Учёба / lab_mat-met

H[b^\Z a i_j_qbke_gbo \bs_ f_lh^•\ \bdhjbklh\mxlv m ijZdlbp• l•evdb ijb j•r_gg• ³\jmqgm´ kbkl_f j•\gygv g_\bkhdh]h ihjy^dm Ijb n! p• f_lh^b ^m`_ ijZp_}fd• lZ g_ _dhghf•qg•

GZc[•evr ihrbj_gbf f_lh^hf j•r_ggy } f_lh^ =ZmkZ F_lh^ =ZmkZ fh`gZ jha[blb gZ ^\Z _lZib

- ijyfbc o•^ ydbc ihey]Z} m ihq_j]h\hfm \bdexq_gg• dhj_g•\ a -]h ih n-bc i_j_l\hj_gv fZljbpv dh_n•p•_gl•\ ^h ljbdmlgh]h \b]ey^m

- a\hjhlgbc o•^ ydbc ihey]Z} m ihq_j]h\hfm \bagZq_gg• dhjg•\ a n-]h ih -bc a i_j_l\hj_gh€ kbkl_fb j•\gyggv

<bdexq_ggy k-]h dhj_gy k = 1, 2,..., n-1 a •-]h j•\gyggy i = k+1, k+2,...,n) \bdhgmxlv reyohf aZf•gb mk•o dh_n•p•}gl•\ i-]h j•\gyggy j•agbp_x f•` ihi_j_^g•fb dh_n•p•}glZfb pvh]h j•\gyggy lZ \•^ih\•^gbfb dh_n•p•}glZfb i-]h j•\gyggy ihfgh`_gbfb gZ fZkrlZ[gbc fgh`gbd

M nhjfmeZo lZ agZd ³ ³ \bdhjbklh\m}lvky yd kbf\he hi_jZp•• ijbk\h}ggy m ijZ\•c qZklbg• \bdhjbklh\mxlvky ihi_j_^g• agZq_ggy dh_n•p•}gl•\ DLM

lZ EL Z m e•\•c - gh\•

Ijb \bdexq_gg• dhj_g•\ f•g•fZevgbc ih]j•o hdjm]e_ggy ijb i_j_jZomgdm dh_n•p•}gl•\ fh`eb\h ^hky]gmlb i_j_klZgh\dhx j•\gygv lZdbf qbghf sh[ fh^me• dh_n•p•}gl•\ DNN ijb \bdexq_gbo dhj_gyo [ N [meb fZdkbfZevgh fh`eb\bfb P_c _lZi f_lh^Z =ZmkZ a\_lvky \b[hjhf ]heh\gh]h _e_f_glZ

<•^ih\•^gh ^h \bs_ih^Zgh]h ko_fZ Ze]hjblfm ijyfh]h oh^m fh`_ fZlb \b]ey^ gZ^Zgbc gZ jbk

M j_amevlZl• ijyfh]h oh^m kbkl_fZ j•\gyggv fZ} \b]ey^

10

<•^ih\•^gh ko_fZ a\hjhlgvh]h oh^m fZ_ \b]ey^ gZ^Zgbc gZ jbk

M ML j•r_ggy kbkl_fb e•g•cgbo j•\gygv fh`_ [mlb ^hky]g_gh aZ ^hihfh]hx e•\hklhjhggvh]h ^•e_ggy Z[h h[_jg_gh€ fZljbp•

X=A\B,

X=inv(A)*B,

X=A^-1*B.

AZ\^Zggy

Jha\¶yaZlb kbkl_fm e•g•cgbo j•\gygv gZ^Zgbo m lZ[ebp• f_lh^hf =ZmkZ f_lh^hf DjZf_jZ Z[h f_lh^hf h[_jg_ggy aZ aZ\^Zggyf \bdeZ^ZqZ Ajh[blb i_j_\•jdm j_amevlZl•\

F_lh^bqg• j_dhf_g^Zp•€ >ey i_j_\•jdb h[qbkeblb • \b\_klb gZ _djZg agZq_ggy g_\’yahd

IL = DL [ + DL [ + + DLQ [Q − EL L = Q

M jZa• \•jgh]h jha\’yaZggy kbkl_fb g_\’yadb ih\bgg• [mlb [ebavdbfb ^h gmey Ijb \bdhjbklZgg• f_lh^m h[_jg_ggy h[qbkeblb kdZeyjgbc ^h[mlhd \bo•^gh€ fZljbp• dh_n•p•}gl•\ : gZ h[_jg_gm J_amevlZlhf ih\bggZ [mlb h^bgbqgZ

^•Z]hgZevgZ fZljbpy

11

j

1 ÷n

Jbkmghd Ijyfbc oL^ f_lh^Z =ZmkZ

12

xn = bn

an ,n

i

n − 1÷ 1 − 1

s = 0

j

i + 1÷ n

s = s+ ai , j x j

xi = bi − s

ai ,i

Jbkmghd A\hjhlgbc oL^ f_lh^Z =ZmkZ

13

EZ[hjZlhjgZ jh[hlZ ‹

JHAJ:AOMGHD ?E?DLJBQGBO E:GPX=1<

F:LJBQGBF F?LH>HF

P•ev jh[hlb gZ\qblbky h[qbkex\Zlb gZ ?HF fZljbqgbf f_lh^hf dhjg• kbkl_f e•g•cgbo j•\gygv

L_hj_lbqg• \•^hfhFl•

J•r_ggy kbkl_f e•g•cgbo j•\gygv \bdhjbklh\m}lvky m _e_dljhl_ogbp• lZ iho•^gbo \•^ g_€ ^bkpbie•gZo ijb jhajZomgdZo klZlbqgbo j_`bf•\ \ jha]Zem`_gbo _e_dljbqgbo eZgpx]Zo

>ey h[qbke_ggy kljmf•\ lZ gZijm] m _e_dljbqgbo dheZo fZljbqgbf f_lh^hf ki_jrm g_h[o•^gh \bagZqblb gZklmig• iZjZf_ljb

q – d•evd•klv \mae•\ m ko_f•

m=q-1 – d•evd•klv g_aZe_`gbo \mae•\

p – d•evd•klv ]•ehd a g_\•^hfbfb kljmfZfb

n=p-m – d•evd•klv g_aZe_`gbo dhglmj•\

I•key pvh]h kdeZ^Z}fh fZljbp• g_aZe_`gbo dhglmj•\ G jhaf•jhf p*n), a¶}^gZgv D jhaf•jhf m*p), Z lZdh` \_dlhjb ?JK E jhaf•jhf p*1), ^`_j_e kljmfm Jk jhaf•jhf m* d\Z^jZlgm ^•Z]hgZevgm fZljbpx hihj•\ Z jhaf•jhf p*p).

?e_f_glb fZljbpv G • D ml\hjxxlvky a]•^gh a nhjfmeZfb

15

AZibr_fh j•\gyggy m fZljbqg•c nhjf• aZ aZdhghf HfZ lZ I i II aZdhgZfb

D•jo]hnZ

hljbfZ}fh kbkl_fm e•g•cgbo j•\gygv : 1 < ydm fh`gZ \bj•rblb ex[bf a \•^hfbo f_lh^•\

AZ\^Zggy

JhajZom\Zlb kljmfb gZijm]b [ZeZgk kljmf•\ lZ gZijm] \ ]•edZo _e_dljbq-gbo eZgpx]•\ gZ^Zgbo gZ jbk IZjZf_ljb ko_f gZ^Zg• \ lZ[e

16

F_lh^bqg• j_dhf_g^Zp•€

H[qbke_ggy kljmf•\ \bdhgZlb aZ ^hihfh]hx hi_jZp•€ e•\h]h ^•e_ggy 1 :\< lZ h[_jg_gh€ fZljbp• 1 Inv(A)*B. Ijhdhgljhex\Zlb qZk g_h[o•^gbc ^ey j•r_ggy dh`gbf f_lh^hf \bdhjbklh\mxqb nmgdp•€ ML etime(t2,t1), tic lZ tok

Jbkmghd

17

EZ[hjZlhjgZ jh[hlZ ‹

HI?J:P12 A IHE1GHF:FB

PLev jh[hlb gZ\qblbky \bjZoh\m\Zlb agZq_ggy klmiLg_\bo ihe•ghf•\ gZc[ievr hsZ^eb\bf aZkh[hf lZ \bdhgm\Zlb jLaghfZgLlgL hi_jZpL€ a gbfb m k_j_^h\bsLiZd_lZ ML.

4.1 L_hj_lbqgi \i^hfhkli

4.1.1AZ]ZevgLihgylly

KlmiLg_\bf iheLghfhf KI m fZl_fZlbpLa\mlv nmgdpLx sh fZ} \b]ey^

L=

^_ n -klmi•gv iheighfm A=[a0 , a1 , … , an ] - \_dlhj dh_n•p•}gl•\ x - g_aZe_`gZ af•ggZ

KI rbjhdh \bdhjbklh\mxlvky m l_hjL€ Z\lhfZlbqgh]h d_jm\Zggy \ i_j_^Zlgbo lZ qZklhlgbo nmgdpLyo

< iZd_lL ML LgnhjfZpLy ijh KI a[_jL]Z}lvky m \b]ey^L fZljbpL-jy^dZ dh_nLpL}glL\ mihjy^dh\Zgbo aZ af_gr_ggyf klmi_gx g_aZe_`gh€ afLggh€ lh[lh \_dlhjm A=[2 5 0 –@\L^ih\L^Z} iheLghf [ + [ − .

Kei^ fZlb gZ m\Zai sh gmf_jZpiy _e_f_gli\ fZkb\i\ m ML aZ\`^b ihqbgZ}lvky a h^bgbpi.

<b\_klb KI gZ _djZg fh`gZ aZ ^hihfh]hx nmgdpi€

P_str = poly2str (A, x_char)

x_char – kbf\hevg_ ah[jZ`_ggy g_aZe_`gh€ afLggh€ P_str – KI m nhjfZli kljhdb kbf\hei\

<b\_klb gZ _djZg ^\• KI m \b]ey^i ^jh[i fh`gZ aZ ^hihfh]hx nmgdpi€

P_str = printsys (B, A, x_char)

^_ qbk_evgbd – ihe•ghf a dh_n•p•}glZfb < Z agZf_ggbd – ihe•ghf a dh_n•p•}glZfb

:.

18

<bjZom\Zggy agZq_ggy klmiig_\h]h iheighfZ

Ihe•ghf fh`gZ i_j_l\hjblb ^h \b]ey^m

:e]hjblf \bjZom\Zggy 3Q ([) kdeZ^_gbc gZ i•^klZ\• \bjZam gZab\Z}lvky ko_fhx =hjg_jZ M \•^ih\i^ghkl• ^h p•}€ ko_fb ihe•ghf i-]h ihjy^dm h[qbkex}lvky aZ nhjfmehx

fh`gZ hljbfZlb [Z`Zg_ agZq_ggy M fZl_fZlbpi ^h\_^_gh sh ^ey ihe•ghf•\ aZ]Zevgh]h \b]ey^m g_ fh`gZ

ih[m^m\Zlb Ze]hjblf [ievr hsZ^gbc m jhamfiggi qbkeZ hi_jZpic n ^h^Z\Zgv lZ n fgh`_gv Zgi` ko_fZ =hjg_jZ

M ML agZq_ggy klmiig_\h]h iheighfZ a dh_n•p•}glZfb A \bjZoh\m} nmgdpLy

Y = polyval (A, X),

X – lhqdb m ydbo lj_[Z h[qbkeblb agZq_ggy KI fh`_ [mlb kdZeyjghx \_ebqbghx

Z[h fZljbp_x

4.1.3 Hi_jZpi€ ai klmiig_\bfb iheighfZfb

Hi_jZpL€ a KI kdeZ^Zxlvky a ^\ho qZklbg nhjfm\Zggy gh\h]h KI ydbc } j_amevlZlhf hi_jZpL€ lh[lh jhajZomghd ch]h dh_nLpL}glL\ \bjZom\Zggy agZq_ggy gh\h]h KI

G_oZc fb fZ}fh ^\Z KI Pn(A,x) lZ Gm(B,x) : j_amevlZlhf ^_ydh€ hi_jZpL€ } s_ h^bg KI Hk(C,x).

>ey Ze]_[jZ•qgh]h kdeZ^Zggy lZ \•^g•fZggy ihe•ghf•\ Pn(A,x) lZ Gm(B,x) lj_[Z kihqZldm \bdhgZlb \L^ih\L^gm hi_jZpLx gZ^ fZkb\Zfb dh_nLpL}glL\ A lZ B. :e_ a\Z`Zxqb gZ l_ sh \hgb fZxlv j•agbc jhafLj ihi_j_^gvh lj_[Z ^hih\gblb f_grbc aZ jhafLjhf fZkb\ ^h [Levrh]h gmeyfb aeL\Z >ey pbo hi_jZpLc jhaf•j j_amevlZlZ Hk(C,x) fh`gZ agZclb yd k=max(m,n) NmgdpLy nhjfm\Zggy dh_nLpL}glL\ C klmiLg_\h]h iheLghfZ ydbc } Ze]_[jZ•qghx kmfhx iheLghfL\ a fZkb\Zfb

19