<MakerFile 3.0F> Aa•ÿ _ ¦ d H H $ d d TableFootnote Ñ ] Ç ] f ¢ Footnote * à * à É TOC g ¤ . . / - Ð + h Heading logon e u l e ž u i j v k u … w u h þ v j ý ý <$lastpagenum> þ <$monthname> <$daynum>, <$year> ÿ "<$monthnum>/<$daynum>/<$shortyear> ;<$monthname> <$daynum>, <$year> <$hour>:<$minute00> <$ampm> "<$monthnum>/<$daynum>/<$shortyear> <$monthname> <$daynum>, <$year> "<$monthnum>/<$daynum>/<$shortyear> <$fullfilename>B <$filename>· <$paratext[Title]><: <$paratext[Heading]> <$curpagenum>—Ï <$marker1> ÿ <$marker2>»c (Continued) Heading & Page Ò<$paratext>Ó on page <$pagenum> Page page <$pagenum> on page <$pagenum>. See Heading & Page %See Ò<$paratext>Ó Table & Page 7Table <$paranumonly>, Ò<$paratext>,Ó on page <$pagenum> + (Sheet <$tblsheetnum> of <$tblsheetcount>) • • • • | |Q A Ž } }Q A • ~ ~PÇ • • •PÉ ‘ € €P He ’ • •P “ ƒ ™Qž A ” ¨ ¨ Au ñ ‘ º ñ = j v ò = ó G u … ô õ ö ÷ ø ù ú û ü ý þ ÿ A B A = M = = = = < < = < < S S = = = = = þ $la > $mo <$d $ye " m>/ /<$ > $mo <$d $ye r>: 0> thn num yea = = me> = >, Q = thn R num = yea Q = ena Q = = = = = = J = e>· $pa tle t[H pag = ker = = >»c - = (Co N ! " # $ % & ' ( ) * + , . / 0 1 2 3 4 5 6 7 8 9 : ; < = > ? @ A B C D E F G H I J K L M N O P Q R S T = = = R = N = = = O T T T = = L = = Q = = = N = = O : : : = N = O O : = = : a = = = = = = = R = = J J N = © age tex ge > e gen Se & Ò< >Ó $pa Pa <$ ly> tex age m> (Sh she <$ oun |Q A • ~ •PÉ He ’ • ™Qž Au ñ = ò = ó G ô A õ B ö A ÷ = ø M ù = ú = û = ü = ý < þ < ÿ = U V W X Y Z [ \ ] ^ _ : : = N O : : N = = = = ` O = a P < < S S = = = = = = Q b P = c = R d e f g h i j k l m n o p q r s t u v w x y z { | } ~ • € • ‚ ƒ „ … † ‡ ˆ ‰ Š ‹ Œ • Ž • • ‘ ’ “ ” • – — ˜ = N = = = J E E = R = = = J J Q Q J J = R = = = = J Q Q Q J Q Q Q Q N : : : = = = = = = = J Q Q Q Q J Q a = Q = Q = = = = = = J = = = = - = N = ! = " = # R $ = % N & = ' = ( = ) O * T + T , T - = . = / L 0 = 1 = 2 Q 3 = 4 = 5 = 6 N 7 = 8 = 9 O : : ; : < : = = > N ? = @ O A O B : C = ™ š › œ • ž Ÿ ¡ ¢ £ ¤ ¥ ¦ § ¨ © ª « ¬ ® ¯ ° ± ² ³ ´ µ ¶ · ¸ ¹ ¿ Ò Ó Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à á â ã ä å a a J Q ^ ^ ^ ^ ^ J = R R Q R R R R R R R R R R R R R R R R R R R Q O Q = N J Q Q Q Q Q Q Q Q Q Q ; ; ; ; ; D E F G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y = : a = = = = = = = R = = J J N = : : = N O : : N = = = O P P = = N = = = J E E = R = = = J J Q Q J J = R = æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù ú û ü ý þ ÿ O N Q Q Q K K K K K Q Q R = = < = J Q [ J Q [ J = R z { | } ~ • € • ‚ ƒ „ … † ‡ ˆ ‰ Š ‹ Œ • Ž • • ‘ ’ “ = = = J Q Q Q J Q Q Q Q N : : : = = = = = = = J Q Q = ” Q = • Q Q – J Q — Q = ˜ a = ™ a = š a Q › J Q œ Q Q • ^ Q ž ^ J Ÿ ^ Q ^ Q ¡ ^ J ¢ J Q £ = Q ¤ R = ¥ R = ¦ Q J § R Q ¨ R Q © R J ª R R « R = ¬ R = - R = ® R ! = ¯ R " J ° R # Q ± R $ Q ² R % Q ³ R & Q ´ R ' J µ R ( C ¶ R ) C · R * C ¸ R + C ¹ R , C ¿ Q - C Ò O . C Ó Q / C Ô = 0 C Õ N 1 = Ö J 2 R × Q 3 = Ø Q 4 = Ù Q 5 = Ú Q 8 = Û Q 9 J Ü Q : Q Ý Q ; Q Þ Q < Q ß Q = Q à Q > Q á ; ? Q © @ Q ã ; A Q ä ; B Q å ; C Q æ O D Q ç N E Q è Q F Q é Q G Q ê Q H Q ë K I O ì K J O í K K O î K L O ï K M O ð Q N O ñ Q O O ò R P O ó = Q O ô = R O õ < S N ö = T O ÷ J U O ø Q V O ù [ W O ú J X O û Q Y O ü [ Z O ý J [ O þ = \ O ÿ R ] O = ^ O = _ O Q ` O Q a O = b O = c O = d O Q e O Q f O Q g O Q h O J i O Q j O Q k O J l Q Q m Q Q n Q = o N = p = J q O Q r O Q s O J t O R u O = v O = w O = x O ! = y O " J z O # Q { O $ Q | O % Q } O & Q ~ O ' J • O ( C € O ) C • O * C ‚ O + C ƒ O , C „ O - C … O . C † O / C ‡ N 0 C ˆ Q 1 = ‰ Q 2 R Š Q 3 = ‹ Q 4 = Œ Q 5 = • Q 8 = Ž Q 9 J • N : Q • J ; Q ‘ Q < Q ’ Q = Q “ Q > Q ” Q ? Q • Q @ Q – Q A Q — N B Q ˜ J C Q ™ Q D Q š Q E Q › Q F Q œ Q G Q • Q H Q ž N I O Ÿ Q J O Q K O ¡ Q L O ¢ Q M O £ Q N O ¤ Q O O Ý = P O ß R Q O à < R O á = S N â = T O ã = U O ä = V O å = W O æ = X O ç R Y O è J Z O é J [ O ê Q \ O ë J ] O ì Q ^ O í Q _ O î J ` O ï J a O ð Q b O ñ Q c O ò J d O ó J e O ô Q f O õ Q g O ö J h O ÷ J i O ø Q j O ù Q k O = l Q = m Q = n Q = o N = p = = q O = r O R s O = t O N u O N v O N w O = x O = y O = z O = { O = | O = } O = ~ O - < • O = € O = • O ! N ‚ O " N ƒ O # = „ O $ N … O % N † O & N ‡ N ' N ˆ Q ( N ‰ Q ) N Š Q * = ‹ Q + = Œ Q , = • Q - = Ž Q . = • N / = • J 0 = ‘ Q 1 < ’ Q 3 R “ Q 4 = ” Q 5 = • Q 7 = – Q 8 D — N = = ˜ J > H ™ Q ? = š Q @ = › Q A = œ Q B H • Q C I ž N D I Ÿ Q E = Q F = ¡ Q M N ¢ Q N N £ Q O N ¤ Q P Q R S T U V W X ` c d f g h o q r u w | } ~ • € ‚ ƒ „ … N N N N J Q Q Q Q = Q J ^ N : J Q Q Q Q Q Q Q C C N N Q Q = † Q Ý ß à á â ã ä å æ ç è é ê ë ì í î ï ð ñ ò ó ô õ ö ÷ ø ù = R < = = = = = = R J J Q J Q Q J J Q Q J J Q Q J J Q Q = ‰ J = m - š 9 d m Ñ = • | = d n Ñ N • } = d o Ñ = p { = H “Ó 3Kº ^¨ p Ò q o = H R H R ¡ N ^¨ q Ò p r o N H z·¸ H z·¸ ¡ N Single LineŠ Q H ' ´ Footnote = H r Ó q t o qùv ?ü- Q s s • ™ ^¨ t Õ r u o Q H °·¸ H Double Line H s × °·¸ º Footnote = r ¡ Q Ô R Q – Q H §ùv D£f u Ó t x o v w Double Line Ô T J è J H x Ó u z o v Ö Ô † Ô w u w Ö v N u Q Ô = Ô J Single Lineî J y y Ô H Z y Ö ´ x z Ó x { o Q Ô TableFootnote E¸¾ GX- RŸb ^¨ { Ó z o E¸¾ P o E¸¾ P o ¡ TableFootnote 6 6 ø ¬ | Ó € m Ñ 6 6 ø ¬ • ñ ™ = 6 6 ø ¬ } Ó ~ ` n Kº 6 6 ø Ž ò ™ü¨ 6 ø ~ Ó • } n ¬ Fo ` ·¸ 6 • Ó ó › s Page = 6 ø • Ó ~ n ø ªª j 11 k ªª UU h R 6 • ô › Q 6 öUV ø € Ó • | m ø £f ªª ªª UU ` in 6 ‘ õ ›in 6 • Ó öUV ø w ø € m ªª ªª UU ` 6 ’ ö › Q ø ªª d ‚ Ñ ªª UU ` ƒ § Ó 6 6 ø ¬ ƒ Ó œ ‚ 6 6 ø ¬ 0 Q “ ÷ ™ ` … ` á ™ $ ` â ™ 2 ` ã ™ @ ™¸¾ - N ™ \ ™ o j ™e x ™ † ™ ” ™ ª ø • ø ¼ ù ™ º Ê ú ™ Ø ü ™ æ û ™ ò - ô ` ` ` ` ` ` ` ` Introduction to Matlab ` ` ` ` ` å ™ Ó ` æ ™ ` # ª * + , R . / ô ™ ™ , ™Pa : ™ H ™ V ™ d ™ r ™ € 0 ™ Ó Ž ` ` ` ` ` ` ` ` ` ™ V œ ` ™ ª h ý ™ w h July 20, 1993 i ¸ ` þ ™ Ó Thomas D. Citriniti Æ ` õ ™ Core Engineering Ô ` ™ ö !Rensselaer Polytechnic Institute â ` ™ § Troy, New York 12180 ð ` ™ þ ` ™ ` ™ ` ™ ÷ ( ` ™ á 6 ` ™ â D ` ™ ã R ™ bThis tutorial was written to introduce MATLAB and provide examples to use in conjunction with the ` ™ kPreface program. It assumes the user has sufficient familiarity with the operating system to logon, create n @ ™ files, and change directories. û | ÿ ™ å Š ™ æ ˜ ` ` ` ™ ¦ ` ä ™ # d „ Ñ 6 … … 6 ø ¬ … Ó „ - 6 6 “ ô 5 ™ 0 UT ø ¬ ƒ ‡ 0 ` UT ªª ` - 1.1 Accessing MATLAB %ÿþ ` ™ 3ÿþ ™ 1 _Once a login session has been established and a UNIX windows has been opened, two commands are Aÿþ @ ™se required to start MATLAB: Oÿþ ` ™ § r ]ÿþ ` ™ % setup matlab-4.0a kÿþ ™ ` yÿþ ` ™ % matlab ‡ÿþ ` ™ •ÿþ ™ fThis begins execution. Note a change in prompt (from % to >>). On line help is available for most MAT £ÿþ @ ™e !LAB commands. To invoke it type: ±ÿþ ` ™ ¿ÿþ ` ™ I ¨ >> help s Íÿþ ` ™fa i Ûÿþ ` ™ra 8which provides a list of help topics to choose from, or s, éÿþ ` - ™or . ÷ÿþ ` ¨ ÿ >> help topic ÿþ ` ™ ÿþ ` ! ™ Jwhich invokes a help file for the topic specified. Try plot ™ . !ÿþ ` " ™ Ó 1UR UT ªª ` # 2.1 Entering Matrices ?ÿü ` ¨ help ™ ô Mÿü ™ gMatrices can be entered into matlab from the command line or thorough data files, we will be doing all ssi [ÿü @ ™is dour work from the command line. Here is how to create a 3x3 matrix and assign it to the variable A. TL iÿü ` $ ™ § wÿü ` % ¨ >> A = [ 1 2 3; 4 5 6; 7 8 9 ] …ÿü ` & ™ÿþ “ÿü ` ' ™at 2This produces the following response from MATLAB: ¡ÿü ` ( ™n. t ¯ÿü ` ) ¬pt A = ½ÿü ` * ¬el 1 2 3 lab Ëÿü ` + ¬ÿþ 4 5 6 @ Ùÿü ` , ¬. 7 8 9 it çÿü ` - ™ õÿü ` . ™ \Note that MATLAB is case sensitive. Thus, when using variables, ÒAÓ is not the same as ÒaÓ. h ÿü ` 0 ™se o ÿü 1 ™ ^Random numbers and matrices. RAND(N) is an N-by-N matrix with random entries. RAND(M,N) is an ÿü 1 ™he bM-by-N matrix with random entries. RAND(A) is the same size as A. RAND with no arguments is a sca -ÿü @ 1 ™ri 4lar whose value changes each time it is referenced. ic ;ÿü ` 5 ™in m Iÿü ` 6 ¨ma >> A = rand(3,2) Wÿü ` 7 ™l d eÿü ` 8 ™ ¬ A = s sÿü ` 9 ¬e m •ÿü ` : ¬ho 0.2342 0.3243 ma •ÿü ` ; ¬ t 0.1334 0.1212 TL •ÿü ` < ¬ 0.8656 0.4543 ¤ ¨ d † Ñ ‡ ‡ $ 6 7 ¬ ‡ Ó † s $ 6 “ ( > ¨ >> A(3,2) = 7 ¬ 0 … ‰ ` ` ? ™el 1 $ ` @ ¬ ans = 2 ` A ¬ @ ` B ¬it 0.4543 N ` C ™ \ D ™t wOnce a matrix has been assigned to variable A, it is stored for theduration of the MATLAB session, or until you assign 1 j @ D ™rs 7new values to A. To clear A of any value, use clear A: tri x ` G ™n ü † ` H ™M- >> clear A th ” ` I ™AN ) ¢ ` J ™as &To empty all variables at once, type: ° ` K ™la h ¾ ` L ™ea >> clear Ì ` M ™ÿü ÛUT UT ªª ` N -ÿü 3.1 Matrix operations éÿþ ` O ™ ÷ÿþ P ™ rWhile matrix operations act upon an entire matrix, MATLABÕs array operators act upon the individual elements of a ÿþ @ P ™ /matrix. For example, using A and B defined as: ÿþ ` 3 ™ !ÿþ ` 4 ™ /ÿþ S ¨ =ÿþ T ™ ‰ Kÿþ U ¬ Yÿþ V ¬ gÿþ W ™ uÿþ X ¨ ƒÿþ Y ¬ 0 ‘ÿþ Z ¬ Ÿÿþ [ ¬On t -ÿþ \ ¨ne ` >> A = [ 1 2; 3 4 ] ` ¬ A = ` 1 2 ` 3 4 ` ` >> B = [ 5 6; 7 8 ] ` B = ` 5 6 ` 7 8 ` >> C = A * B »ÿþ ` ] ™th r Éÿþ ` ^ ™B )produces the Matrix operation result of: ×ÿþ ` _ ™es åÿþ ` ` ¬ny C = óÿþ ` a ¬ F 19 22 or A(1,1)*B(1,1)+A(1,2)*B(2,1) A(1,1)*B(1,2)+A(1,2)*B(2,2) ÿþ ` b ¬ G 43 50 or A(2,1)*B(1,1)+A(2,2)*B(2,1) A(2,1)*B(1,2)+A(2,2)*B(2,2) ¾ ÿþ ` c ™ > where ÿþ ` d ™ÿü +ÿþ ` e ¨ÿü >> C = A .* B ati 9ÿþ ` f ™ O Gÿþ ` g ™ P (produces the Array operation result of: en Uÿþ ` h ™BÕ r cÿþ ` i ©up ¬ C = v qÿþ ` j ¬a ' 5 12 or A(1,1)*B(1,1) A(1,2)*B(1,2) , u •ÿþ ` k ¬ed ( 21 32 or A(2,1)*B(2,1) A(2,2)*B(2,2) •ÿþ ` l ™ ›ÿþ ` ™[ ; d ˆ Ñ A ‰ ‰ 6 6 ø ¬ ‰ Ó ˆ 6 6 ø ¬ ‡ ‹ “ 0B UT UT ªª ` m -ÿþ ,4.1 Statements, expressions, and variables. 5 ÿþ ` n ™ [ n %ÿþ ` o ™ NThe semicolon (;) following a statement will suppress printing of the result. 3ÿþ ` p ™n u Aÿþ ` q © >> A = [ 1 2; 3 4 ] Oÿþ ` r ©ÿþ ]ÿþ ` ©22 ¬ A = ) kÿþ ` s ¬,1 1 2 1 yÿþ ` t ¬2, 3 4 þ ‡ÿþ ` ¤ 5 •ÿþ ` u ©1) >> A = [ 1 2; 3 4 ]; £ÿþ ` v ©) >> ÿþ ±ÿþ ` w ™wh ÁUR UT ªª ` x -ÿü 5.1 Matrix Building Functions. A Ïÿü ` y ™ f Ýÿü ` z ™ KAll zeros. ZEROS(N) is an N-by-N matrix of zeros. ZEROS(M,N) is an M-by-N cÿþ ëÿü ` { ™ ?matrix of zeros. ZEROS(A) is the same size as A and all zeros. ,2) ùÿü ` | ™ k d ÿü ` } ©*B >> A = zeros(2,3) ÿü ` ~ ¬ A = #ÿü ` • ¬ ; 0 0 0 d 1ÿü ` € ¬ 0 0 0 ?ÿü ` • © >> A = zeros(3) Mÿü ` ‚ ¬ A= [ÿü ` ƒ ¬ 0 0 0 iÿü ` „ ¬ m 0 0 0 St wÿü ` … ¬on 0 0 0 ria …ÿü ` † ¨ >> A = rand(3) “ÿü ` g ©se o ¡ÿü ` ‡ ¬a A= men ¯ÿü ` E ¬in 0.3245 0.6457 0.7324 ½ÿü ` ˆ ¬ÿþ 0.4734 0.2333 0.2168 [ Ëÿü ` ‰ ¬ 0.1205 0.3913 0.9573 Ùÿü ` h ¤ ) þ çÿü ` Š ™ 1 1 õÿü ` ‹ ™2, PIf V is a row or column vector with N components, DIAG(V,K) is a square matrix ÿü ` Œ ™ÿþ Mof order N+ABS(K) with the elements of V on the K-th diagonal. K = 0 is the ÿü ` • ™ Mmain diagonal, K > 0 is above the main diagonal and K < 0 is below the main O ÿü ` Ž ™N Odiagonal. DIAG(V) simply puts V on the main diagonal. (note: Your results will ze -ÿü ` • ™ Cdiffer since the rand function returns an array of randum results) ` ~ ;ÿü ` • ™ Iÿü ` ‘ © >> X = diag(A) Wÿü ` R © eÿü ` ’ ¬ze X = sÿü ` “ ¬ 0.3245 ÿü •ÿü ” ¬ 0 ÿü •ÿü • ¬ 0 ÿü •ÿü c ¤ 0 0 ` 0.2333 ` 0.9573 ` d Š Ñ ra g 6 6 ø ‹ ‹ ¬ ‹ Ó Š en 6 “ – © 0 Q 0 — ü ˜ þ © $ ¬73 2 ¬ 6 ø ¬ 32 1ÿü >> X = diag(diag(A)) ` ` X = ` ‰ • ` 0.3245 0 0 Š @ ™ ¬ ` 0 0.2333 0 i N š ¬ve ` 0 0 0.9573 mp \ ` F ¤ # s j ` › ©ÿü - >> B = [ A, zeros(3,2);zeros(2,3), eye(2) ] e x ` d ©Kd † ` œ ¬th B = ü ” ` • ¬ma 0.3245 0.6457 0.7324 0 0 he ¢ ` ž ¬K 0.4734 0.2333 0.2168 0 0 ° ` Ÿ ¬on 0.1205 0.3913 0.9573 0 0 the ¾ ` ¬ot 0 0 0 1.0000 0 l Ì ` ¡ ¬ • 0 0 0 0 1.0000 t Ú ` f ¤et s è ` ¢ ©m ¬ >> ÿü ö o © © ` ü / ™ > gTRIU(X) is the upper triangular part of X. TRIU(X,K) is the elements on and above the K-th diagonal of @ / ™33 fX. K = 0 is the main diagonal, K > 0 is above the main diagonal and K < 0 is below the main diagonal. ` Ô ™ . ` Õ ¨ >> a = rand(5),b = triu(a) < ` Ö ©32 J ` × ¬ – a = > X ` Ø ¬) f ` Ù ¬ 0 $ 0.9103 0.3282 0.2470 0.0727 0.7665 t ` Ú ¬0 $ 0.7622 0.6326 0.9826 0.6316 0.4777 i ‚ ` Û ¬ve $ 0.2625 0.7564 0.7227 0.8847 0.2378 s • ` Ü ¬ÿü $ 0.0475 0.9910 0.7534 0.2727 0.2749 ey ž ` Ý ¬ $ 0.7361 0.3653 0.6515 0.4364 0.3593 ü Þ 0 ß ¬ ` ¬ma º ` ¬4 b = È ` à ¬K 0 Ö ` á ¬8 ( 0.9103 0.3282 0.2470 0.0727 0.7665 13 ä ` â ¬ # 0 0.6326 0.9826 0.6316 0.4777 Ì ò ` ã ¬ 0 - 0 0 0.7227 0.8847 0.2378 ` ä ¬ 0 0 0 0.2727 0.2749 ` å ü R æ n ç ¬ 0 0 0 ¬ t * ¨RI >> a == b el 8 è ¬e F é ¬ ans = T ê ¬ i h b ë ¬ > 1 1 in p ì ¬0 0 1 di ~ í ¬ 0 0 Œ î ¬a 0 0 (a š ï ¬ Ö 0 0 0 ¨ ð ¬ ¤ d 0 ` 0.3593 ` ` ` ` ` 1 ` 1 ` 1 ` 0 ` 1 1 1 1 1 1 1 1 0 1 ` Œ Ñ 0 • • 07 6 6 ø ¬ • Ó Œ 0 6 6 ø ¬ ‹ • “ 0 Û UT UT ªª ` ò -64 6.1 Scalar functions. ÿþ ` ó ™ÿü 0 %ÿþ ` ô ™4 iScalar functions are math functions that operate on a single value at a time throughout the whole array. 3ÿþ ` ö ™ Aÿþ ` ÷ © 0 " >> a = [ 2*pi 3*pi/2 pi pi/2 0 ] Oÿþ ` q ¬ 0 6 ]ÿþ ` ø ¬ â a = kÿþ ` ù ¬ 0 6.2832 4.7124 3.1416 1.5708 0 ¬ 0 yÿþ ` r ¬0. 7 ‡ÿþ ` ú © >> sin(a) 0 •ÿþ ` u ¬ £ÿþ ` ¬ ans = ±ÿþ ` ¬ ! -0.0000 -1.0000 0.0000 1.0000 0 ¿ÿþ ` ¬ Íÿþ ` ý © >> û 0 ü > w é Ûÿþ þ ™ ëUR ÿ ùÿü ` UT ªª ` 7.1 Vector functions. ` ™ ÿü ™1 lVector functions are math functions that operate on a vector, a vector is a single row array. MAX(X) is the ÿü ™ dlargest element in X. For matrices, MAX(X) is a vector containing the maximum element from each col Ó #ÿü ™ dumn. [Y,I] = MAX(X) stores the indices of the maximum values in vector I. MAX(X,Y) returns a matrix 1ÿü @ ™ Jthe same size as X and Y with the largest elements taken from X or Y. £ ?ÿü ` ‚ ¨ou u Mÿü ` ƒ ¨ >> a = [ 1 2 3; 4 5 6; 7 8 9 ] [ÿü ` Ò ¬[ i iÿü ` ¬] a = wÿü ` ¬ÿþ ` ø …ÿü 1 2 3 ` ¬ 4 5 6 ¬ 0 “ÿü ` ¬.1 7 8 9 8 0 ¡ÿü Ó ¬ r . ¯ÿü ` ` š ú >> max(a) (a) ½ÿü ` ¬ u Ëÿü ` ¬ û ans = s = Ùÿü ` ¬ ü 7 8 9 .00 çÿü ` ¬1. 0 õÿü ` š w >> max(max(a)) ÿü ` ¬ÿþ ÿü „ ¬UR ` ÿ ÿü ` ans = ` ¬un s. -ÿü … ¤ ;ÿü 9 ` ™1 gFor vectors, SUM(X) is the sum of the elements of X. For matrices, SUM(X) is a row vector with the sum th Iÿü @ ™ 2over each column. SUM(DIAG(X)) is the trace of X. Wÿü ` ™g eÿü ` ©om sÿü š >> sum(a) ` ¬ [ ] •ÿü ` † ¬he ans = of •ÿü ` ¬ i 12 15 18 •ÿü ` © i d Ž Ñ s • • e 6 6 ø ¬ • Ó Ž . “ ƒ 6 6 ø 0 > UT ¬ • ‘ UT ªª ` - 6 ÿþ 8.1 Matrix functions. ` ™ %ÿþ ` ™ÿü OMatrix functions are math functions that are specific to matrix manipulations. 7 3ÿþ ` ™ bEigenvalues and eigenvectors. EIG(X) is a vector containing the eigenvalues of a square matrix X. Aÿþ ` ! ™ ü Oÿþ ` " © >> y = eig(a) õÿü ]ÿþ ` | ¬ > a kÿþ ` # ¬ yÿþ y = ` $ ¬ a ‡ÿþ 16.1168 ` % ¬ 9 •ÿþ -1.1168 ` & … ü } e ¬ -0.0000 £ÿþ ¬Fo ±ÿþ ` ` ' ©he >> [U,D] = eig(a) of ¿ÿþ ` ~ ¬UM Íÿþ ` ( ¬th m Ûÿþ U = ` ) ¬ 0.2320 ( éÿþ 0 .7858 ` 0.4082 * ¬ac g ÷ÿþ 0.5253 0.0868 -0.8165 ` + ¬ 0.8187 -0.6123 ÿü ÿþ ` • ¬ ] ü ÿþ ` 0.4082 , ¬ a !ÿþ D = ` - ¬ i 16.1168 0 /ÿþ ` 0 . ¬ =ÿþ 0 -1.1168 ` 0 / ¬ 6 Kÿþ € ¬ Yÿþ 0 0 -0.0000 ` ` 0 ¬ gÿþ >> ` 1 ™ ƒ uÿþ ` 5 ™ªª …UR UT ªª ` 2 -un $9.1 Submatrices and colon notation. “ÿü ` 3 ™ix n ¡ÿü 4 ™nc gMatrices can be referenced as whole matrices or submatrices within larger matrices by way of colon ref IG( ¯ÿü 4 ™ta ierences. Colon notation can be used as an implied for loop with the syntax from:step:to which define the ü ½ÿü @ 4 ™ > Iloop contraints. (the transposed) Ëÿü ` ¬ Õ ™ denotes the matrix should be 8 ™ 9 þ Ùÿü ` 9 © o çÿü >> x = [ 0.0:0.1:2.0 ]Õ ` : ¬ ' õÿü ` ; ¬a) þ ÿü x = ` < ¬ þ ÿü ` = ¬ U Ûÿþ ÿü 0 ` > ¬ 0 78 -ÿü 0.1000 ` ? ¬ ac ;ÿü 0.2000 ` @ ¬-0 ÿþ Iÿü 0.3000 ` A ¬ 0 12 Wÿü 0.4000 ` B ¬ ] eÿü 0.5000 ` C ¬ , 0.6000 sÿü ` D ¬ i •ÿü 0.7000 ` E ¬ •ÿü 0.8000 ` F ¬ÿþ 0.9000 / •ÿü ` G ¬00 1.0000 d • Ñ ‘ ‘ 1 6 6 ø ¬ ‘ Ó • UR 6 “ 6 n ø ¬ 1on • “ ` H ¬ 3 ÿü 1.1000 ` I ¬Ma b $ 1.2000 ` J ¬ol o 2 1.3000 ` K ¬in tr @ 1.4000 ` L ¬on ÿü N 1.5000 ` M ¬er on \ 1.6000 ` N ¬se pl j 1.7000 ` O ¬th ro x 1.8000 ` P ¬fi ÿü † 1.9000 ` Q ¬lo nt ” 2.0000 ` R ¬de ¢ ` S ¨d Ëÿü ° >> y = sin(x) ` T ¬ þ ¾ ` U ¬ > Ì y = ` V ¬ÿü Ú 0 ` W ¬ a) è 0.0998 ` X ¬ þ ö 0.1987 ` Y ¬ U ÿü 0.2955 ` Z ¬ 0 ÿü 0.3894 ` [ ¬ 0 ÿü 0.4794 ` \ ¬ 0 ÿü . 0.5646 ` ] ¬ 0 ÿü < 0.6442 ` ^ ¬ 0 ÿü J 0.7174 ` _ ¬ 0 ÿü X 0.7833 ` ` ¬ 0 ÿü f 0.8415 ` a ¬ . 0.8912 ÿü t ` b ¬ 0 ÿü ‚ 0.9320 ` c ¬ 1 • 0.9636 ` d ¬ ž 0.9854 ` e ¬ ¬ 0.9975 ` f ¬ º 0.9996 ` g ¬ È 0.9917 ` h ¬ 0.9738 H Ö ` i ¬ 0.9463 I ä ` j ¬ 0.9093 J ò ` k ¬ ` o ¨00 >> [x y] ` p ™00 ¬ ans = ` q ¬on * 0 0 ` r ¬00 ` 0.1000 0.0998 O 8 ` s ¬ 1 F 0.2000 0.1987 ` t ¬ nt T 0.3000 0.2955 ` u ¬de ` 0.4000 0.3894 S b ` v ¬x) ` 0.5000 0.4794 T p ` w ¬ Ì ~ 0.6000 0.5646 ` x ¬ 0 ` 0.7000 0.6442 W Œ ` y ¬ 0 š 0.8000 0.7174 ` z ¬ ÿü ¨ 0.9000 0.7833 ` { ¬ 0 1.0000 0.8415 d ’ Ñ “ “ 6 6 ø ¬ “ Ó ’ 6 “ 6 ø 0 ¬ 183 ‘ • ` | ¬ ` f 1.1000 0.8912 ` } ¬ $ 1.2000 0.9320 ` ~ ¬ÿü ¬ 1 2 1.3000 0.9636 ` • ¬ 985 @ 1.4000 0.9854 ` € ¬ e ¬ 1.5000 0.9975 N ` • ¬ 0 \ 1.6000 0.9996 ` ‚ ¬ ¬ j 1.7000 0.9917 ` ƒ ¬ 946 x 1.8000 0.9738 ` „ ¬ j ò 1.9000 0.9463 † ` … ¬ 2.0000 0.9093 ¨00 ” ` † ¬ ¢ ` ‡ ¨ = >> a(1:3,2) ° ` ˆ ¬ ¾ ` ‰ ¬00 ` ans = O Ì ` Š ¬ Ú ` ‹ ¬ 2 è ` Œ ¬0. nt ö 5 ` • ¬de 400 8 ` Ž ¬ ` • ¨0. >> a(1:3,3) ` • ©00 . ` ‘ ¬ 0 < ans = ` ’ ¬ J ` “ ¬00 174 X 3 ` ” ¬ z 6 0 f ` • ¬ 9 t ` – ¬0. ‚ ` — ¨ >> a(1:2,2) • ` ˜ © ž ` ™ ¬ ’ ¬ ans = ` š ¬ º ` › ¬ 0 È 2 ` œ ¬ ¬ 5 ` Ö ` • ¬ ä ` ž ¨ 1 >> a(2:3,2) ò ` Ÿ ¬ÿü ` ¬ ` ans = • ` ¡ ¬54 @ 5 ` ¢ ¬ 1 0. * 8 ` £ ¬ 8 ` ¤ ¬96 F >> ` l ¤ 1 0 T ` m ¤ b ` n ¤0. 8 p 2 ¤ ` „ j ~ ¦ ¤ Œ ¿ ¤ 2 0 š ñ ¤ ¨ ‰ © ` ` ` ` d ” Ñ • • 6 6 ø ¬ • Ó ” Š 6 “ 6 ø ¬ ž ž " “ — Œ UT UT ªª ` ç ÿþ 10.1 Output Format. ` Ý ™ %ÿþ ` è © > >> a = 1 3ÿþ ` é ©00 Aÿþ ¬ a = ` ê ¬ = 1 < Oÿþ ` ë © >> a = 1.2 “ ]ÿþ ` ì ¬ ” kÿþ a = ` í ¬ yÿþ 1.2000 ` î © ‡ÿþ >> format short ` ï ©a( >> a = 1.2 •ÿþ ` ð ¬ £ÿþ a = ` ñ ¬ ±ÿþ 1.2000 ` ò © 0 ¿ÿþ >> format long ` ó © >> a = 1.2 Íÿþ ` ô ¬ Ûÿþ a = ` õ ¬3, ¬ÿü éÿþ 1.20000000000000 ` ö © >> format short e ÷ÿþ ` ÷ © >> a = 1.2 ¢ ÿþ ` ø ¬ £ ÿþ a = ` ù ¬ > !ÿþ 1.20000 ` ¤ l 1 /ÿþ ¤ ` m ?UR - UT ªª ` n 11.1 Graphics. Mÿü ` ™ [ÿü ` ™ ¨ >> x = -4:0.01:4; iÿü ` ¨ >> y = sin(x); wÿü ` ¨ , >> plot(x,y), title(ÔExample Sin curve.Õ); …ÿü ` ¨ >> “ÿü ` ™ ž ¡ÿü h ™ e T y • ` - ™10 Fig1: Section 13 example plot. ` Ý ‡ • ™ è • • ™ÿþ £ • = ™ a ` ` ` d – Ñ — — 1. 6 6 ø ¬ — Ó – ÿþ 6 “ 6 ø ¬ ¢ ¢ • ™ î £ ™ÿþ ! ¨ > ` ¨ >> x = -1.5:0.01:1.5; ` >> y = exp(-x.^2); ñ $ ` " ¨ÿþ >> plot(x,y,Õ--gÕ) fo 2 ` $ ¨ >> xx = -2:.1:2; @ ` % ¨ >> yy = xx; N ` & ¨3, >> [x,y] = meshdom(xx,yy); \ ` ' ¨fo >> z = exp(-x.^2 - y.^2); ` ÷ j ( ¨ ` ¢ ' >> mesh(z), title(ÔNeat mesh plot.Õ); ` ù x ) ¨ÿþ h >> © f l N(õ ` 1 ™ Fig2: Section 13 example plot. 1 G ]~I UT ªª ` 3 m~G UT ªª ` ¤ }~E UT ªª ` ¥ -x 4 •~C UT ªª ` § •~A UT ªª ` ¨ -si ) -~? UT ªª ` © ½~= ª -it Ô Í~; « -Õ) Ý~9 ¬ > í~7 - - UT ªª ` UT ªª ` UT ªª ` UT ªª ` ž ý~5 ® - UT ªª ` ~3 UT ¯ - ~1 ° -n e -~/ ± - • =~² - • M~+ ³ - • ]~) ´ m~' µ }~% ¶ •~# · Ó •~! ¸ - ªª ` d UT ªª ` UT ªª ` UT ªª ` UT ªª ` UT ªª ` UT ªª ` UT ªª ` UT ªª ` UT ªª ` ˜ Ñ î £ 6 $ ™ ™ ø Ð ™ Ó ˜ 6 “ $ ø Ð ¤ ¤ — ñ $ UT UT ªª ` ¹ -pl +An additional example of function plotting = - UR UT ªª ` ß ® % 'ÿü 4 ™ } mThe First Derivative test for Relative Extrema. This example will use the derivative of a function to locate 5ÿü @ 4 ™me Yits relative maximum points. The theorem for testing for relative extrema is defined as: Cÿü ` 7 ™ e p Qÿü 8 ™UT ~ If ¦ f ™ has a relative minimum or relative maximum when x = c, then either (i) ¦ fÕ(c) ™ = 0 or (ii) ¦ fÕ(c) ™ is _ÿü 8 ™ªª tundefined. That is, c is a critical number of ¦ f ™ . Based on this theorem, locate all relative extrema for the ªª mÿü @ 8 ™UT function: {ÿü ` = ™ ± • ‰ÿü ` > ™ ² 1 ¦ f(x) = 2x § 3 ¦ - 3x § 2 ¦ - 36x + 14 —ÿü ` ? ™ µ ¥ÿü ` @ ™ ¶ HSolution: By setting the derivative of ¦ f ™ equal to zero, we have Ñ ³ÿü ` A ™ Áÿü ` B ™ & ¦ fÕ(x) = 6x ¯ 2 ¦ -6x - 36 = 0 Ïÿü ` C ¦ 6(x ¯ 2 ¦ - x - 6) = 0 UT Ýÿü ` D ¦dd 6(x - 3)(x + 2) = 0 ëÿü ` E ™UR T ùÿü F ™ÿü ~ Since ¦ f ™ Õ is defined for all real numbers, the only critical numbers of ¦ f ™ are -2 and 3. To test this proof we ÿü F ™me mcan create matrices to represent our function and an interval to operate on, the interval should include the e ÿü F ™ 8 iabove stated critical numbers. To create the interval using arrays, use the colon notation: (remember to #ÿü @ F ™™ ;use the Array operator notation (.^) for the powers of x). nu 1ÿü ` ™ B d ?ÿü M ¨lo >> m Mÿü N ¨ 0 >> ÿü [ÿü O ¨ 5 >> iÿü P ¨ÿü >> grid; wÿü Q ¨ s >> f …ÿü R ¨we >> “ÿü S ¨ 0 ¡ÿü T © 2 ¯ÿü U ¬x 2) ½ÿü V ¬ E 4 >> in Ëÿü W ¬mb n Ùÿü X ¬ s çÿü ` ™ÿü Âûã ` X = -4.0:.1:5.0; ` FofX = 2.0*X.^3 - 3.0*X.^2 - 36.0*X + 14.0; ` plot(X,FofX),title(ÔTest for Relative ExtremaÕ); ` ` [Y,Max_I) = max(FofX); ` [Y,Min_I) = min(FofX); ` >> Relative_Maximum = [ X(Max_I) FofX(Max_I) ] ` ¬ Relative_Maximum = ` -2 58 ` ¨ Relative_Minimum = [ X(Min_I) FofX(Min_I) ] ` Relative_Minimum = ` 3.0000 -67.0000 h g ` à ™te 8Figure 3: Plot of results to Test for Relative Extrema. ra H ™ Ô œ Ø ƒ • ‚ t e H ™ ™ d c Q « ¹ • Ù œ ¦ ‚ er l Q « « ion ±-¸ Ûÿü • É “ ž Ó ” ™ ). Ÿ Ÿ • e or õ€ ½-¨ Ÿ Ó ž – Y<c>Screen1.xwd<U>/afs/rpi.edu/home/62/citrit/consult/projec t/preface93/matlab/Screen1.xwd0 d Ñ ÿü ¨ ¨ > ¯m‘ ² $Ý È(õ ¢ Ó – at £ £ — f ù-• ¼(Ø £ Ó ¢ – > Y<c>Screen2.xwd<U>/afs/rpi.edu/home/62/citrit/consult/projec t/preface93/matlab/Screen2.xwdS ´ ÿü ü Ìûç ¤ Ó ˜ ax xi ¥ ¥ ™ g ÿü ïÿè Àõ ¥ Ó ¤ – ü Y<c>Screen3.xwd<U>/afs/rpi.edu/home/62/citrit/consult/projec t/preface93/matlab/Screen3.xwdn H Ô ¦ Ø • § ‚ at e_ H ~ 3 Q ¹ § Ù ¦ ‚ ÿü Q Z 3: 6 6 ø ¬ ¨ Ó ve 6 6 ø ” ¬ e ` ™ ) n o d • m Right d Reference Ù d Left d ‚ • No Footer d „ d † cr d ˆ me d d Š Ž Ó > Footere d d d s/ No 5 u M #/c /p d ” ˜ d Œ • — d d d0 ’ – æf ™ 5 Body æf ™ 6 c ™ } CellBody 7 æf ™ CellHeading3 8 æf Footnote ™ æf ™ 9 T e Headingd : ¤ Body æf ™ Œ matout ; ¤ H ~ æf Ž ´ ™ ˜ Ø re ™ s/ 5 h H ~ No /c matout < ™ æf Body æf ™ = ™ Body æf ™ > ™ din CellHeading ? ™ æf ™ e æf ™ CellBody @ œ T d TableTitle T:Table <n+>: À @ A › Ž À ü @ to ø ü æf Header B › ø ™ No Footer C ¤ H matout D ™ ´ æf Bo ü ™ Body æf ™ E ¤ æf H ™ Ø matout F ~ æf æf ¢ ™ H œ À ~ @ ¢ matout G › ü æf ø ™ Footer H ™ Z æf ™ Body I ª Z æf ™ Body J © H ™ æf ~ ¢ ™ matout K ¤ H ~ Ø h ™ matout L ™ æf ~ æf ¢ ™ Body N © matout M • æf ™ ™ Body O ¤ æf ™ Body æf ™ P ¤ 9™™ matout Q ¤ æf l æf ™ ™ • H matout R ® æf ~ æf ™ ™ ¢ ato Body S ™ æf ™ Body T ¤ æf ™ H o Body [ ¤ ~ æf ´ ™ B Bo æf ™ ™ z matout ^ ¤ 6 ~ Æ æf 6 ê ™ 2 matout a ¤ ~ ´ æf H matout ™ ¹ ¤ ~ Æ zÙVÓ ™ ;cóo š zÙVÓ › ReadOut îãø œ îãø • ž Emphasis Subscript™ Ÿ Superscript © - ¡ ~ ÿÌà ¢ Titles zÙVÓ £ [ ¤ TypeIn æ_ý‰ ¤ ÿÌà ¥ TypeIn h°„² ¦ Emphasis h°„² § ^ ¤ Emphasis ;cóo ¨ TypeIn ;cóo © ê h°„² ª ÿÌà « ReadOut æ_ý‰ ¬ ReadOut îãø - Titles îãø ® ™ h°„² ¯ Superscript Ñ á Ô ÿÿØ; Õ Ø š Ñ Ò Ö Ó € × Z € Ù Z b d Õ a Å q Medium Thick f € @ € a c e Thin Double Very Thin Å Æ a c @ a > > y Format A a a H > ? > a H H a > ? > ¤ > ? > a a a H H > ? > ? Ç a b @ a b a a a a > y H > ² Format B ) 9 ) * Comment Regular H H > ? > Courier > ? > > ? > Times H H Helvetica > ? > ? Regular Bold Regular Italic VÛÉB«‘œ÷FC.8 [K¶ƒ [£Á@=·d¨ ] tÆa’0§]r©ný kØQK'›5 § <: Ó •e(.Ò•ì 7O‰¹tå$ *à P^nðW¶U,ß )õ°Ü¯ÝÕ > ÿ!™«—Ï &.<ÐÁÕû‡x©{sŒ ÿK™ ÿé Å9 HÀ”‡Ó~, 7âO …«U–$Äœú_QrávÊ~ >ã$è œ¾‘%#