MATLAB Guide

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---' , ... .. In tne IOllowmg example we set up a -oy-,j .& ~..r A'T'T AD

,"

- '" -

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.

..

anrl co"nn,.)

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.,

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',1- J::.

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nf "

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:

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» rm ,n1 = R;7.p.(A) c:

,. , = v

n n U.U. ~,

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r1

'"

.l"1J. J.

P

,

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.,.

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arithmetic. Imnortant snecial cases are matrices (two-dimensional arravs). vectors -' ,

,.1

,

1\;T.

~~rl

1\ ~ A CT'T

•

D

n'

. , ~corl

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,

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,,,h=~ +ho fl,,+~ ' .

rp"l rb,t" hTnp (t.hnlwh fnr rp"l' t..},p im"lTin"r? n"Tt. if' nnt. f'tnrpn). L

.,.u

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,

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!"

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A 1

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.

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.J .J:.

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Vll Wl11Ul IV.Ir"-.L D1"UJ 10 1Ul1111115' .L 11T •• >T ~.

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4.2

39

PRECEDENCE

=•., 4. 9407e-324

» realmin*eps/2 ans = 0 To obtain further insight, repeat all the above computations after typing format hex, which disnlavs the binarv floatin!! noint renresentation of the numbers in hexadecimal tormat. 'l.~.

rreceaence

lVlAILAtl s anthmetIC operators Obey the same preceaence rUles as those m most calCUlaLors anu compuLer languages. .l.Ue rUles are suown in .l.aUle "'l • .L. ~J:'or a mure complete table, showing the precedence of all MATLAB operators, see Table 6.2.) J:;>

lable 4.0 compares the key parameters tor tlle smgle and double precIsIOns.

..

~inQ?7

ni d = "


'lnn the command rand ( , state' . i ). For i =0 the rand g:enerator is set to its initial state f+ha ,,+,,+a:+ h"" n.han l\tfA'T'T.AR "+,, ...+,,\. ~r>...

;

"

, +],.0

t.o it.>:: ith >::t.Rt.P 'T'hp >::t.Rtp of r>'lnnn i>:: >::Pt. in t.hp >::Rmp WRV. 'T'hp ,

,1

I'

'

,.

f'

.1.

.-l 'l1492 ~ 11"1449

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.

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7

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p11U11:::t:i LOi:tU UI::: t:iI:::L up W1LU LUI:::

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of rl'lnn Rnrl

!'i.l l\/f ATIHY

47

\'T'TON

~ ue enu 01 a row can ue speciueu uy a semicolOn insLeau 01 a carriage reLUrn, so a more comnact command with the same effect is

»

A =

nr·c1.·

'0....'

;j b; (

L~

o.,vvv,

vo...

'.

;1'

L c .'

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,;

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. ..... 0""

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.'"

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r 'vu..", va"",

. ml;erprel; l;lle SIgn as an aaaHlOn or suouacl;Ion operal;or. •

•

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I,

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........

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•• 1-

'n

l;lle SIgn, else , . .I. V WlLll ~

11 1;j; I f HI

.1'+,

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.

III .~

-.I.

»

W

=

'';)

~

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~,

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w

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» x x =

2

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4

= [-1 2 - 3 4]

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lVlauices can De consuunea in OlOCK rorm. vv il;ll 0 aennea oy 0 we may create

»

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»

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» Mend: -1: 1 end)

-- -

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shane. Usinl!: this notation another way to define our 3-bv-3 matrix of mimes is

..=.

A

, , 'n' "''''''V'''''''J, .. ,. J ~

~

7

11 19

17

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.In one circumstance--when the right-hand side is a single element-the number or elements In a SUOSCrlptea assIgnment can oe amerent on tne two siaes or tne assignment. In this case the scalar is "expanded" to match the number of elements on the left: ,

A - V"""""'J,

» A(2:3,2:3) = 0 A

= t

t

1

1 1

0 0

0 0

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.

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the functIOn linspace, which accepts the number of POInts rather than the Increment: .ll.nspace ~a, 0, n) generates n eqUallY spacea POIntS oetween a ana D. 1I n IS omlttea LV.lUU.

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Columns 8 through 9 0.7500 1.0000 ~,

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conjugaLe Lranspose 01 LUe maLfix A is oULaineu wiLU A . H A is real, Luis rrn .LU'"

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The functional alternatives ctranspose (A) and transpose (A) are sometimes more convement. rU1- LUI:' O:;.l.ll:'C1cll Ci:ti::il:' U1 .lI. Cll1U y, .lI. "'y 10:; LUI:' 1UUI:'I, O:;CCt1cU, Ul UUL v nroduct which can also be obtained usin!! the dot function as dot (x. v). The vector or cross product ot two is-by-l or I-by-iS vectors (as used III mechamcs) IS produced JOV

~

h" ro ...,., ....

»

x -

»

x'*y

=

L' 1 0 lJ ' ; Y

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=

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» sort(x)

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MATRICES

OU

rr.. hl.. max min mean

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Rm.i" rlnfn

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..... ~~ "~,yo

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Standard deviation

std

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vector contammg the maxImum and Illilllmum element, respectIvely, m each column, l;ne sum rel;urns a vecl;or ,sums, ana sor,; sons l;ne elemenl;S in eacn U~

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To test whether arrays A and B are equal that is of the same size with identical

, +h~ »

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;",,,,,,,,,1 (A B)

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The function isequal is one of many useful logical functions whose names begin with is a selection of which is listed in Table 6.1' for a full list tYoe doc is. For examole isinf (A) returns a logical array of the same size as A containing true where the

.'

Af A "rA ",I"" Ar Tn;n"" ; ",-f "nrl f" l"A ,,,hArA

+hA" -,;

"rA nAt·

» A = [1 info -inf NaN]' » isinf(A) ""''' 0

1

1

0

The function isnan is particularly important because the test x rlll(,P" thp rp"ll]t, 0 (f"],,p) pvpn if y

i"" N"Nl

(A N"N i"

--

tl"\

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and unordered with eyerythinl!.) 1\1 ~to tl.. ,,+ ",., "rr"" A"'" ho ro" 1 ;,., tho

.. n=,.,n= h •• + ,.,~t rAn 1 nn

,

'! 0 i:LVUllRi i:L

'

,

•

-'

'

,

&& sinCl/x) < 0.5 uy

:t.t::1U.

The precedence of arithmetic, relational, and logical operators is summarized in Table o.~ (WhICh 18 based on the mtormatlon prOVIded by heJ.p preceC1enCe). l'or operators of equal precedence MATLAB evaluates from left to right. Precedence can be overridden bv usinl! narentheses. Note in narticular that and has hil!her precedence than or, so a logical expression of the form

f\ 1

RH',T,A'T'TnNAT. ANn T.nnTr'AT

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Unary plus (+), unary minus (-), logical negation (-)

3

lVlultlpllcatlOn l . * J, nght dlvlslOn l . I J, lett dlvlslOn ~. \), maLrix mUlLiplicaLion ~ .. ), maLrix rigm uivision (/) matrix left division (\ ) AddItion (+) subtractlOn (-)

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Although the numenc array i2 has the same elements as the logICal array i 1 (and compares as equal Wll'll 1l'], OillY 11 can oe useu lOr suoscnpl'mg. 1.0 acmeve l'Ue required subscripting effect with a numerical array, i3 must be used. A call to find can sometimes be avoided when its argument is a logical array. in our example on p. 67, x(find(isfinite(x») can be replaced by x(isfinite(x». , . , .. "n~ ""n h" ~rm" rm '

.

"n~ th,,'\T ""n h"

"""~ ;n

arithmetic exnressions containinl!" doubles. The result is always a double: ",

... -

...L ...."" ,

OJ

La.... "''''',

-

"

-

..... a.TOJ,

" ... a."''''\''J

C = .t.

ans = double TT

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Ll.. ~v

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» b/a ??? l:'"nrT; nn

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M A'T'T.A"R h"", frmr fin..., .~~l"

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v . .~

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statements onn

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OPERATORS AND FLOW CONTROL

70

where the statements are executed 11 the elements ot expresswn are all nonzero. t'or exaIl1p1e, Llle cuue ::swap::s x allU y II X 1::S lS''''C>"'''' Lllal1 y.

.. 'J'

~

J

"temp = y; y

.1\.,

= temp;

X

end •

TTn

vv

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. Ull 1"'" ..llllt:: .uy lUI .."llt::l p

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Statements to be executed only if expression is false can be placed after else, as ; .. "h~

e = exn(l)' if 2-e > e-2 -'l~~~('''l-~ ~~ "'~~~~~" ",1",,,, -'.'

,

,

....

-,., .

~ .. -'l

onp. or morp. tp.sts p,an hp. aoop.c1 with elseif (notp. that thp.rp. mnst be no space between else and if): if isnan(x) ni",n('Not a Number') elseif isinf(x) -'I; ",n( '01"",

A1RA

,

,.,.... m;n"",

,

;nT;n;+,,"

.,,,

-,

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~ .. -'l

Tn t.hp t.hircl n; Rn. " nrints ::IS ::I, sinp'lp. cl1Iote '. The for 100D is one of the most useful MATLAB constructs although as discussed Cl

'10 1

.

,

.

.

-.-

r

... h~ n~~

.

"nrl f"",t £'"rlP tr" t" "'TC\irl T"'" ]""n",

1.

.

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'T'h", ",vnt:v i",

for variable = expresswn Ann Usually, expresswn IS a vector ot the torm i:s:; (see ~ectlOn b.:L). The statements are executed with variable equal to each element of expression in turn. For example, t.hp "'11m of thp -flr",t. ?!l t.P.rm", of t.hp h::lrmonic. ",prips 1 Ii is p,omnlltpcl hv

» s = 0; » for i = 1:25, s

= s + 11i; end, s

s = 3.8160 Wi:J,y

LU Ut::l11lt::

"''''1'' "'''''~VI.

i::s U::sill/S Lllt:: ::SllUalt::

6.2

71

FLOW l

=

"'''' .[Or x

LpJ./O PJ.I ~ pJ./.:>J. uJ.sp ~ LX.

0.5236 0.7854 1.0472

sJ.n~XJJJ.

enu

0.5000 0.7071 0.8660

Multiple for loops can of course be nested, in which case indentation helps to improve the readabilitv. The following: code forms the 5-bv-5 svmmetric matrix A with (i,j) element i/j for j > i:

= 0;

A = eye\nJ; = ?'n for i = 1:;-1

n

i'nr ;

AI; ;'\ - ;/..; .

•

= ;/;.•

A(; ;)

end

...

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Th",

nf

-r

thp

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w . .~~~

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,

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, ;~ ...h~nh nn~",

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.,

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An'

,

1\". ATT A n

..:I~nn

nm; nave a repeal>-umir lOOp.) we can rewrhe l>ne previous example less concisely as

"

w . . ~~~

~

x ......

X/~;

A

;

U.

" ....oA. ........

end xlun ~ne oreaK can l:UiSU ve UiSeu LU exn a I:or lOUp. 111 a IleiSl>eU lOUp !:t oreaK exits to the loop at the next higher level. The continue statement causes executIOn ot a for or while lOOp to pass ImmeelY l>0 l>lle neXl> neral>IOn or l>lle lOOp, sKlppmg l>lle remammg sl>al>emenl>s m l>lle lOOp.

AQ >l. trivi>l.l

~

IL-

~

A.

-f,,,-

,u r

~

LVV-

VV. I

~n,VL

;-,1 ·1()

-'.'

f • ,

~.~l",

•.

~~..'l

disolavs the integers 5 to 10. In more comolicated looos the continue statement can .f•• l

l-

'..1 1. ~b

~

..1'

..1

'T'lH' fln,,] "AntrA]

Af "en.d + r n

Tt

'" thp QT.d+rn

..

"

.~

e

"

-'VVuv.".~v

~u~

" "

vJ

-0

l;Ile ca::>es

-' vu~.

"..,1. • Uv u

"

~

a rIlaLCIl MIeIl LIle

.LHe; He;A"

U< a

"He; r

ve;\..o"u
to!

iter • iter + 1: xoi€! • x;

X • Ix + a!xI/Z: xd1:t:t - ab.. (X-xold) lab.. (x) :

......

fprint:t (. %2. Ot: en

!t20.16e

,

'9.2e\n' , iter,

X'I

xditf)

Firure 7.2. MATLAB Editor/Debuaaer.

%}

Here, denotes an arbitrary number of lines of code. MATLAB considers all lines between %{ and %} to be comments even those that are not indiVIdually commented out wIth a leadIng 7. SIgn. !:Slock comments can be nested, so

..

~n

.,.

..

TTT

1""inO't.hp

""n hp

th"t. " hlo"k

• _1,

o

.I.y...- ...,

,1_

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11. Jr T:1!1

w .. " ....

y y v .. n.,u"

U."''''

a ....u

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T1l ...

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,1_

a" ....

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lV.lW1Y

11. Jr "rT1Y

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hlo"k

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1>0 e~g, "u,",u a.n. a..r " "u.',"'.'" eigs needs just the ability to form matrIX-vector products, so A can be gIven eIther as an expnclt matrIX or as a IUllctiOn tnat penorms matrIX vector proaucts. In Its simplest form, eigs can be called in the same way as eig, with [V,D] = eigs(A), when it comnutes the six ehrenvalues of laruest maunitude and the corresnonninu eiJ.!:envectors. See doc eiJZ:s for more details and examples of usage. This function is whu

ADX

.t1X

D

.

real anu symmeLrk poshive "u", .uu

'

"n .

tn tho Ii ~~ ~K

.,J'.

•

1.

.

[I':ill

HI

uu",a.•

• .uU\.'" "U'"

~

lUIS lIS

.

A" "n

u,o fn"'Tn " "n"",,,o ""Tn_

metric matrix and comnute its five al!!ebraicallv laruest eiuenvalues usinu eil!'s. For

.,

~ h.ll

•

» »

n

n

=

-,

~~rl ~ l~'~"D

-"

.•1.. '_1-

-r

= de.lsQ lnumJZ:rl. 'V'... IIV';>.

.

'L U

'Arr~, .

yeal;> JU;>L LV

La"e

"V~~

,

,~

h ....

,~

Irl h~ rir'"

"

"

. '!JUle VUL

V,

Llle

IIaLIIA.

L'";>

h~ ..~~rI ..... ~ rlr~~~~ ~ ..

"

J

-

Star Trek IV: The Voyage Home (Stardate 8390) We share a philosophy about linear algebra: we minK oasls-rree, vvc

vv,

'LC

...,a""",-

'CC,

our wnen me cmps are aown we close rne orrlce uoor anu 'J'

-

IRVING KAPLANSKY Reminiscences rof Paul Halmosl (1991)

The matrix of that equation system is negative definite--which is a positive definite system that has been mUltiplied through by -1. For all practical geometries the common finite difference Laplacian operator gives rise to these, the best of all possible matrices. Just about any standard solution method will succeed, allu ~

c:;

'ally

ale

'VI yvu

fJ'~~~~

Th::>t III/nrk

(1 Q7() ~

_.

vnapter .LU -IVlore on .runctlons

.

...........,

_

~

1n 1

l.,fn~"

"

mpn~ t.n

, n,;+l,.

'T'hp

,

r

..

TT

""" /SUUU ::>~y It:: ~u

..

~He

HL,",vHO

vav

HvOvv,",

Nested functions have two kev oroDerties:

.

~i.J~

UN r

lV~Uttj!;

...

A na"loarl ~

,

h"" "",'a"" lon loha

nf ,,11 ~

",;loh;n ,,,h;nh ;lo

• A function handle for a nested function stores the information needed to access . " ;n " t h" n"",t."r1 ". fl'l'ld t.h" v", I""", nf ",nv loh" , neRted ~ (" ' 1\ that are n""r1r1 to Rconed" it An example of a nested function is given in rationaLex in Listing 10.6 which 11\Ill:tIl subfunctions into nested functions so that the variables are automaticallv available mSlde the suOtunctlons and need not Oe passed. The precise scoping rules of nested functions, can be found in the online MATLAB docnmentation. For further examples of nested functions see Chapters 12 and 22. ~

~u.o.

.ft

.

.

-

r .... "a.,,'" r

~ypical

IVI.ftJ. lJ.ftD

ins~aua~ion con~alns

JU"O u,}' o,}'pU'o

ou'"

.... ...

llunureUs or lVI-IlteS on 1>ue user s pa1>Il, au

- "'" ."

"Q.l"'" V, 0"'" U.L-"''''. ......'" 0'"'' "''''''''' v.

an advantage it can lead to clutter and clashes of names not least due to the oresence OI"nelper IunCtlOns tnat are usea oy otner mnctlons out not mtended to oe called directly by the user. Private functions provide an elegant way to avoid these problems. Anv functions residinl! in a directorv called nrivate are visible onlv to functions in the parent directory. They can therefore have the same names as functions in other ., ," Wh"n M A'T'T. A R Inn],.", fnr '" ~ it th"n ' . the call iR ' functions (relative to the directorv in which the f ,t.. .J ,1.1.1U. 1'l' "

..

n" .... a """

.

.

-..--

~

{auan " h,,;lL;n ~

,\

, loha .

~

,,,;11 ha

10.Q

H;~

V~;

..

~

function rational_ex ex) or neSl;eu

_l:.A

I.

.I.,

C1

U

' nth ~

Hpln for lU.~.

flm ('fin

fl,

ltecurslVe

hp

ll"inp"

hp 1 n nr; VrlT.A \ flm

~'unctlOns

Functions can be recursive, that is, they can call themselves, as we have seen with functIOn gasket m LIstmg 1.7 and functIOn land m Listmg 15.1. J{ecursIOn IS a powerlUI tooL, out not au computatIOns tnat are aescnoea recursIvely are oest programmea t.hi"

Wfl.v

The function koch in Listing 10.7 uses recursion to draw a Koch curve 190 Sec. 2.41. The basic construction in koch is to replace a line by four shorter lines. The upper le>ft_hnrl

.

in H'iITllre> In? "h""'" the> f,,"r line>" tht re>,,"lt fr"rn ~

..

thO . ~

construction to a horizontal line. The unner ri!!ht-hand nicture then shows what ,''v,,

.

~.

U"VUV

r,

"

. . . . .vu W

.~u

ohmn +h~ ~~~+ hn~ l~,,~lo ~f

en1,

y.

"

U"V

~.

"UV

.

~

,1

~u~

,

".

uuv

uuu~,

.

,

~'"

~

.. ~

'0'

"'~,

. u"v Lv 'vL

,

Limes wiMl J.eve.J. one less ana Wil;Il enapoinl;s l;Ilm aenne .I.'16UIl:' .I.V.""

.r.

'v,"'

,. n

r

\

\w,1JJ

uy

Wet" .'

.t"~

r~ ~, LV, v~ ,

pr -

Ll j OJ; 7.

"

v'

~"v'~

TT'

f

u"v

,r ~'~v"

•

T

o

..,

.L.Iv.L

V

R~ght

for k = 1:4 subp.tot,:L,:L,kJ KOCn\pJ.,pr,K) ::IX; !'l ( 'AClll::1l')

." '

""1:'

.

endpo~nt

'0

L,VUI:'.

Mle

mur suoner lines.

,r ~L

... ..

ll.6~~~ ~u

1 " ~v~

~

~U.,

.•

~

~

.

AV\,;ll.

function koch(pl.pr.level) YoKOCH Recursively generated Koch curve. KOCH(PL. PR. LEVEL) recursively generates a Koch curve. Yo where PL and PR are the current left and right endpoints and Yo LEVEL is the level of recursion. Yo .Leve.L ... v plot([pl(1) .pr(l)] • [pl(2) .pr(2)J); ;. Join pl and pro no.LU on else d0"

,1"!..""

"'Vv"

-0.6

~.

(.

f'

~

-0.8

~

~

~'?

':'

'"

~""

u UI • ,1\1

(~haDter

11

-

Numerical Methods: Part I "."

.

,

.,

' I ,,,.,y

.J.H.l"

,

~~,

~,

.

"

"

, .

"

'.lV.l

,

..

~

~

,..

~

.

,

,

'0..~v

~

-0

••

~~

•

~.

~

••

r

-

-J

"'~"J

~~~~

a lUnction Iun mUSL ue passeu as an argument. ftS uescnueu III .,eCtIOn lV.l, Iun can , ,. r aH HaHU.l" • .J.H" lJ" a " V.l a "".l.lH5 MA' AH functions described in this chapter place various demands on the function tnat IS to De passea, out most reqUIre It to return a vector or valUes wnen gIven a vector of inputs. When a function fun is nassed to and evaluated bv another MATLAB function It IS sometImes necessary to commumcate problem parameters to fun. In versIOns ot

.

(.

"J

MA'T'LAR nr;nr tn MA'T'LAR 7 th;" ,,,..,, rlnm> .•

thP

"."

~· ... nk"

1;"t."

to the anonvmous functions and nested functions introduced in MATLAB 7. nassinl!

..

.

I'

All

,

r " , ..

.

,

1

..1

..

•

..1

..1'

moira

."a nf onnn,,_ -J

-0

.

mml" nr np"tprl "

,

n

..

,

d'

H.l "H'"

• l' 1\.". A'T'T AD'

~

,

"

H.a"'" U"" V.l ,,".

10; l'.

'.'-'

"H"H a. to UHto

:1, .l' : UJ r .t'or matnematical oacKgrouna on tne metnoas aescrioea in tnis ana tne next , ,. ., ,-, -, ,.

..1 •

...

..

..1.

cU'" LUj, L'j, L.11j, L""Ij, L""'j, LO"j, L.1U.1j, L.1""Uj •

11 1

••1", ""nrl

n""f""

. .

MATLAB represents a polynomial p(x) = PIX

uy a ruw Vel:LUr p , .

pC>.l " u .~

..

VV'"H "H" ~

• •~

LP~l)

+ P2 X ,..·.1 + ... + Pn X + Pn+l

P~.t:)

.

...

......,n L..i Ol'i....

p~n"l~J

'J

u~

Ul LHe

u,,"u H' .lHaHoY

"

,

.

, 'I

~l'lUte

,,,.,y

tHat l:Ul11~,

"

V"""V'

~'I

nere are tHree proulems relateU to pOlYnomIals:

. .

~

•

£'I'

~

.1-

.

,1-

.. ,

...

~

Root findinll: Given the coefficients find the roots (the noints at which the nolvnomial evaluates to zero). Data fitting: Given a set of data .1

"

{Xi, yil~l' find

a polynomial that "fits" the data.

He stanUaru LecHmque IOr eValUating P~ X) IS norner s meLHou, WIllCH corresponus

,

"U "H" H"" ""U

.

.

(

/

\

\

P(X}-(,,·(h

,

rnL

.

.. 1..

,

..

"

. T-~ l '

'J

" ~~v~

-J'

,

'.-.

,

n)

n

.

.r

,

>I.

,'L...Ii

.

(.J . . \

.\2

1\L'\'2/

i:JZ;

•

..

~

Yi,'

.1.

,

:n ~.;~.

thP' .lPDTPP n "" t h"t n ~

CllaC "", jJ\~i)

t].,D ,jutu

. ,1 11 of npp"rpp Rt

",m

>h

L'

n" 1 " f ; "I- (v "

~ p" UU liLt:;" a l l llHt:;1 4_ OUv

Rnn WP wish t,n finn

~

. ,I J": .

~

• •nit]" ", ...1 ,,-/'; +.

,.un ho

Cf'hP ,nTnt"v i" n

TYIi'ZP.l

uvu

. , fit nf rlpnrpp ~

=

n

C

...---

........

"-,

X)2).

[1 -6 12 -8]; h

Lq ,rJ -

'\

~

Fip"llrp 11,1

»

\

/

\

=

\

./

0.4

-

» g

,

/

l\

.".

!

j

0.8

!./'

°

\

.f

"> '11

T

-0'

! ' V4 J

'0'

v~U

rn

, "U Cllt:; pUly llUllll l",.nrll" tA "n

1

T'7", ... "(m(v),.,,.,(v)-v

•

vn)

2. as the handle to a function M-file: fzeroC\Omvfun xO) where ~

of

.~~

=

mL

-4'

-

'

i:!. .

..1

..... ua uv . . . .-.. '" ""LV ........

",n.,

""''''L

,r .•\

rt1lr .\

-4'.

_~~

,

\,,fA~.

-4',

I.

,I,n fA~

.1-

,. T Y~Uv'-L

~

•

~,

•1. v~.

•

uv~.~

n. " .

~fhA:~~ n nnnln~ ~":n n ...

,

Tf •

,

·0

,

-4'.

,

T Ant

An ;n;ti"l """",.], ;"

"'P"A" Af """n ...1

~

no v, ua..L5 "'. .

, ,,\

= A

r \

t""",(y)-y,

"

.u

m M..........

, .. t:~;l

:11

f"ATYT vn tA finrl "n ~

~

.£ .., v . .L _~.L V

;l

T

L vJ

,

LL .£ U"v .vu ....... V.

11.... 11.1 ~

•

,~

••

' .. \ \ n~;l-4'"~r~,,r'l\\

n ..

nave opposhe sign, l;nen Izero worKS on l;ne imerval oellileo oy xu. r rovioing a , T • T .T· . . .u".

,~.

!U "Ul;>

.....

IZerO~IlI~X)X-1;an~X),J.)

1.5708

!va1.

=

1. 2093e+015 ..L 11

,c.uue;.

,,-

:1

.

, ;"

pf ..\

2

2

~

'''I

'1

~~

[0

olT

l'

, J

"

u.

~

.

.....

'-'"

("I;,,~~ ..h~ r

Tn"",rl{Y\ r. "

we can type

» [x,fval]

= fminsearch(@fquad,ones(2,1),optimset('Disp','final'))

Optimization terminated: Y "''''... ;'''T;'''''' ... h", "'ho OPTIONS.ToIX of 1.000000e-004 ,.,'v\

'.1:.'

iOTl rT";-t:"'T";'" ""';TllJ'

11.4T'HRFAS'T'

1RQ

IIIHlb:HT'RA

UJ:"'l.1UI~".lO.l.run

OI

e

.1.

"UV'±

x -

*

1.0e-004 -0.4582

-0.4717 -1' .• ~,

-

r

r.

-1'.

."

-1'.

rQ~ ~p" 1 n 41 "rlirp"t. ()

A r()lIlTh Wl'lV ()f .

t.hp mhrpn prr()r

t.P~t.

nh()vp

i~

t.hl'lt. Arr(;)

is acceDtablv small if xCi) has as manv correct digits as sDecified bv RelTol or is

err(i) L'"

U11:;1:Ll'\.

LOAL"'V''''L

eacn SUOmterval. .L ney cnoose tne sUOmterva1S accormng to tne lOCal oenavlOr or tne

,

L111:;

V11=

W 111:;1 I:; L111:;

10

-..-

ll1UOL

-

..

Warning messages are produced if the subintervals become very small or if an excesSIve numoer or runctlon evalUatIons IS usea, eltner or wnlcn coUld mdlcate tnat tne integrand has a singularity. To illustrAte how nuad and nuadl work we ron~idpr t.hp int.POTAl (1/

In \

1

\

1

(x - 0.::\)2 -I- 0.01

~

(x - OJ))2 -I- 0.04

) -"

_~.v~~·

...

The integrand is the function humDs orovided with MATLAB which has a large oeak at 0.3 and a smaller one at 0.9. We applied quad to this integral, using a tolerance of 1 ..-4

1Ule l..:< •..: