MATLAB60數(shù)學(xué)手冊復(fù)合版

目錄

第1章矩陣及其基本運(yùn)算.........................................................7

1.1矩陣的表示..............................................................7

1.1.1數(shù)值矩陣的生成....................................................7

1.1.2符號矩陣的生成....................................................8

1.1.3大矩陣的生成......................................................9

1.1.4多維數(shù)組的創(chuàng)建....................................................9

1.1.5特殊矩陣的生成...................................................10

1.2矩陣運(yùn)算...............................................................15

1.2.1力口、減運(yùn)算.......................................................15

1.2.2乘法..............................................................15

1.2.3集合運(yùn)算.........................................................18

1.2.4除法運(yùn)算.........................................................21

1.2.5矩陣乘方.........................................................22

1.2.6矩陣函數(shù).........................................................22

1.2.7矩陣轉(zhuǎn)置.........................................................23

1.2.8方陣的行列式.....................................................23

1.2.9逆與偽逆.........................................................24

1.2.10矩陣的跡........................................................25

1.2.11矩陣和向量的范數(shù)................................................25

1.2.12條件數(shù)...........................................................26

1.2.13矩陣的秩.........................................................26

1.2.14特殊運(yùn)算.........................................................27

1.2.15符號矩陣運(yùn)算....................................................32

1.2.16矩陣元素個(gè)數(shù)的確定..............................................35

1.3矩陣分解...............................................................35

1.3.1Cholesky分解.....................................................35

1.3.2LU分解..........................................................36

1.3.3QR分解..........................................................36

1.3.4Schur分解........................................................38

1.3.5實(shí)Schur分解轉(zhuǎn)化成復(fù)Schur分解...................................38

1.3.6特征值分解.......................................................39

1.3.7奇異值分解.......................................................39

1.3.8廣義奇異值分解...................................................40

1.3.9特征值問題的QZ分解.............................................41

1.3.10海森伯格形式的分解..............................................41

1.4線性方程的組的求解....................................................41

1.4.1求線性方程組的唯一解或特解（第一類問題）.......................41

1.4.2求線性齊次方程組的通解..........................................44

1.4.3求非齊次線性方程組的通解........................................45

1.4.4線性方程組的LQ解法.............................................47

1.4.5雙共舸梯度法解方程組............................................47

1.4.6穩(wěn)定雙共軌梯度方法解方程組......................................48

1.4.7復(fù)共蜿梯度平方法解方程組........................................49

1.4.8共輒梯度的LSQR方法............................................50

1.4.9廣義最小殘差法...................................................50

1.4.10最小殘差法解方程組..............................................51

1.4.11預(yù)處理共舸梯度方法..............................................52

1.4.12準(zhǔn)最小殘差法解方程組............................................52

1.5特征值與二次型.........................................................53

1.5.1特征值與特征向量的求法..........................................53

1.5.2提高特征值的計(jì)算精度............................................54

1.5.3復(fù)對角矩陣轉(zhuǎn)化為實(shí)對角矩陣......................................54

1.5.4正交基...........................................................55

1.5.5二次型...........................................................55

1.6秩與線性相關(guān)性.........................................................56

1.6.1矩陣和向量組的秩以及向量組的線性相關(guān)性.........................56

1.6.2求行階梯矩陣及向量組的基........................................56

1.7稀疏矩陣技術(shù)...........................................................57

1.7.1稀疏矩陣的創(chuàng)建...................................................57

1.7.2將稀疏矩陣轉(zhuǎn)化為滿矩陣..........................................58

1.7.3稀疏矩陣非零元素的索引..........................................59

1.7.4外部數(shù)據(jù)轉(zhuǎn)化為稀疏矩陣..........................................59

1.7.5基本稀疏矩陣.....................................................59

1.7.6稀疏矩陣的運(yùn)算...................................................61

1.7.7畫稀疏矩陣非零元素的分布圖形....................................62

1.7.8矩陣變換.........................................................62

1.7.9稀疏矩陣的近似歐幾里得范數(shù)和條件數(shù)..............................65

1.7.10稀疏矩陣的分解..................................................65

1.7.11稀疏矩陣的特征值分解............................................67

1.7.12稀疏矩陣的線性方程組............................................67

第2章數(shù)值計(jì)算與數(shù)據(jù)分析.....................................................68

2.1基本數(shù)學(xué)函數(shù)...........................................................68

2.1.1三角函數(shù)與雙曲函數(shù)...............................................68

2.1.2其他常用函數(shù).....................................................75

2.2插值、擬合與查表.......................................................82

2.2.1插值命令.........................................................83

2.2.2查表命令.........................................................89

2.3數(shù)值積分...............................................................90

2.3.1一元函數(shù)的數(shù)值積分...............................................90

2.3.2二元函數(shù)重積分的數(shù)值計(jì)算........................................92

2.4常微分方程數(shù)值解.......................................................93

2.5偏微分方程的數(shù)值解....................................................96

2.5.1單的Poission方程.................................................97

2.5.2雙曲型偏微分方程.................................................98

2.5.3拋物型偏微分方程.................................................99

第3章符號運(yùn)算...............................................................101

3.1算術(shù)符號操作..........................................................101

3.2基本運(yùn)算..............................................................103

3.2.1函數(shù)計(jì)算器......................................................114

3.2.2微積分...........................................................115

3.2.3符號函數(shù)的作圖..................................................118

3.2.4積分變換........................................................124

3.2.5Taylor級數(shù).......................................................129

3.2.6其它.............................................................130

第4章概率統(tǒng)計(jì)...............................................................140

4.1隨機(jī)數(shù)的產(chǎn)生..........................................................140

4.1.1二項(xiàng)分布的隨機(jī)數(shù)據(jù)的產(chǎn)生........................................140

4.1.2正態(tài)分布的隨機(jī)數(shù)據(jù)的產(chǎn)生........................................140

4.1.3常見分布的隨機(jī)數(shù)產(chǎn)生............................................141

4.1.4通用函數(shù)求各分布的隨機(jī)數(shù)據(jù).....................................141

4.2隨機(jī)變量的概率密度計(jì)算...............................................142

4.2.1通用函數(shù)計(jì)算概率密度函數(shù)值.....................................142

4.2.2專用函數(shù)計(jì)算概率密度函數(shù)值.....................................143

4.2.3常見分布的密度函數(shù)作圖..........................................144

4.3隨機(jī)變量的累積概率值（分布函數(shù)值）......................................147

4.3.1通用函數(shù)計(jì)算累積概率值..........................................147

4.3.2專用函數(shù)計(jì)算累積概率值（隨機(jī)變量X4K的概率之和）.............147

4.4隨機(jī)變量的逆累積分布函數(shù).............................................149

4.4.1通用函數(shù)計(jì)算逆累積分布函數(shù)值...................................149

4.4.2專用函數(shù)-inv計(jì)算逆累積分布函數(shù)..................................149

4.5隨機(jī)變量的數(shù)字特征....................................................151

4.5.1平均值、中值....................................................151

4.5.2數(shù)據(jù)比較.........................................................153

4.5.3期望.............................................................154

4.5.4方差.............................................................155

4.5.5常見分布的期望和方差............................................157

4.5.6協(xié)方差與相關(guān)系數(shù)................................................158

4.6統(tǒng)計(jì)作圖..............................................................159

4.6.1正整數(shù)的頻率表..................................................159

4.6.2經(jīng)驗(yàn)累積分布函數(shù)圖形............................................160

4.6.3最小二乘擬合直線................................................160

4.6.4繪制正態(tài)分布概率圖形............................................160

4.6.5繪制威布爾（Weibull）概率圖形......................................161

4.6.6樣本數(shù)據(jù)的盒圖..................................................161

4.6.7給當(dāng)前圖形加一條參考線..........................................162

4.6.8在當(dāng)前圖形中加入一條多項(xiàng)式曲線.................................163

4.6.9樣本的概率圖形..................................................163

4.6.10附加有正態(tài)密度曲線的直方圖....................................163

4.6.11在指定的界線之間畫正態(tài)密度曲線................................164

4.7參數(shù)估計(jì)..............................................................164

4.7.1常見分布的參數(shù)估計(jì)..............................................164

4.7.2非線性模型置信區(qū)間預(yù)測..........................................167

4.7.3對數(shù)似然函數(shù)....................................................170

4.8假設(shè)檢驗(yàn)..............................................................172

4.8.1已知，單個(gè)正態(tài)總體的均值口的假設(shè)檢驗(yàn)（U檢驗(yàn)法）...........172

4.8.2未知，單個(gè)正態(tài)總體的均值U的假設(shè)檢驗(yàn)（t檢驗(yàn)法）..............173

4.8.3兩個(gè)正態(tài)總體均值差的檢驗(yàn)（t檢驗(yàn)）...........................173

4.8.4兩個(gè)總體一致性的檢驗(yàn)一秩和檢驗(yàn)............................174

4.8.5兩個(gè)總體中位數(shù)相等的假設(shè)檢驗(yàn)一一符號秩檢驗(yàn)..................175

4.8.6兩個(gè)總體中位數(shù)相等的假設(shè)檢驗(yàn)——符號檢驗(yàn)...................175

4.8.7正態(tài)分布的擬合優(yōu)度測試......................................176

4.8.8正態(tài)分布的擬合優(yōu)度測試......................................176

4.8.9單個(gè)樣本分布的Kolmogorov-Smirnov測試......................177

4.8.10兩個(gè)樣本具有相同的連續(xù)分布的假設(shè)檢驗(yàn)........................178

4.9方差分析..............................................................178

4.9.1單因素方差分析..................................................178

4.9.2雙因素方差分析..................................................180

第5章優(yōu)化問題...............................................................182

5.1線性規(guī)劃問題..........................................................182

5.2foptions函數(shù)...........................................................183

5.3非線性規(guī)劃問題........................................................184

5.3.1有約束的一元函數(shù)的最小值........................................184

5.3.2無約束多元函數(shù)最小值............................................185

5.3.3有約束的多元函數(shù)最小值..........................................187

5.3.4二次規(guī)劃問題....................................................189

5.4“半無限”有約束的多元函數(shù)最優(yōu)解......................................191

5.5極小化極大（Minmax）問題.............................................195

5.6多目標(biāo)規(guī)劃問題........................................................197

5.7最小二乘最優(yōu)問題......................................................200

5.7.1約束線性最小二乘................................................200

5.7.2非線性數(shù)據(jù)（曲線）擬合.........................................201

5.7.3非線性最小二乘..................................................202

5.7.4非負(fù)線性最小二乘................................................204

5.8非線性方程（組）求解....................................................204

5.8.1非線性方程的解..................................................204

5.8.2非線性方程組的解................................................205

第6章模糊邏輯..............................................................207

6.1隸屬函數(shù)..............................................................207

6.1.1高斯隸屬函數(shù)....................................................207

6.1.2兩邊型高斯隸屬函數(shù)..............................................207

6.1.3建立一般鐘型隸屬函數(shù)............................................208

6.1.4兩個(gè)sigmoid型隸屬函數(shù)之差組成的隸屬函數(shù).......................208

6.1.5通用隸屬函數(shù)計(jì)算................................................209

6.1.6建立n型隸屬函數(shù)................................................209

6.1.7通過兩個(gè)sigmoid型隸屬函數(shù)的乘積構(gòu)造隸屬函數(shù)...................210

6.1.8建立Sigmoid型隸屬函數(shù)..........................................210

6.1.9建立S型隸屬函數(shù)................................................211

6.1.10建立梯形隸屬函數(shù)...............................................212

6.1.11建立三角形隸屬函數(shù).............................................213

6.1.12建立Z型隸屬函數(shù)...............................................214

6.1.13兩個(gè)隸屬函數(shù)之間轉(zhuǎn)換參數(shù)......................................215

6.1.14基本FIS編輯器.................................................215

6.1.15隸屬函數(shù)編輯器.................................................217

6.2模糊推理結(jié)構(gòu)FIS.............................................................................................................218

6.2.1不使用數(shù)據(jù)聚類方法從數(shù)據(jù)生成FIS結(jié)構(gòu)...........................218

6.2.2使用減法聚類方法從數(shù)楣生成FIS結(jié)構(gòu).............................219

6.2.3生成一個(gè)FIS輸出曲面............................................219

6.2.4將mamdan型FIS轉(zhuǎn)換為SugenoFIS................................................................220

6.2.5完成模糊推理計(jì)算................................................220

6.2.6模糊c均值聚類..................................................221

6.2.7模糊均值和減法聚類..............................................222

6.2.8繪制一個(gè)FIS..........................................................................................................222

6.2.9繪制給定變量的所有隸屬的曲線...................................223

6.2.10從磁盤裝入一個(gè)FIS...........................................................................................223

6.2.11從FIS中刪除某一隸屬函數(shù).......................................224

6.2.12從FIS中刪除變量...............................................225

6.2.13設(shè)置模糊系統(tǒng)屬性...............................................226

6.2.14以分行形式顯示FIS結(jié)構(gòu)的所有屬性..............................227

6.2.15完成模糊運(yùn)算...................................................228

6.2.16解析模糊規(guī)則...................................................229

6.2.17規(guī)則編輯器和語法編輯器........................................229

6.2.18規(guī)則觀察器和模糊推理框圖......................................230

6.2.19保存FIS到磁盤上...............................................231

6.2.20顯示FIS的規(guī)則.................................................232

6.2.21顯示FIS結(jié)構(gòu)的所有屬性........................................232

第7章繪圖與圖形處理........................................................235

7.1二維圖形..............................................................235

7.1.1基本平面圖形命令................................................235

7.1.2特殊平面圖形命令................................................242

7.1.3二維圖形注釋命令................................................248

7.2三維圖形..............................................................252

7.2.1三維曲線、血填色命令............................................252

7.2.2三維圖形等高線..................................................254

7.2.3曲面與網(wǎng)格圖命令................................................257

7.2.4三維數(shù)據(jù)的其他表現(xiàn)形式命令.....................................261

7.3通用圖形函數(shù)命令......................................................267

7.3.1圖形對象句柄命令................................................267

7.3.2軸的產(chǎn)生和控制命令..............................................278

7.3.3圖形句柄操作命令................................................279

7.3.4圖形窗口的控制命令..............................................281

7.4顏色與光照模式命令...................................................283

7.4.1顏色控制命令....................................................283

7.4.2色圖控制命令....................................................285

第1章矩陣及其基本運(yùn)算

MATLAB,即“矩陣實(shí)驗(yàn)室”，它是以矩陣為基本運(yùn)算單元。因此，本書從最基本的運(yùn)

算單元出發(fā)，介紹MATLAB的命令及其用法。

1.1矩陣的表示

1.1.1數(shù)值矩陣的生成

1.實(shí)數(shù)值矩陣輸入

MATLAB的強(qiáng)大功能之一體現(xiàn)在能直接處理向量或矩陣。當(dāng)然首要任務(wù)是輸入待處理

的向量或矩陣。

不管是任何矩陣（向量），我們可以直接按行方式輸入每個(gè)元素：同一行中的元素用逗

號（，）或者用空格符來分隔，且空格個(gè)數(shù)不限；不同的行用分號（；）分隔。所有元素處于

一方括號（[]）內(nèi)；當(dāng)矩陣是多維（三維以上），且方括號內(nèi)的元素是維數(shù)較低的矩陣時(shí)，

會有多重的方括號。如：

?Time=[111212345678910]

Time=

111212345678910

?X_Data=[2.323.43；4.375.98]

XData=

2.433.43

4.375.98

?vecta=[12345]

vecta=

12345

?MatrixB=[123；

?234；345]

MatrixB=123

234

345

?Null_M=[]%生成一個(gè)空矩陣

2.復(fù)數(shù)矢i陣輸入

復(fù)數(shù)矩陣有兩種生成方式:

第一種方式

例i-i

?a=2.7;b=13/25;

?C=[l,2*a+i*b,b*sqrt(a);sin(pi/4),a+5*b,3.5+1]

C=

1.00005.4000+0.5200i0.8544

0.70715.30004.5000

第2種方式

例1-2

?R=[123;456],M=[ll1213;141516]

R=

123

456

M=

111213

141516

?CN=R+i*M

CN=

1.0000+11.0000i2.0000+12.0000i3.0000+13.0000i

4.0000+14.0000i5.0000+15.0000i6.0000+16.0000i

1.1.2符號矩陣的生成

在MATLAB中輸入符號向量或者矩陣的方法和輸入數(shù)值類型的向量或者矩陣在形式上

很相像，只不過要用到符號矩陣定義函數(shù)sym,或者是用到符號定義函數(shù)syms,先定義一

些必要的符號變量，再像定義普通矩陣一樣輸入符號矩陣。

1.用命令sym定義矩陣：

這時(shí)的函數(shù)sym實(shí)際是在定義一個(gè)符號表達(dá)式，這時(shí)的符號矩陣中的元素可以是任何的

符號或者是表達(dá)式，而且長度沒有限制，只是將方括號置于用于創(chuàng)建符號表達(dá)式的單引號中。

如下例：

例1-3

?symmatrix=sym('[abc；Jack,HelpMc!>NOWAY!]?')

symmatrix=

[abc]

[JackHelpMe!NOWAY!]

?symdigits=sym('[I23；abc；sin(x)cos(y)tan(z)]')

symdigits=

[123]

[abc]

[sin(x)cos(y)tan(z)]

2.用命令syms定義矩陣

先定義矩陣中的每一個(gè)元素為一個(gè)符號變量，而后像普通矩陣一樣輸入符號矩陣。

例1-4

?symsabc；

?Ml=sym('Classical')；

?M2=sym('Jazz')；

?M3=sym('Blues')M

?symsmatrix=[abc；Ml,2,M3；int2slr([2351)1

symsmatrix=

[ab可

Be

[ClassicalJazz

1U司

[23

把數(shù)值矩陣轉(zhuǎn)化成相應(yīng)的符號矩陣

數(shù)值型和符號型在MATLAB中是不相同的，它們之間不能直接進(jìn)行轉(zhuǎn)化。MATLAB提

供了一個(gè)將數(shù)值型轉(zhuǎn)化成符號型的命令，即sym。

例1-5

?Digit_Matrix=[l/3sqrt(2)3.4234；exp(0.23)log(29)23A(-11.23)]

?SymsMatrix=sym(DigitMatrix)

結(jié)果是：

DigitMatrix=

0.33331.41423.4234

1.25863.36730.0000

SymsMatrix=

[-1/3,sqrt(2),17117/5000]

[5668230535726899*2人(-52),7582476122586655*2A(-51),5174709270083729*2A(-103)]

注意：矩陣是用分?jǐn)?shù)形式還是浮點(diǎn)形式表示的，將矩陣轉(zhuǎn)化成符號矩陣后，都將以最接

近原值的有理數(shù)形式表示或者是函數(shù)形式表示。

1.1.3大矩陣的生成

對于大型矩陣，一般創(chuàng)建M文件，以便于修改:

例1-6用M文件創(chuàng)建大矩陣,文件名為examplc.m

exm=[456468873257955

2168754488813

6545678898215

4566845896545987

5488109633771

在MATLAB窗口輸入：

?example;

?size(exm)%顯示exm的大小

ans=

56%表示exm有5行6列。

1.1.4多維數(shù)組的創(chuàng)建

函數(shù)cat

格式A=cat(n,Al,A2,…,Am)

說明n=l和n=2時(shí)分別構(gòu)造[Al；A2]和[Al,A2],都是二維數(shù)組，而n=3時(shí)可以構(gòu)造

出三維數(shù)組。

例1-7

?AI=[1,2,3;4,5,6;7,8,9];A2=Al';A3=A1-A2;

?A4=cat(3,Al,A2,A3)

A4(:,:,l)=

123

456

789

A4(:,:,2)=

147

258

369

A4(:,:,3)=

0-2-4

20-2

420

或用另?種原始方式可以定義：

例1-8

?A1=[1,2,3；4,5,6;7,8,9];A2=Al';A3=A1-A2;

?A5(:,:,1)=A1,A5(:,:,2)=A2,A5(:,:,3)=A3

A5(:,:,l)=

I23

456

789

A5(:,:,2)=

147

258

369

A5(:,:,3)=

0-2-4

20-2

420

1.1.5特殊矩陣的生成

命令全零陣

函數(shù)zeros

格式B=zeros(n)%生成nxn全零陣

B=zeros(m,n)%生成mxn全零陣

B=zeros([mn])%生成mxn全零陣

B=zeros(dl,d2,d3--,)%生成dlxd2xd3x…全零陣或數(shù)組

B=zeros([dld2d3…])%生成dlxd2x(J3x…全零陣或數(shù)組

B=zeros(size(A))%生成與矩陣A相同大小的全零陣

命令單位陣

函數(shù)eye

格式Y(jié)=eye(n)%生成nxn單位陣

Y=eye(m,n)%生成mxn單位陣

Y=eye(size(A))%生成與矩陣A相同大小的單位陣

命令全1陣

函數(shù)ones

格式Y(jié)=ones(n)%生成nxn全1陣

Y=ones(m,n)%生成mxn全1陣

Y=ones([mn])%生成mxn全1陣

Y=ones(dl,d2,d3…)％生成dlxd2xd3x…全1陣或數(shù)組

Y=ones([dld2d3…])%生成d”d2xd3x…全1陣或數(shù)組

Y=ones(size(A))%生成與矩陣A相同大小的全1陣

命令均勻分布隨機(jī)矩陣

函數(shù)rand

格式Y(jié)=rand(n)%生成nxn隨機(jī)矩陣，其元素在(0,1)內(nèi)

Y=rand(m,n)%生成mxn隨機(jī)矩陣

Y=rand([mn])%生成mxn隨機(jī)矩陣

Y=rand(m,n,p,--)%生成mxnxpx…隨機(jī)矩陣或數(shù)組

Y=rand([mnp…])%生成mxnxpx…隨機(jī)矩陣或數(shù)組

Y=rand(size(A))%生成與矩陣A相同大小的隨機(jī)矩陣

rand%無變量輸入時(shí)只產(chǎn)生一個(gè)隨機(jī)數(shù)

s=rand('state')%產(chǎn)生包括均勻發(fā)生器當(dāng)前狀態(tài)的35個(gè)元素的向量

randC^tate*,s)%使?fàn)顟B(tài)重置為s

rand('state',0)%重置發(fā)生器到初始狀態(tài)

rand(lstate,,j)%對整數(shù)j重置發(fā)生器到第j個(gè)狀態(tài)

rand(,state,,sum(100*clock))%每次重置到不同狀態(tài)

例1-9產(chǎn)生一個(gè)3X4隨機(jī)矩陣

?R=rand(3,4)

R=

0.95010.48600.45650.4447

0.23110.89130.01850.6154

0.60680.76210.82140.7919

例1-10