Учёба / lab_mat-met

EZ[hjZlhjgZ jh[hlZ ‹

QBK?EVG? ,GL?=JM<:GGY

P•ev jh[hlb gZ\qblbky h[qbkex\Zlb gZ ?HF \bagZq_g• •gl_]jZeb \•^ nmgdp•c yd• aZ\^Zg• lZ[ebqgh c ZgZe•lbqgh

L_hj_lbqg• \•^hfhkl•

AZ\^Zggy m ydbo ihlj•[gh h[qbke_ggy •gl_]jZe•\ a¶y\eyxlvky ijZdlbqgh m

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\k•o ]Zemayo ijbdeZ^gh€ fZl_fZlbdb Qbk_evg• f_lh^b •gl_]jm\Zggy aZhkgh\Zg• gZ lhfm sh •gl_j\Ze >a, b@ jha^j•[exxlv gZ qZkldb gZ dh`g•c a dhljbo djb\Z ydZ hibkm}lvky i•^•gl_]jZevghx nmgdp•}x f(x aZf•gx}lvky ydhxkv •grhx djb\hx ^ey ydh€ h[qbke_ggy •gl_]jZem ml\hjx}lvky aZ ^hklZlgvh ijhklbfb nhjfmeZfb Z ihl•f mk• iehs• kdeZ^Zxlvky jZahf

M \biZ^dm aZf•gb i•^•gl_]jZevgh€ nmgdp•€ •gl_jihexxqbfb ihe•ghfZfb hljbfmxlv lZd a\Zg• d\Z^jZlmjg• nhjfmeb P• nhjfmeb ^ey j•\gh\•^^Ze_gbo \mae•\ •gl_jiheyp•€ gZab\Zxlv nhjfmeZfb GvxlhgZ-Dhl_kZ

M aZe_`ghkl• \•^ •gl_jihexxqh]h ihe•ghfZ jhaj•agyxlv f_lh^b ijyfhdmlgbd•\ ljZi_p•c lZ d\Z^jZlbqgbo ljZi_p•c qb f_lh^ K•fikhgZ

Hkgh\g• nhjfmeb lZ ihdZagbdb sh oZjZdl_jbamxlv p• f_lh^b ijb

[ = D [Q = E \ = I D \Q = I E

Ihob[dZ f_lh^•\ \bagZqZ}lvky jhaf•jhf •gl_]jZem \•^ hklZlgvh]h qe_gZ •gl_ji ihe•ghfm < nhjfmeZo ^ey hp•gdb ihob[db Mi - fZdkbfZevg_ agZq_ggy •-h€

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Ijb \bdhjbklZgg• f_lh^m K•fikhgZ d•evd•klv qZklhd jha^j•[e_ggy h[h\¶yadh\h ih\bggZ [mlb iZjghx n=2k) • \k• qZkldb ih\bgg• [mlb h^gZdh\bfb (h=const M jZa• g_j•\ghf•jgh]h jha^j•[e_ggy •gl_j\ZeZ •gl_]jm\Zggy \bdhjbklh\mxlv m [•evrhkl• \biZ^d•\ f_lh^b ijyfhdmlgbd•\ lZ ljZi_p•c ^ey ydbo nhjfmeb qbk_evgh]h •gl_]jm\Zggy m pvhfm jZa• ijb^[Zxlv \b]ey^

– ^ey f_lh^Z ljZi_p•c

>ey aZ[_ai_q_ggy aZ^Zgh€ lhqghkl• •gl_]jm\Zggy qZklh \bdhjbklh\mxlv Ze]hjblf a Z\lhfZlbqgbf \b[hjhf djhdm :<D m ydhfm \bdhjbklh\mxlv gZklmigbc aZk•[ H[qbkexxlv agZq_ggy •gl_]jZem aZ ^hihfh]hx h^gh]h a gZ^Zggbo f_lh^•\ a ^_ydbf ihqZldh\bf djhdhf h Z ihl•f ih\lhjxxlv p• ` kZf• h[qbke_ggy a m^\•qb f_gvrbf djhdhf h/ Ydsh \bc^_ lZd sh

^_ ε - ^himklbfZ ihob[dZ •gl_]jm\Zggy lh h[qbkex\Zevgbc ijhp_k aZd•gqmxlv m ijhlbe_`ghfm \biZ^dm a\_jlZxlvky ^h ih^Zevrh]h jha^j•[e_ggy djhdm

H^_j`Zg_ lZdbf qbghf ijb[ebag_ agZq_ggy •gl_]jZeZ fh`gZ mlhqgblb ydsh \bdhjbklh\m\Zlb _dkljZiheypLcgbc i_j_o•^ ^h f_`• aZijhihgh\Zgbc J•qZj^khghf

^_ k= ^ey f_lh^m ijyfhdmlgbd•\ k= ^ey f_lh^m ljZi_p•c k= ^ey f_lh^m K•fikhgZ

>ey h[qbke_ggy \bagZqHgh]h •gl_]jZem m iZd_l• ML \bdhjbklh\mxlvky nmgdp•€ quad, quad8, trapz, cumtrapz.

[z,n]=quad(Fun, a, b, tol, trace, p1...p9) – h[qbkex} •gl_]jZe z j_dmjkb\gbf f_lh^hf K•fikhgZ a :<D gZ •gl_j\Ze• [a,b] a lhqg•klx tol;

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n – d•evd•klv lhqhd m ydbo h[qbkex}lvky agZq_ggy •gl_]jZem g_h[′yadh\bc iZjZf_lj

Fun – •f′y m-nZceZ m ydhfm f•klblvky hibk i•^•gl_]jZevgh€ nmgdp•€ f(x);

trace – \fbdZggy/\bfbdZggy /0 \b\_^_ggy gZ ]jZn•d agZq_gv •gl_]jZem m \k•o lhqdZo h[qbke_ggy;

p1..p9 – ^h^Zldh\• iZjZf_ljb \bdhjbklh\mxlvky ydsh i•^•gl_]jZevgZ nmgdp•y fZ} [•evr h^gh]h Zj]mf_glm f(x, p1...p9);

tol, trace, p1…p9 - g_h[h\′yadh\• Zj]mf_glb

[z,n]=quad8(Fun, a, b, tol, trace, pl...p9)- h[qbkex} •gl_]jZe z gZ hkgh\• •gl_jiheyp•cgh€ nhjfmeb GvxlhgZ-Dhl_kZ -]h ihjy^dm a :<D

z=trapz(X, Y) – h[qbkex} •gl_]jZe z nmgdp•€ Y(X ydZ aZ\^ZgZ lZ[ebqgh f_lh^hf ljZi_p•c

X – \_dlhj Zj]mf_gl•\ Y– \_dlhj Z[h fZljbpy agZq_gv i•^•gl_]jZevgh€ nmgdp•€

Ydsh Zj]mf_gl X \•^kmlg•c – lh nmgdp•y Y aZ\^ZgZ m lhqdZo , 2, 3,…, n.

aZ\^ZgZ lZ[ebqgh f_lh^hf ljZi_p•c X, Y, Z – h^gZdh\h]h jhaf•jm i=1,2,… M ML dj•f h[qbke_ggy \bagZq_gh]h •gl_]jZem fh`gZ hljbfZlb nhjfmeb ^ey

g_\bagZq_gh]h •gl_]jZem \bdhjbklh\mxqb ZiiZjZl kbf\hevgh€ fZl_fZlbdb nmgdp•y int :e_ ihi_jr_ lj_[Z aZy\blb kbf\hevg• af•gg•

Z_s – nhjfmeZ ^ey g_\bagZq_gh]h - Z[h \bagZq_gh]h - •gl_]jZe•\ m kbf\hevghfm \b]^y^•

fun – i•^•gl_]jZevgDnmgdp•y Z[h •f¶yP–nZceZ m ydhfm \hgZ agZoh^blvky var_s – af•ggZ aZ ydhx \•^[m\Z}lvky •gl_]jm\Zggy

A_s, B_s – f_`• •gl_j\Zem •gl_]jm\Zggy

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>ey lh]h sh[ h[qbkeblb agZq_ggy \bagZq_gh]h •gl_]jZem aZ kbf\hevghx nhjfmehx •kgm} nmgdp•y i•^klZgh\db subs.

Z=subs(fun, var_s, var)

fun – kbf\hevgbc \bjZa aZ ydbf lj_[Z jhajZoh\m\Zlb

var_s – i_j_e•d kbf\hevgbo af•ggbo aZf•klv yd•o [m^mlv i•^klZ\eylbky af•gg• var m \bjZa fun (var_s, var – kljmdlmjZ Z[h fZkb\

AZ\^Zggy

>ey gZ^Zgh€ m lZ[ebp• nmgdp•€ f(x) h[qbkeblb \bagZq_gbc •gl_]jZe

E

] = ∫ I [ G[ a lhqg•klx εf_lh^Zfb K•fikhgZ lZ GvxlhgZ-Dhl_kZ -]h ihjy^dm

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<b\_klb gZ _djZg nhjfmeb ^ey g_\bagZq_gh]h lZ \bagZq_gh]h •gl_]jZe•\ m kbf\hevghfm \b]ey^• JhajZom\Zlb agZq_ggy \bagZq_gh]h •gl_]jZem reyohf i•^klZgh\db iZjZf_lj•\ lZ ihj•\gylb ch]h a hljbfZgbf jZg•r_ z. >ey p•}€ `

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jha^j•[e_ggy •gl_j\Zem •gl_]jm\Zggy n = 4, 16, 32, 64 f_lh^hf ljZi_p•c Z[h f_lh^hf ijyfhdmlgbd•\ aZ aZ\^Zggyf \bdeZ^ZqZ Hp•gm\Zlb \ieb\ iZjZf_ljZ n gZ agZq_ggy •gl_]jZem

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EZ[hjZlhjgZ jh[hlZ ‹

=:JFHG1QGBC :G:E1A 1 KBGL?A I?J1H>BQGBO NMGDP1C

P•ev jh[hlb gZ\qblbky \bagZqZlb ]Zjfhg•qgbc kdeZ^ i_j•h^bqgbo nmgdp•c

L_hj_lbqg• \•^hfhkl• Nmgdp•y qZkm f(t gZab\Z}lvky i_j•h^bqghx ydsh ^ey g_€ kijZ\_^eb\• mfh\b

^_ ω =2π /T - djm]h\Z qZklhlZ i_jrh€ ]Zjfhg•db k - ihjy^dh\bc ghf_j ]Zjfhg•db

H[f_`b\rbkv m nhjfme• ^_ydhx d•gp_\hx d•evd•klx ]Zjfhg•d m, h^_j`mxlv Zijhdkbfmxqbc ]Zjfhg•qgbc ihe•ghf Qm(t):

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k=1, 2, ... , m ;

yi= f( ti ); t= i∆ t;

∆ W = 7 Q

Ijb n=2k nmgdp•y Qm(t klZ} ljb]hghf_ljbqgbfLgl_jiheyglhf =Zjfhg•qgbf kbgl_ahf gZab\Z}lvky h^_j`Zggy i_j•h^bqgh€ nmgdp•€ Qm(t)

reyohf i•^kmfm\Zggy €€ ]Zjfhg•qgbo kdeZ^h\bo aZ nhjfmehx 9.4).

AZ\^Zggy

H[qbkeblb dh_n•p•}glb ljb]hghf_ljbqgh]h ihe•ghfZ ydbc Zijhdkbfm} lZ[ebqgm nmgdp•x •a lZ[ebp• ijb j•ag•c d•evdhkl• ]Zjfhg•d m=3, mo, mo+4 lZ ihjZom\Zlb ch]h agZq_ggy (mo – hilbfZevgZ d•evd•klv ]Zjfhg•d mo=(n-1)/2).

Ih[m^m\Zlb j•agbf dhevhjhf ]jZn•db Lgl_jihexxqh€ nmgdp•€ • gZg_klb \bo•^gm lZ[ebqgm nmgdp•x m \b]ey^• a•jhd

F_lh^bqg• j_dhf_g^Zp•€

<•^^•eblb \ hdj_f• nmgdp•€ h[qbke_ggy dh_n•p•}gl•\ lZ jhajZomghd agZq_gv Zijhdkbfxqh]h ihe•ghfZ

Ijhkl_`blb yd \ieb\Z} d•evd•klv ]Zjfhg•d m ijb aZ\^Zg•c d•evdhkl• \•^j•ad•\ jhaljhs_ggy n gZ lhqg•klv ZijhdkbfZp•€

>ey gZhqghkl• s•evg•klv lhqhd gZ ]jZn•dm Zijhdkbfmxqh€ nmgdp•€ ih\bggZ \ 5-10 jZa i_j_\bsm\Zlb s•evg•klv lhqhd sh jhaljhsmxlv i_j•h^ T gZ \•^j•adL ^ey qbk_evgh]hLgl_]jm\Zggy

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