基于matlab的科學計算試驗

1、科學計算與數(shù)據(jù)處理實驗報告學 號S姓 名張園實驗名稱更于MATLAB的科學計算實驗實驗目的1、掌握 MATLAB2、掌握 MATLAB3、掌握 MATLAB中數(shù)組的創(chuàng)建和操作方法 中常用的數(shù)值計算方法 中常用的符號計算方法實驗方案一、一維數(shù)組創(chuàng)建實驗：(1)直接輸入法： test=1 2 3 4 test=1;2;3;4(2)步長生成法： test=1:0.5:10(3)定數(shù)線性采樣法： test = linspace(1,12,5)(4)定數(shù)對數(shù)采樣法： logspace(2,6,4)二、高維數(shù)組創(chuàng)建實驗：(1)直接輸入法：匕一 A=1 2 3;4 5 6;7 8 9(2)使用卜標： cle

2、ar,A(2,3,2)=1(3)使用低維數(shù)組：clear,A=eye(3,4);A(:,:,2)=eye(3,4)*2;A(:,:,3)=eye(3,4)*3;A(:,:,4)=eye(3,4)*4(4)使用創(chuàng)建函數(shù)(cat、repmat、reshape)創(chuàng)建高維數(shù)組： cat(3,1,2,3;4,5,6,eye(2,3)*2,ones(2,3) repmat(1,2;3,4,1,2,3) reshape(1:20,2,5,2)三、標準數(shù)組創(chuàng)建實驗：(1)全 0 矩陣： zeros(3)(2)全 1 矩陣： ones(5)(3)單位矩陣： eye(4)(4) magic 矩陣： magic(4

3、)(5)隨機矩陣： randn(4)四、矩陣變換實驗：令 Data=1,2,3,4;5,6,7,8;9,10,11,12,分別使用 diag、fliplr、flipud、rot90、tril、triu函數(shù)計算 Data的對角、轉(zhuǎn)置、翻轉(zhuǎn)、旋轉(zhuǎn)、 三角矩陣，具體命令如下： Data=1,2,3,4;5,6,7,8;9,10,11,12 diag(Data)(Data) fliplr(Data) flipud(Data) rot90(Data) tril(Data) triu(Data)五、字符串數(shù)組創(chuàng)建與操作實驗：(1)創(chuàng)建字符串數(shù)組： arr=str2mat(I,am,a,student)(2

4、)去掉字符串末尾的空格deblank:：建立字符串，用abs函數(shù)驗證空格的存在；用 deblank去掉空格，用abs已經(jīng)去掉空格 x=a n ;y=abs(x) z=deblank(x);w=abs(z)(3)刪除字符串開頭和結(jié)尾的空格strtrim str1= I am a student ; str2=I am a student; x=strtrim(str1) y=strtrim(str2)(4)執(zhí)行簡單的字符串替代strrep、 str1=I am a student.; str2=student; str3=teacher; str=strrep(str1,str2,str3)(5

5、)規(guī)范格式strread; strread(0.231,%5.3f)(6)函數(shù)strtok找出由特定字符指定的字符串內(nèi)的標記； ar=I am a student strtok(ar,s)六、架構(gòu)數(shù)組的創(chuàng)建與操作實驗：直接創(chuàng)建法： clear x; x.real = 1 2 3 4 5; x.imag = ones(4)命令(struct)創(chuàng)建法 s = struct(name,x,y,id,3,4,w,3,4)Fieldnames 函數(shù)： fieldnames(s)Getfield 函數(shù)： str(1,1).name = x; str(1,1).ID = 5; str(2,1).name =

6、 y; str(2,1).ID = 3; result = getfield(str, 2,1, name)Setfield 函數(shù)： str(1,1).name = x; str(1,1).ID = 5; str(2,1).name = y; str(2,1).ID = 3; str= setfield(str,2,1,name,a); str(2,1).name七、基本運算符號實驗：(1)矩陣加： a=1,2,3;4,5,6;7,8,9; b=3,6,9;1,2,3;2,4,6; a+b(2)矩陣減： a=1,2,3;4,5,6;7,8,9; b=3,6,9;1,2,3;2,4,6; a-b

7、(3)矩陣乘 a=1,2,3;4,5,6;7,8,9; b=3,6,9;1,2,3;2,4,6; a*b(4)數(shù)組乘 a=1,2,3;4,5,6;7,8,9; b=3,6,9;1,2,3;2,4,6; a.*b(5)矩陣乘方 a=1,2,3;4,5,6;7,8,9; aA2(6)數(shù)組乘方 a=1,2,3;4,5,6;7,8,9; b=3,6,9;1,2,3;2,4,6; a.Ab(7)矩陣左除 a=1,2,3;4,5,6;7,8,9; b=2;4;6;ab(8)矩陣右除 a=ones(3); b=1,1,1; a/b(9)數(shù)組左除 a=1,2,3;4,5,6;7,8,9; b=3,6,9;1,

8、2,3;2,4,6; a.b(10)數(shù)組右除 a=1,2,3;4,5,6;7,8,9; b=3,6,9;1,2,3;2,4,6; a./b(11)克羅內(nèi)克張量積 a=1,0,1;1,1,1;1,0,1; b=0,0,1;1,0,1;0,0,1;kron(a,b)(12)邏輯與 a=1,0,1;1,1,1;1,0,1; b=0,0,1;1,0,1;0,0,1;a&b(13)邏輯或 a=1,0,1;1,1,1;1,0,1; b=0,0,1;1,0,1;0,0,1;a|b(14)邏輯非 a=1,0,1;1,1,1;1,0,1; -a(15)邏輯異或 a=1,0,1;1,1,1;1,0,1; b=0,

9、0,1;1,0,1;0,0,1;xor(a,b)八、矩陣分析實驗：(1)范數(shù)(norm): norm(a,1) cond(a)(3)行列式(det): det(a)(4)秩(rank): rank(a)(5)特征值(eig): eig(a) a=pascal(3)(1)導數(shù)(diff)： a=xA3+4*xA2-x+20 a=1,2,3;4,5,6;7,8,9; norm(a,2)(2)條件數(shù)(cond)： V,D=eig(a)(6)化零矩陣(null) Z=null(a)Cholesky 分解( a=pascal(3);chol(a)LU 分解(lu) a=1,2,3;4,5,6;7,8,9

10、; L1,U1=lu(a)(9)正交分解(qr) a=1,2,3;4,5,6;7,8,9; U,S=qr(a)(10)奇異值分解(svd): a=1,2,3;4,5,6;7,8,9; U,S,V=svd(a)九、數(shù)值計算實驗： diff(a)(2)梯度(gradient) a=1,2,3;4,5,6;7,8,9; fx,fy=gradient(a)(3)多項式求根(roots)、 p=1,3,2,5;px=poly2str(p,x);r=roots(p)/ p是多項式的MATLAB描述方法，我們可用poly2str(p,x)函數(shù)，來顯示多項式的形式，px =xA3 +3 xA2 +2 x +5

11、 (4)零點(fzero、fsolve): a=(x)xA2+3*x+2; x=fzero(a,0) x=fsolve(xA2+3*x+2,0)(5)極值(fminbnd、fminsearch、fminunc)、 1; f=(x) xA2-4*x+5; fminbnd(f,0,1) 2; fun=inline(x(1)A2-3*x(1)*x(2)+2*x(2)A2);x0=1,1; fminsearch(fun,x0)3; fun=inline(x(1)A2-3*x(1)*x(2)+2*x(2)A2);x0=1,1; fminunc(fun,x0)(6)積分(quadl)用內(nèi)聯(lián)函數(shù)定義被積函數(shù)

12、： fun=inline(-x.*x,x); y=quadl(fun,0,1)十、符號計算實驗：(1)將cosx + J-sin2 x化簡：先用syms定義符號變量，再用 simplify函數(shù)進行化簡，具體命令如下： simplify(cos(x)+sqrt(-sin(x)A2)(2)求(x+2)x=2的解：用solve函數(shù)求解，命令如下：x=solve(x+2)Ax=2,x)實驗記錄 test=1 2 3 4 test =1234 test=1;2;3;4test =1234(2),. test=1:0.5:10test =Columns 1 through 41.00001.50002.0

13、0002.5000Columns 5 through 83.00003.50004.00004.5000Columns 9 through 125.00005.50006.00006.5000Columns 13 through 167.00007.50008.00008.5000Columns 17 through 199.00009.500010.0000(3) test = linspace(1,12,5)test =Columns 1 through 41.00003.75006.50009.2500Column 512.0000(4) logspace(2,6,4)ans =1.0e

14、+006 *0.00010.00220.04641.0000二、 (1)直接輸入法 A=1 2 3;4 5 6;7 8 9 A =123456789(2)使用卜標 clear,A(2,3,2)=1 A(:,:,1)=000000A(:,:,2)=000001(3)使用低維數(shù)組 ：.-clear,A=eye(3,4);A(:,:,2)=eye(3,4)*2;A(:,:,3)=eye(3,4)*3;A(:,:,4)=eye(3,4)*4 A(:,:,1)= 100001000010A(:,:,2)= 200002000020A(:,:,3)= 300003000030A(:,:,4)= 40000

15、4000040(4)使用創(chuàng)建函數(shù)(cat、repmat、reshape)創(chuàng)建高維數(shù)組。1, cat(3,1,2,3;4,5,6,eye(2,3)*2,ones(2,3) ans(:,:,1)=123456ans(:,:,2)=200020ans(:,:,3)=1111112, repmat(1,2;3,4,1,2,3) ans(:,:,1)=12123434ans(:,:,2)=12123434ans(:,:,3)=121234343, reshape(1:20,2,5,2)ans(:,:,1)ns(:,:,2)=11131517191214161820三、zeros

16、(3)ans =000000000(2)ones(5)ans =1111111111111111111111111eye(4)ans =1000010000100001(4)magic(4)ans =16231351110897612414151 randn(4)ans =1.06680.2944-0.6918-1.44100.0593-1.33620.85800.5711-0.09560.71431.2540-0.3999-0.83231.6236-1.59370.6900四、 Data=1,2,3,4;5,6,7,8;9,10,11,12Data =12345678910 diag(Dat

17、a) ans =1611(2)(Data)ans =111215926103711-J48 fliplr(Data) ans =12432187651211(4) flipud(Data) ans =1099101112567812 rot90(Data) ans =3448123711261015(6) tril(Data)9ans =10005600910110 triu(Data)ans =12340678001112五、 arr=str2mat(I,am,a,student) arr =Iam astudent x=a n ;y=abs(x)y =973211032 z=deblank

18、(x);w=abs(z) w =9732110 str1=I am a student ; str2=I am a student ; x=strtrim(str1)x =I am a student y=strtrim(str2) y =I am a student str1=I am a student.; str2=student; str3=teacher; str=strrep(str1,str2,str3) str =I am a teacher. strread(0.231,%5.3f) ans =0.2310 ar=I am a studentar =I am a studen

19、t strtok(ar,s)ans =I am a六；(1)直接法 clear x; x.real = 1 2 3 4 5; x.imag = ones(4) x =real: 1 2 3 4 5imag: 4x4 double(2)struct 函數(shù) s = struct(name,x,y,id,3,4,w,3,4) s =1x2 struct array with fields:nameidfieldnames 功能演示 fieldnames(s) ans =nameidgetfield功能演示 str(1,1).name = x; str(1,1).ID = 5; str(2,1).na

20、me = y; str(2,1).ID = 3; result = getfield(str, 2,1, name) result =ysetfield功能演示 str(1,1).name = x; str(1,1).ID = 5; str(2,1).name = y; str(2,1).ID = 3; str= setfield(str,2,1,name,a); str(2,1).nameans =a七：(1)矩陣加 a=1,2,3;4,5,6;7,8,9; b=3,6,9;1,2,3;2,4,6; a+bans = TOC o 1-5 h z 812791215(2)矩陣減 a=1,2,3

21、;4,5,6;7,8,9; b=3,6,9;1,2,3;2,4,6; a-bans = TOC o 1-5 h z -2-4-6333543(3)矩陣乘 a=1,2,3;4,5,6;7,8,9; b=3,6,9;1,2,3;2,4,6; a*bans = TOC o 1-5 h z 1122332958874794141(4)數(shù)組乘 a=1,2,3;4,5,6;7,8,9;L-./ jr y| 產(chǎn) b=3,6,9;1,2,3;2,4,6; a.*bans =12271018143254(5)矩陣乘方 a=1,2,3;4,5,6;7,8,9; aA2ans = TOC o 1-5 h z 303

226)數(shù)組乘方 a=1,2,3;4,5,6;7,8,9; b=3,6,9;1,2,3;2,4,6; a.Abans =16419683425216494096531441(7)矩陣左除 a=1,2,3;4,5,6;7,8,9; b=2;4;6;abWarning: Matrix is close to singular or badly scaled.Results may be inaccurate. RCOND = 2.203039e-018.ans =-0.46670.93350.1999(8)矩陣右除 a=ones(3); b=1,1,1; a/ba

23、ns =111(9)數(shù)組左除 a=1,2,3;4,5,6;7,8,9; b=3,6,9;1,2,3;2,4,6; a.bans =3.00003.00003.0000.0.25000.40000.50001 10.28570.50000.6667- 1-I-(10)數(shù)組右除 a=1,2,3;4,5,6;7,8,9; b=3,6,9;1,2,3;2,4,6; a./bans =0.33330.33330.33334.00002.50002.00003.50002.00001.5000(11)克羅內(nèi)克張量積 a=1,0,1;1,1,1;1,0,1; b=0,0,1;1,0,1;0,0,1;kron

24、(a,b) ans =001000001101000101001000001001001001101101101001001001001000001101000101001000001(12)邏輯與 a=1,0,1;1,1,1;1,0,1; b=0,0,1;1,0,1;0,0,1;a&b ans =0010100(13)邏輯或 a=1,0,1;1,1,1;1,0,1; b=0,0,1;1,0,1;0,0,1;a|b ans = TOC o 1-5 h z 101111101(14)邏輯非 a=1,0,1;1,1,1;1,0,1; -a ans =010000010(15)邏輯異或 a=1,0,

25、1;1,1,1;1,0,1; b=0,0,1;1,0,1;0,0,1;xor(a,b)ans = TOC o 1-5 h z 00，丈11r / y . i010I ./ J _/ . jf f /00八、(1)范數(shù)(norm)用來度量矩陣或者向量在某種意義下的長度 a=1,2,3;4,5,6;7,8,9; norm(a,1) ans =18 norm(a,2)ans =16.8481(2)條件數(shù)(cond)可以描述矩陣為良性矩陣還是病態(tài)矩陣的一個參數(shù) cond(a)ans =5.0524e+016(3)行歹U式(det) det(a)ans =0(4)秩(rank) rank(a) ans

26、=2(5)特征值(eig) eig(a) ans =16.1168-1.1168-0.0000 V,D=eig(a) V = -0.2320-0.78580.4082-0.5253-0.0868-0.8165-0.81870.61230.4082D =16.1168000-1.1168000-0.0000(6)化零矩陣(null) Z=null(a) Z = -0.4082 0.8165 -0.4082(7) Cholesky分解(chol)把對稱正定矩陣表示成上三角矩陣的轉(zhuǎn)置與其本身的乘積a=pascal(3)/pascal 矩陣a =111123136a=pascal(3);chol(a)

27、ans =111012001(8) LU分解(lu)a=1,2,3;4,5,6;7,8,9;L1,U1=lu(a)L1=10.14291.0000010.57140.50001.00001.000000U1=7.00008.00009.000000.85711.714300-0.0000(9)正交分解(qr) a=1,2,3;4,5,6;7,8,9; U,S=qr(a)U =-0.12310.90450.4082-0.49240.3015-0.8165-0.8616-0.30150.4082S =-8.1240-9.6011 -11.078200.90451.809100-0.0000(10)

28、奇異值分解(svd) a=1,2,3;4,5,6;7,8,9; U,S,V=svd(a)U =-0.21480.88720.4082-0.52060.2496-0.8165-0.8263-0.38790.4082S =16.84810001.06840000.0000V =0.4797-0.7767-0.40820.5724-0.07570.81650.66510.6253-0.4082九(1)導數(shù)/ J/ : 5 /.r r V- J . J r 產(chǎn). J 卡I a=xA3+4*xA2-x+20 a =xA3+4*xA2-x+20 diff(a) ans =3*xA2+8*x-1 diff(

29、a,2) ans =6*x+8(2)梯度 a=1,2,3;4,5,6;7,8,9; fx,fy=gradient(a) fx = TOC o 1-5 h z 111111111fy =333333333(3)多項式求根 p=1,3,2,5;px=poly2str(p,x);r=roots(p)/ p是多項式的MATLAB描述方法，我們可用 poly2str(p,x)函數(shù)，來顯示 多項式的形式，px =xA3 +3 xA2 +2 x +5r =-2303/793-205/4278+ 851/649i-205/4278- 851/649i a=(x)xA2+3*x+2; x=fzero(a,0)-

30、1 x=fsolve(xA2+3*x+2,0)Optimization terminated: first-order optimality is less than options.TolFun.-1(5)極值1; f=(x) xA2-4*x+5; fminbnd(f,0,1)ans =0.99992；fun=inline(x(1)A2-3*x(1)*x(2)+2*x(2)A2);x0=1,1;fminsearch(fun,x0)Exiting: Maximum number of function evaluations has been exceededans =1.0e+042 *1.89461.40903； fun=