Marshak_Elektron_tabl

lh kbkl_fZ j•\ghkl_c hagZqZ} sh \•^ih\•^g• _e_f_glb fZljbpv lZ

j•\g• f•` kh[hx : •a pvh]h aZ hagZq_ggyf j•\ghkl• ^\ho fZljbpv \bieb\Z} sh fZljbpy AX ^hj•\gx} fZljbp• B.

F_lh^b jha\ yaZggy kbkl_f e•g•cgbo Ze]_[jZ€qgbo j•\gygv

<k• f_lh^b jha\ yaZggy KE:J fh`gZ ih^•eblb gZ ^\• ]jmib lhqg• lZ •l_jZp•cg•

Lhqgbc f_lh^ jha\ yaZggy ^ha\hey} h^_j`Zlb lhqg• agZq_ggy g_\•^hfbo \ j_amevlZl• ijh\_^_ggy kd•gq_ggh]h qbkeZ Zjbnf_lbqgbo hi_jZp•c gZijbd-

eZ^ f_lh^ DjZf_jZ 1l_jZp•cg• f_lh^b ^ha\heyxlv h^_j`Zlb rmdZgbc jha-

\ yahd m \b]ey^• ]jZgbp• ihke•^h\ghkl• gZ[eb`_gv ih[m^h\Z ydbo a^•ckgx}lvky aZ ^hihfh]hx •l_jZp•cgh]h ijhp_km

F_lh^ DjZf_jZ (IjZ\beh DjZf_jZ). F_lh^ DjZf_jZ8 aZklhkh\m}lvky ^ey jha\ yaZggy d\Z^jZlgbo KE:J a \bagZqgbdhf hkgh\gh€ fZljbp• sh g_ ^hj•\-

gx} gmex

>h\_^_gh sh lZdZ kbkl_fZ fZ} jha\ yahd ijbqhfm l•evdb h^bg G_oZc ^ZgZ d\Z^jZlgZ KE:J

Jha\ yahd lZdh€ kbkl_fb agZoh^blvky aZ nhjfmeZfb DjZf_jZ

8 F_lh^ jhajh[e_gh r\_cpZjkvdbf fZl_fZlbdhf DjZf_jhf m j

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\bagZqgbdZ D aZf•ghx i-]h klh\ipy klh\ip_f •a \•evgbo qe_g•\ [_a af•gb

•grbo klh\ip•\

GZijbdeZ^ D1 – \bagZqgbd ydbc ml\hjx}lvky a \bagZqgbdZ D aZf•ghx i_jrh]h klh\ipy klh\ip_f •a \•evgbo qe_g•\

AZ ZgZeh]•}x fh`gZ agZclb \k• •gr• Dι (i = 1, 2,….,n).

Nhjfmeb fh`gZ aZibkZlb \ jha]hjgmlhfm \b]ey^•

Kmdmig•klv agZq_gv [ [ [Q } jha\ yadhf kbkl_fb

Q_j_a \bkhdm h[qbkex\Zevgm kdeZ^g•klv f_lh^Z – ihlj•[gh h[qbke_ggy n + 1 \bagZqgbdZ jhaf•jghkl• n ´ n \•g g_ aZklhkh\m}lvky ^ey fZrbggh]h

jha\ yaZggy \_ebdbo KE:J Ijhl_ \•g •gh^• aZklhkh\m}lvky ^ey jmqgh]h jhajZomgdm lZ \ l_hj_lbqgbo \bdeZ^dZo

ì [ - [ + [ =

ï + + =

IjbdeZ^ 12 Jha\ yaZlb kbkl_fm e•g•cgbo Ze]_[jZ€qgbo j•\gygv í[ [ [ aZ nhj-

ï - + = î [ [ [

fmeZfb DjZf_jZ

Jha\¶yahd F_lh^ DjZf_jZ aZklhkh\m}lvky ^ey jha\ yaZggy d\Z^jZlgbo KE:J a \bagZqgbdhf hkgh\gh€ fZljbp• sh g_ ^hj•\gx} gmex AZ^ZgZ KE:J kdeZ^Z}lvky a ljvho j•\gygv a ljvhfZ g_\•^hfbfb lh[lh m = n • \•^ih\•^Z} hagZq_ggx d\Z^jZlgh€ L_i_j ke•^ i_j_\•jblb qb

^hj•\gx} \bagZqgbd hkgh\gh€ fZljbp• gmex

>ey pvh]h aZibr_fh hkgh\gm fZljbpx KE:J m ^•ZiZahg dhf•jhd < D5 lZ[ebqgh]h ijhp_khjZ yd ihdZaZgh gZ jbk . ihke•^h\g•klv hi_jZp•c h^gZdh\Z • ^ey MS Excel • ^ey

OO Calc).

AgZc^_fh \bagZqgbd D hkgh\gh€ fZljbp• A AZibr_fh hkgh\gm fZljbpx \ ^•ZiZahg < D Nhjfmem ^ey h[qbke_ggy \bagZqgbdZ \\_^_fh m dhf•jdm )

^ey MS Excel: FHIJ?> < D10);

^ey OO Calc: =MDETERM < D10).

42

Nmgdp•y FHIJ?> ^ey HH &DOF – MDETERM ih\_jlZ} \bagZqgbd fZljbp• KbglZdkbk nmgdp•c

^ey MS Excel: FHIJ?> fZkkb\ ,

^ey OO Calc: 0'(7(50 fZkkb\ ,

^_ fZkkb\ – p_ d\Z^jZlgbc fZkb\ ^ey hljbfZggy \bagZqgbdZ

Jbk 1

<bagZqgbd hkgh\gh€ fZljbp• D = 10 ¹ 0 Hl`_ px KE:J fh`gZ jha\ yaZlb aZ

nhjfmeZfb DjZf_jZ

AgZc^_fh D1 jbk >ey pvh]h \\_^_fh \ ^•ZiZahg dhf•jhd < D14 dh_n•p•}glb hkgh\gh€ fZljbp• lZ aZf•gbfh i_jrbc klh\i_pv klh\ip_f •a \•evgbo qe_g•\ m dhf•jdm ) \\_^_fh nhjfmem ^ey jhajZomgdm \bagZqgbdZ

^ey MS Excel: FHIJ?> < D10);

^ey OO Calc: =MDETERM < D10).

Jbk 2

:gZeh]•qgh agZoh^bfh •gr• \bagZqgbdb jbk

43

FZljbqg_ j•\gyggy jha\ yam}lvky lZdbf qbghf H[b^\• qZklbgb j•\gyggy ihfgh`mxlvky ae•\Z9 gZ fZljbpx A - 1 h[_jg_gm ^h fZljbp• AaZ mfh\hx \bagZqgbd D fZljbp• A \•^f•ggbc \•^ gmey hl`_ fZljbpy A fZ}

h[_jg_gm fZljbpx A - 1).

Hl`_ fZ}fh A – 1 (AX) = A – 1 B <bdhjbklh\mxqb \eZklb\•klv ^h[mldm ljvho fZljbpv fZ}fh A – 1 A)X = A – 1 B Hkd•evdb A – 1 A = E ^_ E – h^bgbqgZ fZljbpy h^_j`m}fh j•\g•klv EX = A – 1 B :e_ EX = X a\•^kb

X = A – 1 B (3.13)

Nhjfmem gZab\Zxlv fZljbqgbf aZibkhf jha\ yadm KE:J Z[h jha\ yadhf fZljbqgh]h j•\gyggy

ì [ - [ + [ =

ï + + =

IjbdeZ^ 13 Jha\ yaZlb kbkl_fm e•g•cgbo Ze]_[jZ€qgbo j•\gygv í[ [ [ aZ

ï - + = î [ [ [

^hihfh]hx h[_jg_gh€ fZljbp•

Jha\¶yahd Yd \•^hfh a IjbdeZ^m , aZ^ZgZ KE:J ljvho e•g•cgbo j•\gygv a ljvhfZ g_\•^hfbfb fZ} \bagZqgbd hkgh\gh€ fZljbp• sh g_ ^hj•\gx} gmex

Hl`_ KE:J fh`gZ aZf•gblb _d\•\Ze_glgbf €c fZljbqgbf j•\gyggyf LZdbf qbghf

AZibr_fh p• fZljbp• yd ihdZaZgh gZ jbk .5.

Jbk 5

Jha\ yadhf fZljbqgh]h j•\gyggy } X = A – 1 B.

9 Fgh`_ggy j•\gyggy gZ fZljbpx A-1 g•qh]h g_ ^Z}, lhfm sh i_j_klZ\eylb \ ^h[mldm fZljbpv fZljbp• f•kpyfb g_ fh`gZ hl`_ (AX) A-1= AXA-1• AA-1X.

45

AgZc^_fh h[_jg_gm fZljbpx A – 1 jbk >ey pvh]h ke•^ \b^•eblb ^•ZiZahg dhf•jhd < D \ ydbc [m^_ ih\_jg_gZ h[_jg_gZ fZljbpy gZlbkgmlb agZd j•\ghkl• lZ

\\_klb nhjfmem

^ey MS Excel: FH;J < D5);

^ey OO Calc: = 0,19(56(< D5).

Nmgdp•y FH;J ^ey HH &DOF – MINVERSE ih\_jlZ} h[_jg_gm fZljbpx KbglZdkbk nmgdp•c

^ey MS Excel: FH;J fZkkb\ ,

^ey OO Calc: 0,19(56( fZkkb\ ,

^_ fZkkb\ – p_ d\Z^jZlgbc fZkb\ ^ey hljbfZggy h[_jg_gh€ fZljbp•

AZ\_jrm}lvky \\_^_ggy nhjfmeb fZkb\m gZlbkdZggyf ih}^gZggy deZ\•r

SHIFT+CTRL+ENTER.

Jbk 6

>Ze• agZoh^bfh fZkb\ g_\•^hfbo X \b^•ey}fh ^•ZiZahg dhf•jhd J8: J1 lZ \\h^bfh nhjfmem

^ey MS Excel: FMFGH@ < D10; J3:J5); ^ey OO Calc: 008/7< D10; J3:J5),

aZ\_jrm}fh \\_^_ggy nhjfmeb gZlbkdZggyf dhf[•gZp•€ deZ\•r SHIFT+CTRL+ENTER. Nmgdp•y FMFGH@ ^ey HH &DOF – MMULT \bdhgm} fgh`_ggy ^\ho fZljbpv KbglZdkbk nmgdp•c

^ey MS Excel: FMFGH@ fZkkb\ 1; fZkkb\ 2), ^ey OO Calc: 008/7 fZkkb\ fZkkb\ .

D•evd•klv klh\ip•\ \ fZkb\• ih\bggZ ^hj•\gx\Zlb d•evdhkl• jy^d•\ fZkb\m

AZ\_jrm}lvky \\_^_ggy nhjfmeb fZkb\m gZlbkdZggyf ih}^gZggy deZ\•r SHIFT+CTRL+ENTER •gZdr_ [m^_ ih\_jlZlbky l•evdb agZq_ggy \_jogvh€ e•\h€ dhf•jdb

fZkb\m >ey HH &DOF ijb kl\hj_gg• nhjfmeb fZkb\m aZ ^hihfh]hx FZkl_jZ nmgdpbc ke•^ dh`gh]h jZam \klZgh\ex\Zlb FZkb\ sh[ j_amevlZlb ih\_jlZebky m \b]ey^• fZkb\m

FZkb\ g_\•^hfbo agZc^_gh jbk

Jbk 7

46

IjbdeZ^ 14. I_j_dhgZlbky sh agZc^_g• jha\¶yadb } \•jgbfb

Sh[ ^•agZlbky qb } h^_j`Zg• agZq_ggy g_\•^hfbo jha\¶yadhf KE:J ke•^ \bdhgZlb

i_j_\•jdm >ey pvh]h g_h[o•^gh i•^klZ\blb hljbfZg• agZq_ggy g_\•^hfbo m j•\gyggy

lh[lh ijb fgh`_gg• hkgh\gh€ fZljbp• A gZ fZljbpx g_\•^hfbo X ^h[mldhf fZ} [mlb fZljbpy \•evgbo qe_g•\ B.

<b^•ey}fh ^•ZiZahg F F lZ \\h^bfh nhjfmem ^ey MS Excel: =FMFGH@ < D5; J8:J10);

^ey OO Calc: 008/7< D5; J8:J10).

AZ\_jrm}fh \\_^_ggy nhjfmeb fZkb\m gZlbkdZggyf ih}^gZggy deZ\•r

SHIFT+CTRL+ENTER.

J_amevlZlb gZ\_^_gh gZ jbk

Jbk 8

>H>:LHD

AZ\^Zggy ^h eZ[hjZlhjgbo jh[•l

EZ[hjZlhjgZ jh[hlZ ‹

L_fZ: Ih[m^h\Z ]jZn•d•\ nmgdp•c

F_lZ <b\qblb fh`eb\hkl• lZ[ebqgbo ijhp_khj•\ 06 ([FHO lZ

2SHQ2IILFH RUJ&DOF ^ey ih[m^h\b ]jZn•d•\ nmgdp•c

AZ\^Zggy

1. <bagZqblb h[eZklv •kgm\Zggy nmgdp•€ y = f (x) (LZ[ebpy 1).

GZ •gl_j\Ze• ydbc gZe_`blv ^h h[eZkl• •kgm\Zggy nmgdp•€ jha\¶yaZlb aZ^Zqm lZ[mex\Zggy nmgdp•€ y = f (x) (LZ[ebpy 1 lZ ih[m^m\Zlb €€

]jZn•d 1gl_j\Ze lZ djhd lZ[mex\Zggy \b[jZlb ^h\•evgh

3. IjhlZ[mex\Zlb nmgdp•x LZ[ebpy 2 ydZ aZ^ZgZ \ iheyjgbo dhhj-

^bgZlZo ^ey ljvho agZq_gv dh_n•p•}gl•\ lZ ih[m^m\Zlb ljb ]jZn•dZ nmgdp•c \ h^g•c ^_dZjlh\•c kbkl_f• dhhj^bgZl

EZ[hjZlhjgZ jh[hlZ ‹

L_fZ Qbk_evg• f_lh^b jha\¶yaZggy g_e•g•cgbo j•\gygv a h^g•}x af•gghx

F_lZ <b\qblb fh`eb\hkl• lZ[ebqgbo ijhp_khj•\ 06 ([FHO lZ

2SHQ2IILFH RUJ&DOF ^ey jha\¶yaZggy g_e•g•cgbo j•\gygv a h^g•}x af•gghx \bdhjbklh\mxqb qbk_evg• f_lh^b

AZ\^Zggy

<bagZqblb •gl_j\Zeb •aheyp•€ dhj_g•\ j•\gyggy f (x) = 0 (LZ[ebpy 3).

2.Mlhqgblb h^bg a ^•ckgbo dhj_g•\ j•\gyggy ε = 10 - 4 \bdhjbklh-

\mxqb

'f_lh^ ^•e_ggy \•^j•adm gZ\i•e

'f_lh^ ohj^

'f_lh^ ^hlbqgbo

'f_lh^ ijhklh€ •l_jZp•€

'\[m^h\Zgbc •gkljmf_gl _e_dljhggbo lZ[ebpv Ih^[hj iZjZf_ljZ.

3.Ihj•\gylb hljbfZg• j_amevlZlb lZ ajh[blb \bkgh\db

48

EZ[hjZlhjgZ jh[hlZ ‹

L_fZ Jha\¶yaZggy kbkl_f e•g•cgbo Ze]_[jZ€qgbo j•\gygv

F_lZ <b\qblb fh`eb\hkl• lZ[ebqgbo ijhp_khj•\ 06 ([FHO lZ

2SHQ2IILFH RUJ&DOF ^ey jha\¶yaZggy kbkl_f e•g•cgbo Ze]_[jZ€qgbo

j•\gygv

AZ\^Zggy

1.Jha\¶yaZlb kbkl_fm e•g•cgbo j•\gygv LZ[ebpy 4):

'aZ nhjfmeZfb DjZf_jZ

'fZljbqgbf kihkh[hf

2.I_j_dhgZlbky sh gZc^_g• jha\¶yadb } \•jgbfb

3.Ihj•\gylb hljbfZg• j_amevlZlb lZ ajh[blb \bkgh\db

LZ[ebpy 1

49

Ijh^h\` lZ[e 1

50