiation

• Raising a matrix to a power is equivalent to repeatedly multiplying the matrix by itself.

• For example, A²= AA.

• The matrix exponential¹⁵is a matrix function on square matrices analogous to the ordinary exponential function, more explicitly,

e^A=

∞

X

n=0

Aⁿ n!.

• However, it is not allowedto perform A^B.

15Seematrix exponentialsandPauli matrices.

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Determinants

• Consider the matrix

A =a b c d

 .

• Then det(A) = ad − bc is called the determinant of A.

• The method of determinant calculation in high school is a wrong way but produces correct answers for all 3 × 3 matrices.

• Let’s try the minor expansion formula for det(A).¹⁶

16See http://en.wikipedia.org/wiki/Determinant.

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Recursive Algorithm for Minor Expansion Formula

1 function y = myDet(A)

2

3 [r, ~] = size(A);

4

5 if r == 1

6 y = A;

7 elseif r == 2

8 y = A(1, 1) * A(2, 2) - A(1, 2) * A(2, 1);

9 else

10 y = 0;

11 for i = 1 : r

12 B = A(2 : r, [1 : i - 1, i + 1 : r]);

13 cofactor = (-1) ˆ (i + 1) * myDet(B);

14 y = y + A(1, i) * cofactor;

15 end

16 end

17 end

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• It needs n! terms in the sum of products, so this algorithm runs in O(n!) time!

• Use det for determinants, which can be done in O(n³) time by using LU decomposition or alike.¹⁷

17See https://en.wikipedia.org/wiki/LU_decomposition. Moreover, various decompositions are used to implement efficient matrix algorithms in numerical analysis. See

https://en.wikipedia.org/wiki/Matrix_decomposition.

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Linear Systems (Transformation/Mapping)

• A linear system is a mathematical model of a system based on linear operators satisfying the property of superposition.

• For simplicity, Ax = y for any input x associated with the output y .

• Then A is alinear operatorif and only if

A(ax1+ bx2) = aAx1+ bAx2= ay1+ by2

for a, b ∈ R.

• For example, ^{d (x2}_dx^{+3x )} = ^dx_dx² + 3^dx_dx = 2x + 3.

• Linear systems typically exhibit features and properties that are much simpler than the nonlinear case.

• What about nonlinear cases?

18See https://en.wikipedia.org/wiki/Linear_system.

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First-Order Approximation: Local Linearization

• Let f (x ) be any nonlinear function.

• Assume that f (x ) is infinitely differentiable at x₀.

• By Taylor’s expansion¹⁹, we have

f (x ) = f (x₀) + f⁰(x₀)(x − x₀) + O (x − x₀)²
, where O (x − x0)²
is the collection of higher-order terms, which can be neglected as x − x₀ → 0.

• Then we have a first-order approximation f (x ) ≈ f⁰(x0)x + k, with k = f (x₀) − x₀f⁰(x₀), a constant.

19See https://en.wikipedia.org/wiki/Taylor_series.

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Two Observations

• We barely feel like the curvature of the ground; however, we look at Earth on the moon and agree that Earth is a sphere.

• Newton’s kinetic energy is a low-speed approximation (classical limit) to Einstein’s total energy.

• Let m be the rest mass and v be the velocity relative to the inertial coordinate.

• The resulting total energy is

E = mc²

p1 − (v/c)².

• By applying the first-order approximation,

E ≈ mc²+1 2mv².

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Example: Kirchhoff’s Laws

• The algebraic sum of currents in a network of conductors meeting at a point is zero.

• The directed sum of the potential differences (voltages) around any closed loop is zero.

20See https://en.wikipedia.org/wiki/Kirchhoff’s_circuit_laws.

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General Form of Linear Equations

• Let n be the number of unknowns and m be the number of constraints.

• A general system of m linear equations with n unknowns is











a₁₁x₁ +a₁₂x₂ · · · +a_1nx_n = y₁ a21x1 +a22x2 · · · +a2nxn = y2

... ... . .. ... = ... a_m1x₁ +a_m2x₂ · · · +a_mnx_n = y_m where x1, . . . , xn are unknowns, a11, . . . , amn are the coefficientsof the system, and y1, . . . , ym are theconstant terms.

21See https://en.wikipedia.org/wiki/System_of_linear_equations.

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Matrix Equation

• Hence we can rewrite the aforesaid equations as follows:

Ax = y . where

A =











a11 a12 · · · a1n

a₂₁ a₂₂ · · · a_2n ... ... . .. ... am1 am2 · · · amn









 ,

x =





 x₁

... x_n





, and y =





 y₁

... y_m





.

• Finally, x can be done by x = A⁻¹y , where A⁻¹ is called the inverse of A.

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Inverse Matrices

• For simplicity, let A ∈ M_n×n(R) and x, y ∈ Rⁿ.

• Then A is called invertible if there exists B ∈ Mn×n(R) such that

AB = BA = In, where I_n denotes a n × n identity matrix.

• We use A⁻¹ to denote the inverse of A.

• You can use eye(n) to generate an identity matrix In.

• Use inv(A) to calculate the inverse of A.

22See https://en.wikipedia.org/wiki/Invertible_matrix#The_

invertible_matrix_theorem.

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• However, inv(A) may return a weird result even if A is ill-conditioned, indicates how much the output value of the function can change for a small change in the input

argument.²³

• For example, calculate the inverse of the matrix

A =





1 2 3 4 5 6 7 8 9



.

• Recall the Cramer’s rule²⁴: A is invertible iff det(A) 6= 0.

(Try.)

• If these constraints cannot be eliminated by row reduction, they are linearly independent.

23You may refer to thecondition numberof a function with respect to an argument. Also try rcond.

24See https://en.wikipedia.org/wiki/Cramer’s_rule.

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Linear Independence

• Let K = {a1, a2, . . . , an} for each a_i ∈ R^m.

• Now consider this linear superposition

x₁a₁+ x₂a₂+ · · · + x_na_n= 0, where x₁, x₂, . . . , x_n∈ R are the weights.

• Then K islinearly independent iff

x1 = x2= · · · = xn= 0.

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Example: R

• Let

K₁=









 1 0 0



,



 0 1 0



,



 0 0 1









 .

• It is clear that K₁ is linearly independent.

• Moreover, you can represent all vectors in R³ if you collect all linear superpositions from K1.

• We call this new set a spanof K₁, denoted by Span(K₁).²⁵

• Clearly, Span(K₁) = R³.

25See https://en.wikipedia.org/wiki/Linear_span.

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• Now let

K₂=









 1 0 0



,



 0 1 0



,



 0 0 1



,



 1 2 3









 .

• Then K₂ is not a linearly independent set. (Why?)

• If you takeone or more vectors out of K₂, then K₂ becomes linearly independent.

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Basis of Vector Space & Its Dimension

• However, you can take only one vector out of K₂ if you want to represent all vectors in R³. (Why?)

• Thedimensionof R³ is exactly the size (element number) of K2.

• We say thatthe basis of Rⁿ is a maximally linearly independent set of size n.

• Note that the basis of R³ is not unique.

• For example, K₁ could be also a basis of R³.

26See https://en.wikipedia.org/wiki/Basis_(linear_algebra), https://en.wikipedia.org/wiki/Vector_space, and

https://en.wikipedia.org/wiki/Dimension_(vector_space).

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Linear Transformation (Revisited)

27See https://en.wikipedia.org/wiki/Linear_map; also see https://kevinbinz.com/2017/02/20/linear-algebra/.

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Example: Vector Projection (R

→ R

)

• Let u ∈ R³ and v ∈ R².

• We consider the projection matrix (operator),

A =1 0 0 0 1 0



so that Au = v .

• For example,

1 0 0 0 1 0





 1 2 3



=1 2

 .

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Solution Set to System of Linear Equations

• Recall that m is the number of constraints and n is the number of unknowns.

• Now consider the following cases.

• Ifm = n, then there exists a uniquesolution.

• Ifm > n, then it is called an overdetermined system and there is no solution.

• Fortunately, we can find aleast-squares error solutionsuch that k Ax − y k²is minimal, shown later.

• Ifm < n, then it is called a underdetermined system which hasinfinitely many solutions.

• Become an optimization problem?

• For all cases,

x = A \ y .

28See https://www.mathworks.com/help/matlab/ref/mldivide.html.

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Case 1: m = n

• For example,







3x + 2y − z = 1

x − y + 2z = −1

−2x + y − 2z = 0

1 >> A = [3 2 -1; 1 -1 2; -2 1 -2];

2 >> b = [1; -1; 0];

3 >> x = A \ b

4

5 1

6 -2

7 -2

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Case 2: m > n

• For example,







2x − y = 2

x − 2y = −2

x + y = 1

1 >> A = [2 -1; 1 -2; 1 1];

2 >> b = [2; -2; 1];

3 >> x = A \ b

4

5 1

6 1

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Case 3: m < n

• For example,

 x + 2y + 3z = 7

4x + 5y + 6z = 8

1 >> A = [1 2 3; 4 5 6];

2 >> b = [7; 8];

3 >> x = A \ b

4

5 -3

6 0

7 3.333

• Note that this solution is a basic solution, one of infinitely many.

• How to find the directional vector? (Try cross.)

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Gaussian Elimination Algorithm

• First we consider the linear system is represented as an augmented matrix [ A | y ].

• We then transform A into an upper triangular matrix

[ ¯A | y ] =











1 a¯₁₂ · · · a¯_1n y¯₁ 0 1 · · · a¯2n y¯2

... ... . .. ... ... 0 0 · · · 1 y¯_n









 .

where ¯aij’s and ¯yi’s are the resulting values after elementary row operations.

• This matrix is said to be in reduced row echelon form.

29See https://en.wikipedia.org/wiki/Gaussian_elimination.

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• The solution can be done by backward substitution:

x_i = ¯y_i −

n

X

j =i +1

¯ a_ijx_j,

where i = 1, 2, · · · , n.

• Time complexity: O(n³).

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Exercise

1 clear; clc;

2

3 A = [3 2 -1; 1 -1 2; -2 1 -2];

4 b = [1; -1; 0];

5 A \ b % check the answer

6

7 for i = 1 : 3

8 for j = i : 3

9 b(j) = b(j) / A(j, i); % why first?

10 A(j, :) = A(j, :) / A(j, i);

11 end

12 for j = i + 1 : 3

13 A(j, :) = A(j, :) - A(i, :);

14 b(j) = b(j) - b(i);

15 end

16 end

17 x = zeros(3, 1);

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18 for i = 3 : -1 : 1

19 x(i) = b(i);

20 for j = i + 1 : 1 : 3

21 x(i) = x(i) - A(i, j) * x(j);

22 end

23 end

24 x

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Selected Functions of Linear Algebra

• Matrix properties: norm, null, orth, rank, rref, trace, subspace.

• Matrix factorizations: lu, chol, qr.

30See https://www.mathworks.com/help/matlab/linear-algebra.html.

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Numerical Example: 2D Laplace’s Equation

• A partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.³¹

• Let Φ(x , y ) be a scalar field on R².

• Consider Laplace’s equation³²as follows:

∇²Φ(x , y ) = 0,

where ∇² = _∂x^∂22 +_∂y^∂22 is the Laplace operator.

• Consider the system shown in the next page.

31See

https://en.wikipedia.org/wiki/Partial_differential_equation.

32Pierre-Simon Laplace (1749–1827).

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

V1 V2 V3 V4 V5

V6 V7 V8 V9 V10

V11 V12 V13 V14 V15

V16 V17 V18 V19 V20

V21 V22 V23 V24 V25

• Consider theboundary condition:

• V1= V2= · · · = V4= 0.

• V21= V22= · · · = V24= 0.

• V1= V6= · · · = V16= 0.

• V5= V10= · · · = V25= 1.

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An Simple Approximation

• As you can see, we partition the region into many subregions by applying a proper mesh generation.

• Then Φ(x , y ) can be approximated by

Φ(x , y ) ≈ Φ(x + h, y ) + Φ(x − h, y ) + Φ(x , y + h) + Φ(x , y − h)

4 ,

where h is small enough.

33See

https://en.wikipedia.org/wiki/Finite_difference_method#Example:

_The_Laplace_operator.

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Matrix Formation

• By collecting all constraints, we have Ax = b where

A =





























4 −1 0 −1 0 0 0 0 0

−1 4 −1 0 −1 0 0 0 0

0 −1 4 0 0 −1 0 0 0

−1 0 0 4 −1 0 −1 0 0

0 −1 0 −1 4 −1 0 −1 0

0 0 −1 0 −1 4 −1 0 −1

0 0 0 −1 0 0 4 −1 0

0 0 0 0 −1 0 −1 4 −1

0 0 0 0 0 −1 0 −1 4





























and

b =

0 0 1 0 0 1 0 0 1 T

.

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Dimension Reduction by Symmetry

• As you can see, V₇= V₁₇, V₈ = V₁₈ and V₉ = V₁₉.

• So we can reduce A to A⁰

A⁰ =

















4 −1 0 −1 0 0

−1 4 −1 0 −1 0

0 −1 4 0 0 −1

−2 0 0 4 −1 0

0 −2 0 −1 4 −1

0 0 −2 0 −1 4

















and

b⁰=

0 0 1 0 0 1 T

.

• The dimensions of this problem are cut to 6 from 9.

• This trick helps to alleviate the curse of dimensionality.³⁴

34See https://en.wikipedia.org/wiki/Curse_of_dimensionality.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

V1 V2 V3 V4 V5

V6 V7 V8 V9 V10

V11 V12 V13 V14 V15

V16 V17 V18 V19 V20

V21 V22 V23 V24 V25

0.0000 0.0000 0.0000 0.0000 1.0000

0.0000 0.0714 0.1875 0.4286 1.0000

0.0000 0.0982 0.2500 0.5268 1.0000

0.0000 0.0714 0.1875 0.4286 1.0000

0.0000 0.0000 0.0000 0.0000 1.0000

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Remarks

• This is a toy example for numerical methods of PDEs.

• We can use the PDE toolbox for this case. (Try.)

• You may consider thefinite element method(FEM).³⁵

• The mesh generation is also crucial for numerical methods.³⁶

• You can use the Computational Geometry toolbox for triangular mesh.³⁷

35See https://en.wikipedia.org/wiki/Finite_element_method.

36See https://en.wikipedia.org/wiki/Mesh_generation.

37See https:

//www.mathworks.com/help/matlab/computational-geometry.html.

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