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Lecture 5II

Lecture 5II. Roots of nonlinear functions Nested FOR loops. Nonlinear system. A set of nonlinear functions. Function evaluation. function F = myfun(x) F(1) = x(1).^2 + x(2)^2-4; F(2) = x(1) - x(2); return. x=linspace(-2,2); plot(x,x); hold on; plot(x,sqrt(4-x.^2))

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Lecture 5II

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  1. Lecture 5II • Roots of nonlinear functions • Nested FOR loops 軟體實作與計算實驗

  2. Nonlinear system 軟體實作與計算實驗

  3. A set of nonlinear functions 軟體實作與計算實驗

  4. Function evaluation function F = myfun(x) F(1) = x(1).^2 + x(2)^2-4; F(2) = x(1) - x(2); return 軟體實作與計算實驗

  5. x=linspace(-2,2); plot(x,x); hold on; plot(x,sqrt(4-x.^2)) plot(x,-sqrt(4-x.^2)) 軟體實作與計算實驗

  6. Least square of nonlinear functions 軟體實作與計算實驗

  7. Find a zero x0= ones(1,2)*2; x = lsqnonlin(@myfun,x0) y=myfun(x); sum(y.^2) CHECKING 軟體實作與計算實驗

  8. Multiple roots v=[] demo_lsq.m for i=1:n x0= ones(1,2)*2; x = lsqnonlin(@myfun,x0) v=[v;x]; plot(x(1),x(2),'o'); 軟體實作與計算實驗

  9. 軟體實作與計算實驗

  10. Function evaluation function F = myfun2(x) F(1) = 2*x(1).^2 + x(2)^2-24; F(2) = x(1).^2 - x(2).^2+12; return 軟體實作與計算實驗

  11. Find a zero x0= ones(1,2)*2; x = lsqnonlin(@myfun2,x0) y=myfun2(x); sum(y.^2) CHECKING 軟體實作與計算實驗

  12. Multiple roots v=[] for i=1:n x0= ones(1,2)*2; x = lsqnonlin(@myfun2,x0) v=[v;x]; plot(x(1),x(2),'o'); 軟體實作與計算實驗

  13. x=linspace(-5,5); plot(x,sqrt(24-2*x.^2),'r');hold on; plot(x,-sqrt(24-2*x.^2),'r') plot(x,sqrt(12+x.^2),'b') plot(x,-sqrt(12+x.^2),'b') 軟體實作與計算實驗

  14. Demo_lsq_2c source code 軟體實作與計算實驗

  15. 軟體實作與計算實驗

  16. Function evaluation function F = myfun3(x) F(1) = x(1).^2 -2* x(2)^2-2; F(2) = x(1).*x(2) -2; return 軟體實作與計算實驗

  17. x=linspace(-3,3); x=x(find(abs(x) > 0.1)); plot(x,sqrt((x.^2-2)/2),'r');hold on; plot(x,-sqrt((x.^2-2)/),'r') x=linspace(0.5,3); plot(x,2./x,'b'); x=linspace(-0.5,-3); plot(x,2./x,'b') 軟體實作與計算實驗

  18. Find a zero x0= ones(1,2)*2; x = lsqnonlin(@myfun3,x0) y=myfun3(x); sum(y.^2) CHECKING 軟體實作與計算實驗

  19. Multiple roots v=[] for i=1:n x0= ones(1,2)*2; x = lsqnonlin(@myfun3,x0) v=[v;x]; plot(x(1),x(2),'o'); 軟體實作與計算實驗

  20. demo_lsq_hyperbola.txt two_hyperbola.txt 軟體實作與計算實驗

  21. Matrix Multiplication • C=A*B • C(i,j)=A(i,:) *B(:,j) for all i,j 軟體實作與計算實驗

  22. Nested FOR Loops [n,d1]=size(A);[d2,m]=size(B) for i=1:n for j=1:m C(i,j)=A(i,:) *B(:,j) 軟體實作與計算實驗

  23. [n,d1]=size(A);[d2,m]=size(B); for i=1:n for j=1:m C(i,j)=A(i, :) *B(:,j); end end 軟體實作與計算實驗

  24. Constrained optimization Inequality constraint Nonlinear equality 軟體實作與計算實驗

  25. Constrained optimization 軟體實作與計算實驗

  26. Constraints • Lower bound and upper bound • Equality constraint • Inequality constraint 軟體實作與計算實驗

  27. Constraints Lower bound Upper bound Inequality constraints 軟體實作與計算實驗

  28. Demo_conop2 demo_conop2.m function demo_conop2() lb =[-2 0]; ub =[10,10]; Aeq=[-1 1];beq=[0]; A=[];b=[]; x = fmincon(@fun,[1 1],A,b,Aeq,beq,lb,ub) x(1).^2+x(2).^2 return function y = fun(x) y = (x(1).^2+x(2).^2-13).^2; return 軟體實作與計算實驗

  29. Objective function function y = fun(x) y = (x(1).^2+x(2).^2-13).^2; return 軟體實作與計算實驗

  30. Calling fmincon x = fmincon(@fun,[1 1],A,b,Aeq,beq,lb,ub) objective function initial guess linear equality inequality Lower and upper bound 軟體實作與計算實驗

  31. Random variable • Sample space S={A,T,C,G} • X is a discrete random variable with sample space S. 軟體實作與計算實驗

  32. Flip-flop pa pt pg pc T C G A Generative model gatcacaggt 軟體實作與計算實驗

  33. Partition 0 1 Knots partition [0,1] into four intervals respectively with lengths, 軟體實作與計算實驗

  34. r=rand; Tag= (r<=c(1))+ 2*(r<=c(2) & r > c(1))+3*(r<=c(3) & r > c(2))+4*(r<=c(4) & r > c(3)) switch Tag case 1 s=[s 'A']; case 2 s=[s 'T']; case 3 s=[s 'C']; case 4 s=[s 'G']; end 軟體實作與計算實驗

  35. Sequence generation • Emulation of a generative model for creation of an atcg sequence 軟體實作與計算實驗

  36. FOR Loops input p,m n=length(p) p_sum=0;s=[]; for j=1:m for i=1:n r=rand; switch r … end p_sum=p_sum+p(i); c(i)=p_sum; 軟體實作與計算實驗

  37. Flip-flop Flip-flop pa pa pt pt pg pg pc pc T T C C G G A A Mixture model 軟體實作與計算實驗

  38. Mixture model • According to probabilities, and each time one of two joined models is selected to generate a character • Created characters are collected to form an atcg sequence 軟體實作與計算實驗

  39. pa pa pt pt pg pg pc pc T T C C G G A A Hidden Markov model 1 2 軟體實作與計算實驗

  40. Transition probability 軟體實作與計算實驗

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