上海復(fù)旦大學(xué)matlab課件

MatlabMathCleveMorler著陳文斌（wbchen@)復(fù)旦大學(xué)2002年MatlabMathCleveMorler著1/class/cs138/mlmath.html/class/2黃金分割黃金分割3p=[1–1–1];r=roots(p);r=-0.618033988749891.61803398874989C(1)*X^N+...+C(N)*X+C(N+1)=0p=[1–1–1];r=C(1)*X^N+4r=solve(‘1/x=x-1’);r=[1/2*5^(1/2)+1/2][1/2-1/2*5^(1/2)]符號工具箱Mapplephi=r(1)vpa(phi,50)1/2*5^(1/2)+1/21.6180339887498948482045868343656381177203091798058phi=double(phi)1.61803398874989r=solve(‘1/x=x-1’);r=符號5圖形的方法f=inline('1/x-(x-1)')ezplot(f,0,4)phi=fzero(f,1)holdonplot(phi,0,’o’)圖形的方法f=inline('1/x-(x-1)')600.511.522.533.54-3-2-101234567x1/x-(x-1)結(jié)果圖00.511.522.533.54-3-2-101234567ff-111黃金分割圖ff-111黃金分割圖8%GOLDRECTGoldenRectangle%GOLDRECTplotsthegoldenrectanglephi=(1+sqrt(5))/2;x=[0phiphi00];y=[00110];u=[11];v=[01];plot(x,y,'b',u,v,'b--');text(phi/2,1.05,'\phi')text((1+phi)/2,-.05,'\phi-1');text(-.05,.5,'1');text(.5,-.05,'1')axisequalaxisoffset(gcf,'color','white')%GOLDRECTGoldenRectangle9連分式連分式10連分式Goldfract.mg=1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1))))))g=21/13g=1.61538461538462err=0.0026字符串有效數(shù)字2位連分式Goldfract.mg=字符串有效數(shù)字2位11Fibonacci

LeonardoPisanoFibonacciwasbornaround1170anddiedaround1250inPisainwhatisnowItaly.HetraveledextensivelyinEuropeandNorthernAfrica.Hewroteseveralmathematicaltextsthat,amongotherthings,introducedEuropetotheHindu-Arabicnotationfornumbers.Eventhoughhisbookshadtobetranscribedbyhand,theywerestillwidelycirculated.Inhisbestknownbook,LiberAbaci,publishedin1202,heposedthefollowingproblem.Amanputapairofrabbitsinaplacesurroundedonallsidesbyawall.Howmanypairsofrabbitscanbeproducedfromthatpairinayearifitissupposedthateverymontheachpairbegetsanewpairwhichfromthesecondmonthonbecomesproductive?FibonacciLeonardoPisanoFibo12兔子的故事兔子從出生要一個月，從出生到成熟要過一個月月123456繁殖111235出生011235成熟001123總數(shù)1235813兔子的故事兔子從出生要一個月，從出生到成熟要過一個月13Fibonacci序列Fibonacci.m123581321345589144233遞歸實現(xiàn)：Fibnum.mFibonacci序列Fibonacci.m1遞歸實現(xiàn)：Fi14遞歸實現(xiàn)遞歸實現(xiàn)是elegantbutexpensivetic,fibnum(24),toctic,fibonacci(24),toc試一下，比較時間！遞歸實現(xiàn)遞歸實現(xiàn)是elegantbutexpensive15黃金分割和Fibonacci數(shù)比較一下goldfract(6)和fibonacci(7)連分式：g=g+f;黃金分割：f(k)=f(k-1)+f(k-2);n=40;f=fibonacci(n);f(2:n)./f(1:n-1)f(2:n)./f(1:n-1)-phi黃金分割和Fibonacci數(shù)比較一下goldfract(616Fibonacci的兔子Fibonacci的兔子以黃金分割的速度增長。＋有兩個解phi和(1-phi)由初始條件：是有理分式Fibonacci的兔子Fibonacci的兔子以黃金分割的17Fibonacci的兔子注意：沒有半只兔子

formatlongen=(1:40)';f=(phi.^(n+1)-(1-phi).^(n+1))/(2*phi-1)

f=round(f)Fibonacci的兔子注意：沒有半只兔子formatl18魔方陣A=magic(3)sum(A)sum(A’)sum(diag(A))sum(diag(flipud(A)))sum(1:9)/3816357492魔方陣A=magic(3)sum(1:9)/3819魔方陣的八種組合fork=0:3rot90(A,k)rot90(A’,k)end逆時陣旋轉(zhuǎn)k*90度魔方陣的八種組合fork=0:3逆時陣旋轉(zhuǎn)k*90度20基本代數(shù)運(yùn)算X=inv(A)det(A)-3600.1472-0.14440.0639-0.06110.02220.1056-0.01940.1889-0.1028formatrat53/360-13/9023/360-11/1801/4519/180-7/36017/90-37/360基本代數(shù)運(yùn)算X=inv(A)det(A)-360021基本代數(shù)運(yùn)算r=norm(A)e=eig(A)s=svd(A)e=15.00004.8990-4.8990s=15.00006.92823.4641r=15NORM(X)isthelargestsingularvalueofX,max(svd(X)).基本代數(shù)運(yùn)算r=norm(A)e=s=r=NO22符號計算A=sym(A)sum(A)sum(A’)’det(A)inv(A)eig(A)svd(A)符號計算A=sym(A)23100200300400500100200300400500600圖象顯示loaddurerwhosimage(X)colormap(map)axisimageMelancolia,aRenaissanceetchingbyAlbrectDurer1002003004005001002003004005002450100150200250300350501001502002503003504階魔方陣loaddetailimage(X)colormap(map)axisimage501001502002503003505010015020254階魔方陣16231351110897612414151A=magic(4)A=A(:,[1324])4階魔方陣：880個5階魔方陣：275305224個6階魔方陣：？4階魔方陣16226秩和奇異陣A=magic(4)det(A)inv(A)矩陣的秩是矩陣中線性無關(guān)的行和列的個數(shù)，一個n階矩陣是奇異的當(dāng)且僅當(dāng)它的秩小于n

rank(magic(4))秩和奇異陣A=magic(4)矩陣的秩是矩陣中線性27秩和奇異陣forn=1:24,r(n)=rank(magic(n));end[(1:24)’r’]bar(r)tiltle(‘魔方陣的秩’)秩和奇異陣forn=1:24,r(n)=2805101520250510152025n:奇數(shù)，滿秩n=4k:秩是3

n=2k!=4k:秩是n/2+205101520250510152025n:奇數(shù)，滿秩29魔方陣的三維顯示surf(magic(n))axisoffset(gcf,’doublebuffer’,’on’)cameratoolbarforn=8:11

subplot(2,2,n-7),surf(magic(n))axisoff,view(30,45)%axistightend魔方陣的三維顯示surf(magic(n))forn30圖形對象圖形對象31上海復(fù)旦大學(xué)matlabppt課件32小鴨子小鴨子33加密－－Matlab字符能力x=reshape(32:127,32,3)’c=char(x)char(32)c=!"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~加密－－Matlab字符能力x=reshape(334字符串d=char(48:57)double(d)–‘0’d=0123456789ans=0123456789字符串d=char(48:57)d=35字符串（高八位）char(reshape(160:255,32,3)’)char(160)char(160:161)牎ⅲぅΗī氨渤吹斗腹夯冀究懶旅吶魄壬仕掏蝸醒矣哉腫剄譙茌捱噌懺溴驍栝覿祉鉺瘃蝮趲鲼?字符串（高八位）char(reshape(160:255,36模運(yùn)算x=[37–3737–37]’;y=[1010-10-10]’;r=[xyrem(x,y)mod(x,y)]371077-3710-7337-107-3-37-10-7-7模運(yùn)算x=[37–3737–37]’;337字符加密取素數(shù)：P=97(要表示95個字符）x=double(‘TV’)’-32y=AxmodPA=[712226]‘TV’’1U’

Ify=AxmodPthenx=AymodPmod(A^2,P)=I

Crypto.m‘helloworld’

,?{p]>~Y&=字符加密取素數(shù)：P=97(要表示95個字符）x=38分形蕨Fern.m分形蕨Fern.m39數(shù)論中未解決的3n+1問題如果n=1，停止如果n是偶數(shù)，n=n/2;如果n是奇數(shù)，n=3n+1問題：這個過程是否一定會終止？綜述文章：AmericanMathematicalMonthly,92(1985),3-23

http://www.cecm.sfu.ca/organics/papers/lagarias數(shù)論中未解決的3n+1問題如果n=1，停止問題：這個過程40threenplus1.mthreenplus1.m41functionthreenplus1(n)%"Threenplus1".%Studythe3n+1sequence.%threenplus1(n)plotsthesequencestartingwithn.%threenplus1withnoargumentsstartswithn=1.%uicontrolsdecrementorincrementthestartingn.%Isitpossibleforthistorunforever?functionthreenplus1(n)42if~isequal(get(gcf,'tag'),'3n+1')shgclfresetuicontrol('position',[26052522],...'string','<',...'callback','threenplus1(''<'')');uicontrol('position',[30052522],...'string','>',...'callback','threenplus1(''>'')');set(gcf,'tag','3n+1');endif~isequal(get(gcf,'tag'),'3n43ifnargin==0n=1;elseifisequal(n,'<')n=get(gcf,'userdata')-1;elseifisequal(n,'>')n=get(gcf,'userdata')+1;endifn<1,n=1;endset(gcf,'userdata',n)ifnargin==044y=n;whilen>1ifrem(n,2)==0n=n/2;elsen=3*n+1;endy=[yn];endy=n;45semilogy(y,'.-')axistightymax=max(y);ytick=[2.^(0:ceil(log2(ymax))-1)ymax];iflength(ytick)>8,ytick(end-1)=[];endset(gca,'ytick',ytick)title(['n='num2str(y(1))]);semilogy(y,'.-')46x=32768

y=0L:lcadyshiftright5bitsaddxstoreinxchangesignshiftright5bitsaddystoreinyplotxygotoL畫園h=1/32;x=1;y=0;while1x=x+h*y;y=y-h*x;plot(x,y,’.’)drawnowendCirclegen.mx=32768畫園h=1/32;47畫園畫園48畫園中的特征值[V,E]=eig(A)E是對角陣h=2*rand,A=[1h;-h1-h^2],lambda=eig(A),abs(lambda)畫園中的特征值[V,E]=eig(A)E是對角陣h=49特征值的符號運(yùn)算symshA=[1h;-h1-h^2]lambda=eig(A)d=det(A)d=simple(prod(lambda))設(shè)theta=acos(1-h^2/2）;Lambda=[cos(theta)-i*sin(theta);cos(theta)+i*sin(theta)]diff=simple(lambda-Lambda)特征值的符號運(yùn)算symshd=det(A)設(shè)th50浮點數(shù)1985年，IEEE標(biāo)準(zhǔn)委員會和美國國家標(biāo)準(zhǔn)局對二進(jìn)制浮點數(shù)運(yùn)算建立ANSI/IEEE745-1985標(biāo)準(zhǔn)。

Matlab采用IEEE雙精度浮點運(yùn)算。f最多用52位表示規(guī)范數(shù)表示精度與f有關(guān),表示的數(shù)的范圍與e有關(guān)指數(shù)e是整數(shù)浮點數(shù)1985年，IEEE標(biāo)準(zhǔn)委員會和美國國家標(biāo)準(zhǔn)局對二進(jìn)制51雙精度浮點數(shù)雙精度浮點數(shù)以64位存儲，f:52位,e:11位,數(shù)的符號：1位。指數(shù)以e+1023存儲(1,211-2)整個浮點數(shù)的分?jǐn)?shù)部分不是f,而是1+f，IEEE標(biāo)準(zhǔn)在64位中存儲了65位的信息。Floatgui.meps雙精度浮點數(shù)雙精度浮點數(shù)以64位存儲，f:52位,e:11位52精度和舍入誤差eps是1到一個最近浮點數(shù)的距離,eps=2^(-t)對IEEE雙精度：eps=2^(-52)≈2.2204*10^(-16)最大的相對誤差是eps/2，數(shù)之間的最大的相對距離是epst=0.1不能被被精確存儲第1項的1不用存儲，所以重復(fù)出現(xiàn)1001，知道第53項，而54項以后的級數(shù)大于eps/2,舍入到53項精度和舍入誤差eps是1到一個最近浮點數(shù)的距離,eps=253精度和舍入誤差t=0.1formathext3fb999999999999a3fbh=10191019-1023=-4試一下：-0.1是如何存的？看一下：0.3/0.1和3的存儲0:0.1:1的問題formatlongx=4/3–1y=3*xz=1-y精度和舍入誤差t=0.13fb9999999999954浮點數(shù)

BinaryDecimalHexeps2^(-52)2.2204e-163cb0000000000000realmin2^(-1022)2.2251e-308