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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
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OPERATORS AND FLOW CONTROL
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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;
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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.
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.
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Script M-files (or command files) have no input or output arguments and operate on variables in the worksnace. r!pfinit.inIl linp >Jnr! r>Jn accent innut arffUments and return output arguments and their internal variables are local to M_Alp,;:
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To Q"et an idea of the nrobabilitv of each of the five outcomes vou can run the scrint , T • .., 1 T. 1 nnn .1 . ., "',' ..,.r • ..,vv _.Vo..J y'''''u ..'h", .. ~
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same wav as eXlstmg MA' ,A H tunctlOns such as sin. eve size etc. Llsung (.~ snows a SImple Iuncl;lOn l;nal; evamal;es l;ne largesl; elemenl; m aosoml;e value of a matrix. This example illustrates a number of features. The first line beQ"ins with the keyword function followed bv the outnut arQ"ument v. and the = symbol. On the right of = comes the function name, maxentry, followed by the input A ",Hhin
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including automatic indentation of loops and i f structures, commenting out blocks ot cOd.e, color syntax hIghlightmg, and. bracket and. quote matchmg. These and. other features can be turned off or customized via the File-Preferences menu of the editor. A verv useful feature introduced in MATLAB 7 is block commentinrr: a block of code can be commented out (no matter what editor you are using) by surrounding it h"
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M-FILES
function [x,iter] III sqrtn(s;tol)
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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
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the MATLAB interpreter. The MATLAB search path is a list of directories that specIhes where M A' .A K looks tor M-hles. An M-hle IS avaHable only It it IS on the searcn pal;n. .Lype pal;n l;0 see l;ne currenl; searcn pal;n. .L ne pal;n can oe sel; ana added to with the nath and addnath commands or from the Path browser that h'l Invoked by the l"lle-:;et Path menu optIOn or by tYPIng pathtool. n
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format lon£. format('lon£') a~sp ~ 'tte.l.lO ), a~sp tte.l.lO diary mydiary, diary ( 'mydiary') warning off warning ( 'off') l,"Ol;e, nowever, l;nal; l;ne IOrID snOUlQ oe usea onlY lOr Iuncl;ions l;nal; take string arguments. In the example
a.uo
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mterpretS:l as a strmg and mean IS applIed to the A::;Gll value 01 :l, namely ' f"",.,... ""n ho .",or! "n1" ",han n" ""t....."t
l\T"to "1",,, th"t tho
is renuested. Thus X
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OJ
on il;.
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Keplace repetItIve expressIOns oy calfs to a common runctlon. -
It
BRIAN W. KERNIGHAN and P. J. PLAUGER, The Elements of Programming Style (1978)
Much of MATLAB's power is derived from its extensive set of functions. .. Some of the functions are intrinsic, or "built-in" to the MATLAB processor itself. Others are available in the library of external M-files distributed with MATLAB . .. IS LransparenL LO LIe user wUeLUer a /uncLfon IS mLrtnSlc or conLameU m an /v/- 1ft::. -:>0"_"1\,1 1\
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y-axes, the spacmg of the axIS tIck marks, and the color and type of the line used tor tne plOt. More generally, we could replace plot (x, y) with plot (x, y , string), where string combines un to three elements that control the color marker and line stvle. For example plot tx, Y, 'r*--') speClhes tnat a rOO astensK IS to be placea at each nn;nt .... (;)
u(;) n,-l that tho r>nh.,t" a ..." tn }." ;n;n",-I }." a ..."rI rla"j.,,,rI Hn"
,
GRAPHICS
88
A
7
•
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....
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Table 8.1. Ovtions for the nlot command. ... ····Marker .... ;~~
:
~olor
r g b c
n.ea Green Blue Cyan
m
'
v
Yellow
T.T
:
• ,1.
.
~
nl. ~
~
.~u
Line style
("'t~""'QQ
"quare
::I
.
,
... . Solid line i default)
--
Dashed line
-
Dash-dot line
~
-
K
01 •
••
..
'"
A
~1.
UTh;+"
..
n"tt"rl ];n"
Upward tnangle uownwara tnangle
V
"
8 or xd), yd) anu od), c ~i) Wil>ll 801iu green ana aa8llea rea . ,
,.
uu'" "L'y ", UV"U ,o-u.y-,o, }'.LV" ,.n., 1 J "U'" olots created bv corresoonding columns 01 X and Y. 11 nonreal numbers are suoolied to Pl.ot then ImagInary parts are generally Ignorea. The Only exceptIon to thIS rUle
.
.
Q 1
'T'n,r.
~
r< ~ ,~n""w
,T
Q()
8
• ,,
5
4
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,, ,, ,
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,,
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7
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, , ,
...:
,-
.
I
I
,
I
3
5
4
~
6
-
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,
8
Figure 8.2. Two nondefault x-y plots.
anses when plot IS gIven a smgle argument. It Y IS nonreal, plot ~
'T'hA "VA" "1'A
,.
rn,.
~~ .. hn~ .. h.~
.
....
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:~
,
~
,I .
.f
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-1
0.5
1
0.5
0
"!V" v> ... >Ouoc; U.U.
--
J..... 'J
r> ~c;-
• '0
tne aOOltlOnal commano
.
, rA
• _1.
1.
,~
~h~
~~~,
-' ~lA~ A& "[;':~ .. _~
nf th., fir",t. nlnt. T
for a
..
Q
0;
:ro
, ...h:~h
",(t\ -
(n ...l h\ f'n",(t\ _ hl'n",((n /h...l- 1\t\
y(t) =
a + b) sm(t) - bsml(a/b + l)t)
= 12 and b = 5.
-
; ... = v:v.VO:J.V"pJ.; A-\.o.
A~ ~h~ .
,
r ' ..
-
r
UJ ""'V\."J
U ·",V\.\.o.l U'''J
-"J.
Otr1ng ,s, 'Pos1t10n
~
...
,L~
-
-
L·(
.t'
.t'> ......
,
4UJ)
Note the use of the cell array options to avoid repeatedly having to type the arguments ' Interpreter' , ' latex' , ' FontSize' ,18. The fill function works in a similar manner to plot. Tvping fill(x,v, [r j;! hl)d1"r1",,,,, UThA"'" "r",'r> h"th",nA;nt"v(;) ,,(;) 'T'hQnr.;nt" ,. -, ,,' ,an'! taken in order. and the last vertex is ioined to the first. 'T'he r.olor of the , , , 1.. ,1.. ,1..' .... , '1"1. .l .... c. , h", or>~l~ro;n th", r~n~", rn 11 t h", l",u",l nl' rnrl ~rr>r>n nnrl hI, ,n
.
.
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~,
.L .L
vvv-
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,, ~u
"'
r.~
• •
,
~ue
1.",.~ 'r> t], r • T",., (see Section 10.1),
tr. hI"
I"
0-
"""
"tr,no- r.r "
],,,nrllp
• The x and/or u hmits are given bv lims. • tol IS a relatIve error tOlerance, the aerault value or 0.2% accuracy. -
.cc>.v ,,,uou n
.
... pvu,ua
'u,
OJ" '-La"....
uu" p,vu.
vv
VJY
•
:
x IV
correspondmg to
o
rn,
1
v.~
'-'-
~
T>
r1
AA
vv
12
1
U.,
~
4
A
o
V
I I
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o. r
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-
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I \ 1/ \ I \ '/ \ I
_0
\
-10
00
0
v·
, ...
-1.5
nQ
0
ULL''"'J,
L..La. ...
title('\bf{surf} shading flat' ,FS,14) SUOp.10"t\LLQ), wa"terla.1.1\L.), "t1"t.1e\ \OIi,wa"terla.1.1j ,.l'::i,lQ) ()
thp
r
f'nlnr nf "
ever. mesh surf ann n~laten
mav also he lisen in the form mp",h (y . v . Z . W)
1'.
•
r
1'.
angle. .L nis can oe overriuuen wicn cne Iunccion V1ew. .Lyping V1ew ~a, ,
,
VVl"C
aVUUL LHC ",-aAl" LU a.
.~.
0)
,
aHU LHC
,Cl LILal
sees cne LU U
degrees. The default IS view( -37.5,30), while view(2) is equivalent to view (0,90) and gIves a ~u VIew ot a surtace lookmg down trom above. The ro"ta"te ;;U toOl on Llle
LUUlval
Vl Lne llgure
Llle 1I1vu::;e LV ue u::;eu LV
~
Llle allgle Vl
view by clicking and dragging within the axis area. It is Dossible to view a 2D Dlot as a 3D one bv using the view command to sDecifv
~l _+(-f'-f'+( " . r OJ
Hl Llle UeXL
(17\\\. ,
•
we
,(':1\ • ,
O'
a
• "
uaCLal
.
WILU Lue
. lCC.UlN 'C
land shown in Listing 8.1, which uses a variant of the random midpoint displacement algontnm I\}V, ~ec. 1.01; see 1:' Igure b.l\). ttecurSlOn IS Q!SCUSSed rurtner m ~ectlOn lV.\}. rile uasic sLep LaKen uy .LanU is LO upUaLe an l~-UY-l~ maLrix WiLll nonzeros onlY in each corner bv fillinl! in the entries in nositions (1 d) (d 1) (d d) (d N) and
~
.LVV
~
Q~"~
Q~"
0.8 1
1"
)1#.>_ ~
0.6 0.4
01"
40
20
~20
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1
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0.5
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0.5
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0
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20
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,
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..
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40 .............
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.......
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~/~~....
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---::..---'""'40
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20
0 0
0 0
Figure 8.17. Surface vlots with surf surfc and waterfall.
.............
----- ---
~
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n
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108
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are oelOw l;ne average valUe. .L nis resuning aal;a manix, laS.Lana, is alSO ~u",
~lU1U
.LU'"
Vl"''' •
.
UN::: V ..LI::lW, L'
lVU1~U
"'UU
~VJ
'0
65] ), respectIvely. For these two subplots we also control the
axIS
.
whn
J W1U V ..LI::lW, L.o!:,*V
limIts.
randn('state',10); ?~h ... 1·
1.- -
A
= zeros(k)' -, ,
r.
1
r.
r.
_
,_"
n~
1
n1.
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1
,
4
=
ri:l
. rOnl;i:l1Ze ; l;1l;J.e,·UeJ:aUJ.l; V1eW ,ri:l, J.'/;)
Bisland = max(B,mean(mean(B»); tsm1n = m1n~m1n~ts1S.Lana)); Bmax = max(max(Bisland)); subnlot(222) meshz(Bisland) t1tleVUe:tault V1ew' ,F'~,l:l) subnlot(223) r._'7"
.1
meshz(Bisland)
Jln"
",v;C!(rn 1.- n 1.- Rm; ..... Rm",v"
ti tIe ('ViAT.d r-7fi 401)' . FS 12)
..
~ ,,",_,~ V1ew~
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_~~"_(TH~' ~~..1'
L:.I:':lV OOJ)
.
~.
CUI...Ll:>'LV
,
v
A .
~~w
A
,, DW..Lll
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DWCUl.JJ
, , .. -, .....
VV~
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.LaDle ~.O summarIzes l;ne mosl; popUlar ;jU plottmg runctIOns. 1\S tne taDle mdicates, several of the functions have "easy-to-use" alternative versions with names bel1'inninl1' ez; Section 8.3 discusses some of these functions. A feature common to alllITanhics functions is that NaNs are internreted as "miss-
r-
,,,,
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.
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,
,
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,
115W.r1rl'c
(lCl8:~) hr>r>kc
up until recent years, have never had a good picture4 of a cardioid. .. o"or know
"h::>t
Innvorl Iiko
::>
IIhon I tnnk
because the illustrations were done by graphic artists whn wprp trvinn tn imitatp
hv
artic;tc;
without seeing the real thing. ~~
4ezpolar('1+cos(t)')
~
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.
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MATLAB was ori!!inallv desi!!ned for linear al!!ebra comnutations, so it not surmising M,-,m; nf thp linp'-'T "]",oh",, "
frnm t.hp LA PA(;K f~l
,-,rp h,-,,
""
Fhr rA.QPQ in whirh
oo.n
.t:>
nnnn
"
,~ ....... '
thp
of thp ?-norm of A. mAtriy il'l too
function normest can be used to obtain an estimate. The call normest (A. tol) uses the power method on A* A to estimate IIAlb to within a relative error tol; the default ;" + ... 1 -
1 .. -1':
·L'I..
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..
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> 1 is the condition
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I
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a singUlar
01
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T.
•
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l;~~n¥ nnn"~~ A ~
.£ " v.
IIAIIIIA- I ll
lar snuare matrix A ",fA)
For A,
cumpu~t:u
p
"'~''''U''''
l'
v.~~.
...... u
..
uy
p
.... e
.£ v •
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,
,
with default p - 2. 1''or p - 2, rectangular matnces are allowed, m which case the conUltlon number IS aennea Dy "'2I.li) - 1I.1ill211.1i 112, wnere .Ii 18 tne pseuao-mverse (see Section 9.3). Comouting the exact condition number is exoensive so MATLAB orovides two tunCtlOns tor estlmatmg the I-norm condItIOn number ot a square matrIX A, rcond Hnth
1rr11.
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vnolesKy ractOrIZe tne matrIx. 11 tne vnolesKy ractOrIZatlon succeeas It IS usea . . LV "VIVIO ~lllO - J ' 'etll LlU WH11 lJetl~~letl r - o 1" out.
use tne
ve runCtIOn In place or tne oacKslasn operator..I.ue syntax IS A -
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A = 'trl.u\.ranQ.\.n)); 0Pl'16 -
............... \
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X = linsolve(A,B,opts); res - norm\.ti·-A*A) Give LINSOLVE incorrect onts structure. opts = struct('LT',true);
Yo
= '';_~~' ..o(A 1:1 ~~+~,. ·hY) ....... = y
./ TT~~o_ +_.; ~_~,' ~_ "rf ~
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-
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A = rand(n); A = A*A' ; opts = struct l 'SYM " true, • ____ ,true); A
=
res
,D,OP-':;SJ,
= normCB-A*X)
The output
res
IS
=
~.",o~ut::-v~o
res
= 10.7664
res
=
1.9675e-016 res = .IL ~')')~",,_n1l';
The residuals in this example should be of order eps for a correct solution. The second call to linsolve wronglv soecities the matrix A as lower trianl!Ular. 1::lince linsolve performs no checks on the matrix properties it gives an incorrect answer without any or p.rror. Hp.nr.p. r.A.TP. il'l
in t.hp. 111'lP. of t.hil'l ~
'Thp.
time saved bv usiUl! linsolve instead of backslash is highlv deoendent on the matrix n~..:I th~
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lj.1.
1 t)!::
i)ome examptes OJ now to set op'ts structure m J.J.nsoJ.ve.
lVlatrix property
op'ts structure
.1.JUW\:::l'
Opl;5
-
Upper triangular upper nessenoerg Symmetric/Hermitian and nositive definite Use (conju~ate) transpose
opts opts
= struct('UT',true) = struct (. 'UHe::>::> ,true)
onts
= struct ( , SYM'
r.~ ~o+p;~
~~
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Q.2.2
., "H,:a 0;:; ,~
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To
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a." ...
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SOlutIOn 01 mlmmal ~-norm, whlch can be computed as pinv tA) *b. 11 the system has no sOlU~ion \ ~na~ is, i~ is inconsis~em) ~nen A\0 is a leas" squares sOlUuon. nere IS an Cll(,"
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r.
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r~
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1'.J.ll v
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• ,L
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= A\b;
» x ""
LX
=
ans
Y - p1nvtAJ*b;
YJ
2.0000
1.0000 1
()
1.0000 ........
()()()()
1.0000 ~~~( .. ' l
r~~~(~,
;on'" ",.",vV,L
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...
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matrix (evA (n) ). A matrix
"U~"H~
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;~ ~~~~ o~rl ;+ ho~
OJv
U" ""v
~,,11 ,,~ ...+~~
0
U.
.fiV -
~.
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= pascal (3)
-"'
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1
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n
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= inv(A)
.
n.
r.f "n 'YI_h,,_'YI TYl"tr;v LI ;" "TYl"tr;"
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e
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ue
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.LU
lu(A) produces an m-by-n Land n-by-n U if m
• .Ll n 1" ,.-vy-'. LUell L.L.,VJ 2: n and an m-by-m Land m-by-n U
It m < n. Using x = A\b to solve a linear system Ax = b with a square A is equivalent to LU factorizinO' the matrix and then solvinO' with the factors: LL,UJ A
• •v
. .~
n
.-I' 'V~
.LU~A);
-
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11 If A 'T'T A D
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.
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illunuer linear sysLems involving VOOB
a. "a."
BL5
lI.
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l'.
v . . . . ~. ov
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T
. . .~,
~.v
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1 ~
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are LU ue ::;UlVeu LUen Lue l.U laCLor::; Ctln ue reuseu,
BL
Any HermItIan posItIve dehmte matnx has a Gholesky tactonzatlOn A - 1C X, wnere n IS upper l;nangUlar wIl;n real, posil;Ive aiagonal elemenl;S. .1 ne vnOl€SKY factor is computed by R = chol (A). For example:
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pasca.l Vi)
A
" ~
~
1
:L
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4
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0
.LV
.L
~
~
9.5 OR
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FACTORIZATION
.L
» R R=
'±
-
.LV
L.V
ehol(A)
0
1
1 1
1 2
1 3
()
()
1
':l
0
0
0
1
I'Wte tnat enol. lOOKS only at tne elements m tne upper tnangle or 11 . it.
ttl
it helps to reveal near rank-deticiency. Roughly speaking, if A is near a matrix ot ranK r < n tnen tne laSt n r magonal elementS or n, Wlli De or oraer eps*norm'A). 1'1 LUUU VULjJUL
nermutation matrix:
IVlce::>
ro
R Pl = ar (A).
y'.L
LV
u::>e
jJ'
''-'"'HIS
CLUU leLUlH LHe
If the economv size factorization with
l.lU
r
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.
;'"
-0
.1
UVVV
~1
=
''';
1 0
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n, .11
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9.6
131
SINGULAR VALUE DECOMPOSITION
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D
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, LL
1..L J.W
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=
B
1 0
»
-2
-2
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1
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-2
=
[U,8, V]
svd(B, 'econ')
TT -
-R 1194A-001 ~
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c
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.
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=
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g
e~g\a,D);
Y,
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V = 0.55335 0.15552
0.23393 -0.57301
2.3747 1.2835
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() Q()Q~R
eivals = .f\
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.... ~
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~~~~
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ans= 1 z . .LO'±'::1e-v.LO
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after iter iterations.
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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
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D
.
real anu symmeLrk poshive "u", .uu
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tn tho Ii ~~ ~K
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.
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metric matrix and comnute its five al!!ebraicallv laruest eiuenvalues usinu eil!'s. For
.,
~ h.ll
•
» »
n
n
=
-,
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= de.lsQ lnumJZ:rl. 'V'... IIV';>.
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h ....
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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
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Here, f is a function handle to the anonymous function. The @ character, which ,t.hp function handle is followed bv a list of innut arP1lments to the function in narentheses and then bv a single MATLAB expression. Being a function handle "n
~
Thp'J ".
,
--1
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v.v "",."" ""'" ••", J'
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= fd_deriv(f,x,h)
function y YoFD_DERIV %
Finite difference approximation to derivative. FD_DERIV(F,X,H) is a finite difference approximation to the derivative of function F at X with difference parameter H. H defaults to SQRT(EPS).
7.
%
oJ, .u. .u.cu./5".u. "'''!... " '''''pOl, y - t!lX+h) - !lX))/hj .~
",.u.u
... Y IQ..Qerlvo
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;
» alpha - l', » g = @(x,y,z) x-2+y-2-alpha*z-2; » 0'(1 ?~) ans = .11
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%
P. M. Maurer A rose is a rose ...
%
pp. 631-645.
if nargin < 2, b ... 1; end i f nargin < 1 a" l' end c - O' d-l·t)'" a*b' lX, yJ ... sp~rola, a, c, P.LOl;\X,YJ axis sauare
/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
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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' "
..
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.
.
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~
{auan " h,,;lL;n ~
,\
, loha .
~
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10.Q
H;~
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..
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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,
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for k = 1:4 subp.tot,:L,:L,kJ KOCn\pJ.,pr,K) ::IX; !'l ( 'AClll::1l')
." '
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mur suoner lines.
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ll.6~~~ ~u
1 " ~v~
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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'
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~
~
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0.5
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1"igure 10.3. Koch snowjtake created with function koch.
....
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10.10.
'U'''
...
(
'1'1 it Vl'tTiables within a fnnr.tion A.rP. local to that is convenient to create variables that exist in more than one worksDace includiW!.
."
tho TTla;n
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allU ylt::aLt::1
.
::>L"',
AIIr::IIC::TIIC::
allU
::>u UI • ,1\1
(~haDter
11
-
Numerical Methods: Part I "."
.
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~~~~
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> .•
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to the anonvmous functions and nested functions introduced in MATLAB 7. nassinl!
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11 1
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pC>.l " u .~
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nere are tHree proulems relateU to pOlYnomIals:
. .
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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
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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\
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e
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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..:< •..:
