lecture 5ii

軟體實作與計算實驗 1

Markov ChainRoots of nonlinear functionsNested FOR loops

Lecture 5II

軟體實作與計算實驗 2

State

1S 3S2S

軟體實作與計算實驗 3

Initial population

in denotes the initial population of being state iS

ni

i n

軟體實作與計算實驗 4

State transition

A person has a current state, e.g. Si Its next state may be any one of possible

statesState transition from Si to Sj

軟體實作與計算實驗 5

Transition probabilities

ijp denotes the probability of transition from toiS

1p

conditionUnitary

j

ij

jS

軟體實作與計算實驗 6

State

1S 3S2S

31p

12p

軟體實作與計算實驗 7

State transition

Populations after one time of state transition

Pv

mmmmm

m

m

n

nn

ppp

pppppp

2

1

21

22221

11211

, vP

軟體實作與計算實驗 8

State transition

Populations after one time of state transition

Populations after k times of state transition

vP'

vPP ''...k

軟體實作與計算實驗 9

Exercise I

Draw a flow chart to calculate populations after k times of state transitions using for-loops

Write a matlab function to implement the flow chart

Find the result forP =

0.1000 0.4500 0.4500 0.3333 0 0.6667 0.5000 0 0.5000

v =

100 600 300

k=1k=10k=20k=100

軟體實作與計算實驗 10

Convergence

Change

Norm

new

new

vvvPv

'

)( newnorm vv

軟體實作與計算實驗 11

Convergence

The answer converges if

8

6

10)(

10)(

e.g.number positive small a than less is )(

new

new

new

norm

norm

norm

vv

vv

vv

軟體實作與計算實驗 12

for i=1:k

v_new=P'*v

norm(v-v_new)<10^-6

Yes:converge

break v=v_new

軟體實作與計算實驗 13

for i=1:kv_new=P' *v;

if norm(v_new-v)<10^-6 break else v=v_new

endend

vP k)'(lim k

軟體實作與計算實驗 14

Exercise II

Draw a flow chart to illustrate finding

by a for-loopImplement your flow chart by a matlab

function

vP k)'(lim k

軟體實作與計算實驗 15

for i=1:k

v_new=P'*v

norm(v-v_new)<10^-6

Yes:converge

break v=v_new

change>10^-6

change=1

v_new=P'*v

change=norm(v-v_new)v=v_new

軟體實作與計算實驗 16

while-loop

change>10^-6

change=1

v_new=P'*v

change=norm(v-v_new)v=v_new

軟體實作與計算實驗 17

change = 1while change < 10^-6

v_new=P' *v;change = norm(v-v_new);v=v_new

end

軟體實作與計算實驗 18

Exercise III

Draw a flow chart to illustrate finding

by a while-loopImplement your flow chart by a matlab

function

vP k)'(lim k

軟體實作與計算實驗 19

Nonlinear system

004

21

22

21

xxxx

軟體實作與計算實驗 20

A set of nonlinear functions

21212

22

21211

211

21121

),(4),(

),(),(

),F(

xxxxfxxxxf

xxfxxf

xx

軟體實作與計算實驗 21

function F = myfun(x) F(1) = x(1).^2 + x(2)^2-4; F(2) = x(1) - x(2);return

Function evaluation

軟體實作與計算實驗 22

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 23

Least square of nonlinear functions

i

ixxxxf 2

21,),(min

21

0),(04),(

SOLVE

21212

22

21211

xxxxfxxxxf

軟體實作與計算實驗 24

x0= ones(1,2)*2;x = lsqnonlin(@myfun,x0)

Find a zero

y=myfun(x);sum(y.^2)

CHECKING

軟體實作與計算實驗 25

Multiple roots

x0= ones(1,2)*2;x = lsqnonlin(@myfun,x0)v=[v;x];plot(x(1),x(2),'o');

v=[]

for i=1:n

demo_lsq.m

軟體實作與計算實驗 26

012),(

0242),(

SOLVE

22212

22

21211

21

xxxxf

xxxxf

軟體實作與計算實驗 27

function F = myfun2(x) F(1) = 2*x(1).^2 + x(2)^2-24; F(2) = x(1).^2 - x(2).^2+12;return

Function evaluation

012),(

0242),(22

212

22

21211

21

xxxxf

xxxxf

軟體實作與計算實驗 28

x0= ones(1,2)*2;x = lsqnonlin(@myfun2,x0)

Find a zero

y=myfun2(x);sum(y.^2)

CHECKING

軟體實作與計算實驗 29

Multiple roots

x0= ones(1,2)*2;x = lsqnonlin(@myfun2,x0)v=[v;x];plot(x(1),x(2),'o');

v=[]

for i=1:n

軟體實作與計算實驗 30

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 31

Demo_lsq_2c

source code

軟體實作與計算實驗 32

02),(022),(

SOLVE

21212

22

21211

xxxxfxxxxf

軟體實作與計算實驗 33

function F = myfun3(x) F(1) = x(1).^2 -2* x(2)^2-2; F(2) = x(1).*x(2) -2;return

Function evaluation

02),(022),(

SOLVE

21212

22

21211

xxxxfxxxxf

軟體實作與計算實驗 34

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 35

x0= ones(1,2)*2;x = lsqnonlin(@myfun3,x0)

Find a zero

y=myfun3(x);sum(y.^2)

CHECKING

軟體實作與計算實驗 36

Multiple roots

x0= ones(1,2)*2;x = lsqnonlin(@myfun3,x0)v=[v;x];plot(x(1),x(2),'o');

v=[]

for i=1:n

軟體實作與計算實驗 37

demo_lsq_hyperbola.m

two_hyperbola.m

軟體實作與計算實驗 38

Matrix Multiplication

C=A*BC(i,j)=A(i,:) *B(:,j) for all i,j

軟體實作與計算實驗 39

Nested FOR Loops

C(i,j)=A(i,:) *B(:,j)

[n,d1]=size(A);[d2,m]=size(B)

for i=1:n

for j=1:m

軟體實作與計算實驗 40

[n,d1]=size(A);[d2,m]=size(B);for i=1:nfor j=1:m C(i,j)=A(i, :) *B(:,j);endend

軟體實作與計算實驗 41

Constrained optimization

13

100102

22

21

12

2

1

xx

xxxx

Inequality constraint

Nonlinear equality

軟體實作與計算實驗 42

12

2

1

100102

xxxx

222

21,

)13(min21

xxxx

Constrained optimization

軟體實作與計算實驗 43

Constraints

Lower bound and upper boundEquality constraintInequality constraint

軟體實作與計算實驗 44

Constraints

Lower bound

Upper bound

],[]0,2[ 21 xx

]10,10[],[ 21 xx

Inequality constraints

0112

1

12

xx

xx

軟體實作與計算實驗 45

Demo_conop2

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).^2returnfunction y = fun(x)y = (x(1).^2+x(2).^2-13).^2;return

demo_conop2.m

軟體實作與計算實驗 46

function y = fun(x)y = (x(1).^2+x(2).^2-13).^2;return

Objective function

軟體實作與計算實驗 47

Calling fmincon

x = fmincon(@fun,[1 1],A,b,Aeq,beq,lb,ub)

objective function initial

guesslinearequality

inequalityLower and upperbound

軟體實作與計算實驗 48

Random variable

Sample space S={A,T,C,G}X is a discrete random variable with

sample space S.

apAX )Pr(

cpCX )Pr(tpTX )Pr(

gpGX )Pr(

軟體實作與計算實驗 49

Generative model

pa

pt pc

pg

A T C G

Flip-flop1 gcta pppp

gatcacaggt

軟體實作與計算實驗 50

Partition

ap0 1cta

ta

a

pppnppn

pn

3

2

1

1n 2n 3n

Knots partition [0,1] into four intervals respectively with lengths,

321 and , nnn

gcta pppp and ,,

tp cp gp

軟體實作與計算實驗 51

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 Tagcase 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 52

Sequence generation

Emulation of a generative model for creation of an atcg sequence

軟體實作與計算實驗 53

FOR Loops

r=rand;switch r…end

input p,mn=length(p)p_sum=0;s=[];

for i=1:n for j=1:m

p_sum=p_sum+p(i);c(i)=p_sum;

軟體實作與計算實驗 54

Mixture model

pa

pt pc

pg

A T C G

Flip-flop

pa

pt pc

pg

A T C G

Flip-flop1 2

軟體實作與計算實驗 55

Mixture model

According to probabilities, and each time one of two joined models is

selected to generate a characterCreated characters are collected to form

an atcg sequence

1 2

軟體實作與計算實驗 56

Hidden Markov model

1 2

pa

pt pc

pg

A T C G

pa

pt pc

pg

A T C G

11 21

22

12

軟體實作與計算實驗 57

Transition probability

2221

1211