Getting started with Matlab

1
Getting started with Matlab

• Numerical Methods
• Appendix B
• http//www.mathworks.com/access/helpdesk/help/tech
  doc/learn_matlab/learn_matlab.html

2
Linear Algebra with Matlab

• Introduction of Basic Matlab functions

3

• Solve
• Axb

• Matlab
• x A\b

4
trace(A)

• gtgt A1 3 62 1 9 3 6 1
• A
• 1 3 6
• 2 1 9
• 3 6 1
• gtgt trace(A)
• ans
• 3

5
rank()

• gtgt B1 2 3 3 4 7 4 -3 1-2 5 3 1 -7 6
• B
• 1 2 3
• 3 4 7
• 4 -3 1
• -2 5 3
• 1 -7 6
• gtgt rank (B)
• ans
• 3

Number of independent rows
6
Reduced row echelon formrref(B)

• gtgt B1 2 3 3 4 7 4 -3 1-2 5 3 1 -7 6
• gtgt rref(B)
• ans
• 1 0 0
• 0 1 0
• 0 0 1
• 0 0 0
• 0 0 0

7
inv(A)

• gtgt Arand(3,3)
• A
• 0.1389 0.6038 0.0153
• 0.2028 0.2722 0.7468
• 0.1987 0.1988 0.4451
• gtgt inv(A)
• ans
• -0.8783 -8.5418 14.3617
• 1.8694 1.8898 -3.2348
• -0.4429 2.9695 -2.7204
• gtgt Ainv(A)
• ans ???

8
det(A)

• gtgt Arand(3,3)
• A
• 0.9318 0.8462 0.6721
• 0.4660 0.5252 0.8381
• 0.4186 0.2026 0.0196
• gtgt det(A)
• ans
• 0.0562

9
Axb xA\b

• gtgt Arand(3,3)
• A
• 0.9318 0.8462 0.6721
• 0.4660 0.5252 0.8381
• 0.4186 0.2026 0.0196
• gtgt brand(3,1)
• b
• 0.1509
• 0.6979
• 0.3784

• gtgt x A\b
• x
• 3.4539
• -5.4966
• 2.3564
• gtgt Ax
• ans
• 0.1509
• 0.6979
• 0.3784

10
tic, toc, elapsed_time

• gtgt tic
• gtgt toc
• elapsed_time
• 2.1630

• gtgt tic
• gtgt xtoc
• x
• 2.5240

11
time.m

• Arand(1000,1000)
• tic
• inv(A)
• time_to_inverse_Atoc

12
output functions

• disp(strings to be shown on screen)
• fprintf(As C language 8.2f\n, ver1)
• gtgt ver11.3333
• gtgt fprintf('As C language 8.2f\n', ver1)
• As C language 1.33

13
norm(V,n)

• v3, 4
• norm(v,1)
• ans
• 7
• norm(v)
• ans
• 5
• norm(v,3)
• ans
• 4.4979
• norm(v,inf)
• ans
• 4

• V1(v1V2Vn )
• V2(v12V22Vn2 ) -2
• V3(v13V23Vn3 ) -3
• Vinf(v18V2 8 Vn 8) - 8

14
(No Transcript)
15
If Axb has solution

• Then
• Ax 0 only when x0
• det(A) ?0
• reff(A) I
• rank(A)n

16
LU decomposition

• gtgt Arand(3)
• A
• 0.9991 0.8848 0.4642
• 0.3593 0.4178 0.2477
• 0.3566 0.0836 0.1263
• gtgt L1,Ulu(A)
• L1
• 1.0000 0 0
• 0.3596 -0.4291 1.0000
• 0.3569 1.0000 0
• U
• 0.9991 0.8848 0.4642
• 0 -0.2322 -0.0394
• 0 0 0.0638

• gtgt L,U,Plu(A)
• L
• 1.0000 0 0
• 0.3569 1.0000 0
• 0.3596 -0.4291 1.0000
• U
• 0.9991 0.8848 0.4642
• 0 -0.2322 -0.0394
• 0 0 0.0638
• P
• 1 0 0
• 0 0 1
• 0 1 0

17
11.1.2 Cholesky decomposition

• gtgt A2 3 4 3 6 7 4 7 10
• gtgt Pchol(A)
• P
• 1.4142 2.1213 2.8284
• 0 1.2247 0.8165
• 0 0 1.1547
• gtgt P'P-A
• ans
• 1.0e-015
• 0.4441 0 0
• 0 0 0
• 0 0 0

• A is positive definite symmetric
• APTP

18
QR decomposition

• A
• 0.8138 0.7576 0.2240
• 0.1635 0.0536 0.8469
• 0.0567 0.5092 0.0466
• gtgt Q,Rqr(A)
• Q
• -0.9781 -0.0246 -0.2065
• -0.1965 -0.2161 0.9564
• -0.0682 0.9761 0.2065
• R
• -0.8319 -0.7862 -0.3887
• 0 0.4667 -0.1430
• 0 0 0.7734

• Q
• Orthogonal matrix
• R
• Upper triangle matrix

19
Singular Value Decomposition

• gtgt USV'
• ans
• 1.0000 2.0000 3.0000 4.0000
  5.0000
• 5.0000 9.0000 2.0000 3.0000
  4.0000
• 2.0000 2.0000 3.0000 4.0000
  2.0000
• AUSVT
• A .. m x n
• U .. m x m
• S .. m x n
• (singular value)
• V .. n x n

• gtgt A 1 2 3 4 5 5 9 2 3 4 2 2 3 4 2
• gtgt U,S,Vsvd(A)
• U
• -0.4581 0.6831 -0.5688
• -0.7993 -0.5966 -0.0727
• -0.3890 0.4213 0.8193
• S
• 14.0087 0 0 0
  0
• 0 5.1976 0 0
  0
• 0 0 1.9346 0
  0
• V
• -0.3735 -0.2803 0.3650 -0.4827
  -0.6447
• -0.6344 -0.6080 -0.0793 0.2604
  0.3920
• -0.2955 0.4079 0.3133 -0.5529
  0.5852
• -0.4130 0.5056 0.4052 0.6063
  -0.2051
• -0.4473 0.3601 -0.7734 -0.1609
  -0.2149

20
Reduce 3 columns

• V
• -0.3735 -0.2803 0 0
  0
• -0.6344 -0.6080 0 0
  0
• -0.2955 0.4079 0 0
  0
• -0.4130 0.5056 0 0
  0
• -0.4473 0.3601 0 0
  0
• gtgt USV'
• ans
• 1.4016 1.9127 3.3447 4.4458
  4.1490
• 5.0514 8.9888 2.0441 3.0570
  3.8912
• 1.4215 2.1258 2.5035 3.3579 3.2258

21
Pseudo-inverse

• A
• 0 0
• 0 0
• 0 0
• 1 0
• 0 1
• gtgt Apluspinv(A)
• Aplus
• 0 0 0 1 0
• 0 0 0 0 1
• gtgt AplusA
• ans
• 1 0
• 0 1

• gtgt AAplus
• ans
• 0 0 0 0 0
• 0 0 0 0 0
• 0 0 0 0 0
• 0 0 0 1 0
• 0 0 0 0 1
• Since A is not n by n, there is no inverse A-1.
• Axb can be solved by pseudo-inverse A.
• x A b

22
cond(A), rcond(A)

• gtgt Aeye(3).20
• A
• 20 0 0
• 0 20 0
• 0 0 20
• gtgt cond(A) rcond(A)
• ans
• 1 1

• cond(A) 1
• A is perfectly conditioned
• stable
• cond(A) gt large number
• A is ill-conditioned
• too sensitive
• rcond is a rapid version of cond
• rcond1/cond

23
Condition of a Hilbert matrix

• gtgt hilb(4)
• ans
• 1.0000 0.5000 0.3333 0.2500
• 0.5000 0.3333 0.2500 0.2000
• 0.3333 0.2500 0.2000 0.1667
• 0.2500 0.2000 0.1667 0.1429
• gtgt cond(hilb(4))
• ans
• 1.5514e004

24
Matlab Programming
25
if, else, and elseif
if A gt B 'greater' elseif A lt B 'less'
elseif A B 'equal' else error('Unexpected
situation') end
26
switch and case
switch (rem(n,4)0) (rem(n,2)0) case 0 M
odd_magic(n) case 1 M single_even_magic(n)
case 2 M double_even_magic(n) otherwise
error('This is impossible') end
27
for
for n 332 r(n) rank(magic(n)) end r
r Display the result
28
While
a 0 fa -Inf b 3 fb Inf while b-a gt
epsb x (ab)/2 fx x3-2x-5 if
sign(fx) sign(fa) a x fa fx else b
x fb fx end end x
(bisection method)
The result is a root of the polynomial x3 - 2x -
5 x 2.09455148154233
29
continue
fid fopen('magic.m','r') count 0 while
feof(fid) line fgetl(fid) if
isempty(line) strncmp(line,'',1) continue
end count count 1 end disp(sprintf('d
lines',count))
30
break
a 0 fa -Inf b 3 fb Inf while b-a gt
epsb x (ab)/2 fx x3-2x-5 if fx
0 break elseif sign(fx) sign(fa) a
x fa fx else b x fb fx end end
x
31
try - catch
try statement ... statement catch
statement ... statement end
32
return
function d det(A) DET det(A) is the
determinant of A. if isempty(A) d 1 return
else ... end
return terminates the current sequence of
commands and returns control to the invoking
function
33
GUIDE
34
GUIDE
35
Blank GUI
36
M-file editor
37
Tools -gtRun
38
GUI with Axes and Menu
39
(No Transcript)
40
Run
41
M-file editor
42
Try it yourself, and think about how to use it.