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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.

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Selected Algebraic System Examples from Lectures

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  1. Selected Algebraic System Examples from Lectures

  2. 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

  3. Gaussian Elimination Example • Form augmented matrix • Eliminate x1 from second and third equations

  4. Gaussian Elimination Example • Eliminate x2 from third equation • Solve triangular system

  5. 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

  6. 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

  7. Gauss-Jordan Elimination • Eliminate x2 entry from third row • Make the diagonal elements unity

  8. Gauss-Jordan Elimination cont. • Eliminate first two entries in third column • Obtain identity matrix • Matrix inverse

  9. Gauss-Jordan Elimination cont. • Verify result

  10. 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

  11. Ill-Conditioned Matrix Example • Example • e represents measurement error in b2 • Two rows (columns) are nearly linearly dependent • Analytical solution • 10% error (e = 0.1)

  12. 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

  13. 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

  14. Condition Number • Definition: k(A) = ||A||||A-1|| • A “large” condition number indicates an ill-conditioned matrix • Well conditioned matrix • Ill-conditioned matrix

  15. Condition Number cont. • Effect of data errors • Small data errors can lead to large solution errors • Bound can be very conservative • Example

  16. 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

  17. 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

  18. Underdetermined Systems cont. • Example

  19. 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

  20. Matlab: Linear Algebraic Systems cont. >> A = [1 2; 2 -1; -2 -1]; >> b = [25 -25 -25]'; >> x = linsolve(A,b) x = -1.0000 17.0000 >> A=[1 2 -2; 2 -1 -1]; >> b=[25 -25]'; >> x = linsolve(A,b) x = x = -5.0000 -7 15.0000 not equal to 13.5 0 2.5

  21. Plotting a Function » x = [0.1:0.1:10]; » y1 = 7*x./(0.6 + x); » y2 = 5*x ./ (0.08+x); » subplot(2,1,1) » plot(x,y1) » ylabel('y1') » subplot(2,1,2) » plot(x,y2) » ylabel('y2') » plot(x,y1,x,y2) » xlabel('x') » ylabel('y') » legend('y1','y2') » figure

  22. 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

  23. Non-Square Systems >> A = [1 2; 2 -1; -2 -1]; >> b = [25 -25 -25]'; >> x=A\b x = -1.0000 17.0000 >> x = linsolve(A,b) x = -1.0000 17.0000

  24. Distinct Real Eigenvalue Example • Characteristic matrix • Characteristic equation • Eigenvalues: l1 = -5, l2 = 2

  25. Distinct Real Eigenvector Example • Eigenvalues • Determine eigenvectors: Ax = lx • Eigenvector for l1 = -5 • Eigenvector for l1 = 2

  26. Repeated Real Eigenvalue Example • Characteristic matrix • Characteristic equation • Eigenvalues: l1 = 2, l2 = 2

  27. Repeated Real Eigenvector Example • Eigenvalues • Determine eigenvectors: Ax = lx • Eigenvectors for l = 2 • Eigenvectors are linearly dependent

  28. 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

  29. Complex Eigenvalue Example • Characteristic matrix • Characteristic equation • Eigenvalues: l1 = -1+i, l2 = -1-i

  30. Complex Eigenvector Example • Eigenvalues • Determine eigenvectors: Ax = lx • Eigenvector for l = -1+i • Eigenvector for l = -1-i

  31. 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

  32. Matrix Diagonalization Example

  33. 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

  34. Exit Gas Flow Fresh Media Feed (substrates) Agitator Exit Liquid Flow (cells & products) Continuous Bioreactor Example • ODE model

  35. Bioreactor: Fixed-Point Solution • Steady-state biomass equation • Iterative equation • Initialization:

  36. Bioreactor: Newton-Raphson Solution • Iterative equation • Function • Derivative

  37. Bioreactor: Secant Solution • Iterative equation • Function • Initialization:

  38. Matlab Example #1 • 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: >> xo = 0; >> fzero('fun',xo) ans = 0.5376 >> fzero('fun',4) ans = 3.4015 >> fzero('fun',1) ans = 1.2694

  39. Matlab Example #2 • Solution of biomass equation • Parameter values: D=0.04 h-1, mm=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);

  40. 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)

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