Selected Algebraic System Examples from Lectures

Selected Algebraic System Examples from Lectures

Matlab Examples

Matrix addition

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

>> B=[3 -4 6; 1 -2 5];

>> D=A+B

D = 4 -1 8

3 2 10 Matrix multiplication

>> C=[2 3; -1 2; 4 -3];

>> E=A*C

E = 7 3

20 -1

Gaussian Elimination Example

Form augmented matrix

Eliminate x1 from second and third equations

1234

3345

1223~

1

3

1

234

345

223

3

2

1

Α

x

x

x

3/13/23/10

3/43/13/20

1223

3/413/823/830

3/533/1033/1040

1223

Gaussian Elimination Example

Eliminate x2 from third equation

Solve triangular system

12/100

3/43/13/20

1223

6/43/16/13/200

3/43/13/20

1223

2

3

3

1

3/4

1

2/100

3/13/20

223

3

2

1

3

2

1

x

x

x

x

x

x

Matlab Example

Ax = b x = A-1b (discuss next lecture)

>> A=[3 -2 2; -5 4 -3; -4 3 -2];

>> rank(A)

ans = 3

>> b=[-1; 3; 1];

>> x=inv(A)*b

x = 3.0000

3.0000

-2.0000

Determinant Examples

By hand

Using Matlab

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

>> det(A)

ans = -2

>> A=[1 2 3;4 5 6;7 8 9];

>> det(A)

ans = 0

2)3)(2()4)(1(43

21

0)7)(5)(3()9)(4)(2()8)(6)(1(

)8)(4)(3()7)(6)(2()9)(5)(1(

987

654

321

Gauss-Jordan Elimination

Eliminate x2 entry from third row

Make the diagonal elements unity

114500

013720

001211

011031722200

013720

001211

2.02.08.0100

05.05.15.310

001211

)1(2.0)1(2.0)4(2.0)5(2.000

0)1(5.0)3(5.0)7(5.0)2(5.00

00121)1(

Gauss-Jordan Elimination cont.

Eliminate first two entries in third column

Obtain identity matrix

Matrix inverse

2.02.08.0100

7.02.03.1010

4.04.06.0011

2.02.08.0100

)2.0(5.30)2.0(5.35.0)8.0(5.35.1)1(5.35.3)0(5.310

)2.0(20)2.0(20)8.0(21)1(22)0(21)0(21

1

2.02.08.0100

7.02.03.1010

3.02.07.0001

2.02.08.0100

7.02.03.1010

7.04.0)2.0(4.0)3.1(6.00111

AI

2.02.08.0

7.02.03.1

3.02.07.01A

Gauss-Jordan Elimination cont.

Verify result

0)2.0)(2()7.0)(1()3.0)(1(

0)2.0)(2()2.0)(1()2.0)(1(

1)8.0)(2()3.1)(1()7.0)(1(2.02.08.0

7.02.03.1

3.02.07.0

431

113

211

13

12

11

1

c

c

c

AAC

IAAC

100

010

0011

Matlab Examples

>> A=[-1 1 2; 3 -1 1; -1 3 4];

>> inv(A)

ans =

-0.7000 0.2000 0.3000

-1.3000 -0.2000 0.7000

0.8000 0.2000 -0.2000

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

>> b=[1; 2];

>> x=inv(A)*b

x =

-1.0000

1.0000

Ill-Conditioned Matrix Example

Example

» represents measurement error in b2» Two rows (columns) are nearly linearly dependent

Analytical solution

10% error ( = 0.1)

1

1

11

0001.19999.0x

5.0

5.0

5.49995.0

5.50005.0 0 xx

45.499

55.500

)1.0)(5.4999(5.0

)1.0)(5.5000(5.0x

Matlab Example

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

>> cond(A)

ans = 38.9743 (well conditioned)

>> A=[0.9999 -1.0001; 1 -1];

>> cond(A)

ans = 2.0000e+004 (poorly conditioned)

>> b=[1; 1.1]

>> x=inv(A)*b

x =

500.5500

499.4500

Matlab: Vector and Matrix Norms

>> x=[2 -3 0 1 -4]';

>> norm(x,2)

ans = 5.4772

>> norm(x,inf)

ans = 4

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

>> norm(A,1)

ans = 7

>> norm(x,inf)

ans = 7

>> norm(A,'fro')

ans = 8.3066

Condition Number

Definition: (A) = ||A|| ||A-1|| A “large” condition number indicates an ill-conditioned

matrix Well conditioned matrix

Ill-conditioned matrix

75.3)30(56

1)7()(

1992

9192

2212

56

1

421

241

115

1

1

1

1

AAA

AA

20002)0001.2(0002.0

1)0001.2()(

0001.11

10001.1

0002.0

1

0001.11

10001.1

1

1

1

1

AAA

AA

Condition Number cont.

Effect of data errors

» Small data errors can lead to large solution errors

» Bound can be very conservative

Example

b

bA

x

xbbAx

)(

0143.0

0143.0

0143.0

0143.9

9857.4

0143.2

1.0

1.0

1.0

28

0

14

421

241

1159

5

2

28

0

14

421

241

115

xxx

xx

0429.0429.0)16(42

3.0)75.3()(

11

11

xb

bAx

Matlab: Condition Number

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

>> AI = inv(A)

AI =

0.2143 -0.0357 -0.0357

-0.0357 0.3393 -0.1607

-0.0357 -0.1607 0.3393

>> norm(A,1)*norm(AI,1)

ans = 3.7500

>> cond(A,1)

ans = 3.7500

>> norm(A,inf)*norm(AI,inf)

ans = 3.7500

>> cond(A,inf)

ans = 3.7500

Overdetermined Systems cont.

Normal equations: (ATA)x = ATb

Solution: x = (ATA)-1ATb» ATA must be nonsingular» (ATA)-1AT is called the left inverse matrix

Example

17

1

25

25

25

112

221

62

29)(

62

29

12

12

21

112

221

25

25

25

12

12

21

1

1 bAAAx

AAx

TT

T

Underdetermined Systems cont.

Example

5.2

5.13

7

25

25

62

29

12

12

21

)(

62

29

12

12

21

112

221

25

25

112

221

1

1bAAAx

AAx

TT

T

Matlab: Linear Algebraic Systems

>> A=[-1 1 2; 3 -1 1; -1 3 4];

>> b=[1 2 3]';

>> x=inv(A)*b

x =

0.6000

0.4000

0.6000

>> x = linsolve(A,b)

x =

0.6000

0.4000

0.6000

>> x=A\b

x =

0.6000

0.4000

0.6000

Matlab: Linear Algebraic Systems cont.

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

>> b = [25 -25 -25]';


x =

-1.0000

17.0000

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

>> b=[25 -25]';


x = x =

-5.0000 -7

15.0000 not equal to 13.5

0 2.5

Plotting a Function

y17x

0.6 xy2

5x

0.08 x

» x = [0.1:0.1:10];» y1 = 7*x./(0.6 + x);» y2 = 5*x ./ (0.08+x);

» plot(x,y1,x,y2)» xlabel('x')» ylabel('y')» legend('y1','y2')» figure

» subplot(2,1,1)» plot(x,y1)» ylabel('y1')» subplot(2,1,2)» plot(x,y2)» ylabel('y2')

Square Systems

>>A=[-1 1 2; 3 -1 1; -1 3 4];

>> b=[2 6 4]';

>> x=inv(A)*b;

>> x=A\b;

>> x=linsolve(A,b)

x =

1.0000

-1.0000

2.0000

x1 x2 2x3 2

3x1 x2 x3 6

x1 3x2 4x3 4

Non-Square Systems

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

>> b = [25 -25 -25]';

>> x=A\b

x =

-1.0000

17.0000


x =

-1.0000

17.0000

252

252

252

21

21

21

xx

xx

xx

Distinct Real Eigenvalue Example

Characteristic matrix

Characteristic equation

Eigenvalues: 1 = -5, 2 = 2

43

21

10

01

43

21IA

0103)3)(2()4)(1( 2 IA

Distinct Real Eigenvector Example

Eigenvalues

Determine eigenvectors: Ax = x

Eigenvector for 1 = -5

Eigenvector for 1 = 2

3

1

03

0261

21

21

xxx

xx

1

2

063

022

21

21

xxx

xx

2

5

43

21

2

1

A

0)4(3

02)1(

43

2

21

21

221

121

xx

xx

xxx

xxx

Repeated Real Eigenvalue Example



Eigenvalues: 1 = 2, 2 = 2

20

52

10

01

20

52IA

0)0)(5()2)(2( IA

Repeated Real Eigenvector Example

Eigenvalues


Eigenvectors for = 2

Eigenvectors are linearly dependent

0

1

0

1

000

05021

21

21

xxxx

xx

2

2

20

52

2

1

A

0)2(0

05)2(

20

52

21

21

221

121

xx

xx

xxx

xxx

Matlab Example

>> A=[2 5; 0 2];

>> e=eig(A)

e =

2

2

>> [X,e]=eig(A)

X =

1.0000 -1.0000

0 0.0000

e =

2 0

0 2

Complex Eigenvalue Example



Eigenvalues: 1 = -1+i, 2 = -1-i

11

11

10

01

11

11IA

01)1()1)(1()1)(1( 2 IA

Complex Eigenvector Example

Eigenvalues


Eigenvector for = -1+i

Eigenvector for = -1-i

1

0

01

21

21

iixx

xixx

i

i

1

1

11

11

2

1

A

0)1(

0)1(

21

21

221

121

xx

xx

xxx

xxx

1

0

02

21

21

iixx

xixx

Matlab Example

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

>> e=eig(A)

e =

-1.0000 + 1.0000i

-1.0000 - 1.0000i

>> [X,e]=eig(A)

X =

0.7071 0.7071

0 - 0.7071i 0 + 0.7071i

e =

-1.0000 + 1.0000i 0

0 -1.0000 - 1.0000i

Matrix Diagonalization Example

20

05

215

45

13

21

7

1

13

21

43

21

13

21

7

1

13

21

7

1

13

211

2,2

3

1,5

43

21

1

1

121

2211

AXXD

AXXD

XxxX

xxA

Matlab Example

>> A=[-1 2 3; 4 -5 6; 7 8 -9];

>> [X,e]=eig(A)

X =

-0.5250 -0.6019 -0.1182

-0.5918 0.7045 -0.4929

-0.6116 0.3760 0.8620

e =

4.7494 0 0

0 -5.2152 0

0 0 -14.5343

>> D=inv(X)*A*X

D =

4.7494 -0.0000 -0.0000

-0.0000 -5.2152 -0.0000

0.0000 -0.0000 -14.5343

Continuous Bioreactor Example

ODE model

0

0

0

)0()(

)0()(1

)(

)0()(

PPXSqPV

F

dt

dP

SSXSY

SSV

F

dt

dS

XXXSXV

F

dt

dX

f

Fresh Media Feed (substrates)

Exit Gas Flow

Agitator

Exit Liquid Flow(cells & products)

Bioreactor: Fixed-Point Solution

Steady-state biomass equation

Iterative equation

Initialization:

)(00

)()()(02

SDXKSSK

SSXSXDXSX

V

F

im

m

ninnm

nmnnn S

KSSK

SDSSDS

SSDS

21 )(

)(

0S

Bioreactor: Newton-Raphson Solution

Iterative equation

Function

Derivative

innm

nmn KSSK

SDSfSDSf

2)()()(

dSSdf

SfSS

dxxdf

xfxx

n

nnn

n

nnn )(

)(

)(

)(11

22

2

)(

)21()()()(

innm

mnnmminnmnn

KSSK

KSSKSSK

dS

Sd

dS

Sdf

Bioreactor: Secant Solution

Iterative equation

Function

Initialization:

innm

nmn KSSK

SDSfSDSf

2)()()(

)()()(

)()()(

1

11

1

11

nn

nnnnn

nn

nnnnn SfSf

SSSfSS

xfxf

xxxfxx

10 , SS

Solution of a single nonlinear algebraic equation:

Create Matlab m-file function: fun.m:

Call fzero from the Matlab command line to find the solution:

Different initial guesses, xo, give different solutions:

Matlab Example #1

0604.0)9.0(

1

01.0)3(

1)(

22

xxxf

>> xo = 0;>> fzero('fun',xo)ans = 0.5376

>> fzero('fun',1)ans = 1.2694

>> fzero('fun',4)ans = 3.4015

Matlab Example #2

Solution of biomass equation

Parameter values: D=0.04 h-1, m=0.48 h-1, Km=1.2 g/L, Ki=22 g/L

Create Matlab M-file: substrate.m

function f = substrate(x)

D = 0.04;

mm = 0.48;

Km = 1.2;

Ki = 22;

f = -D+mm*x/(Km+x+x^2/Ki);

0)(2

im

m

KSSK

SDSf

Matlab Example #2 cont.

Guess solution and call function fzero

>> xo=0;

>> fzero('substrate',xo)

>> ans = 0.1091 (meaningful)

Solution obtained depends on initial guess

>> xo=10;

>> fzero('substrate',xo)

>> ans = -20.7263 (not meaningful)

Selected Algebraic System Examples from Lectures

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