MMO / mmouchpos

30

ydZ ijbcfZ} f•g•fZevg_ agZq_ggy m \gmlj•rg•c lhqp• •gl_j\Zem ih\bggZ [mlb ijbgZcfg• d\Z^jZlbqghx Ydsh ` nmgdp•y e•g•cgZ lh €€ hilbfZevg_ agZq_ggy fh`_ ^hky]Zlbky l•evdb \ h^g•c •a ^\ho ]jZgbqgbo lhqhd •gl_j\Zem

LZdbf qbghf ijb j_Ze•aZp•€ f_lh^m hp•gx\Zggy a \bdhjbklZggyf d\Z^jZlbqgh€ ZijhdkbfZp•€ i_j_^[ZqZ}lvky sh \ h[f_`_ghfm •gl_j\Ze• fh`gZ Zijhdkbfm\Zlb nmgdp•x d\Z^jZlbqgbf ihe•ghfhf Z ihl•f \bdhjbklh\m\Zlb ih[m^h\Zgm ZijhdkbfZp•cgm ko_fm ^ey hp•gx\Zggy dhhj^bgZlb lhqdb •klbgb f•g•fmfm nmgdp•€

Ydsh aZ^ZgZ ihke•^h\g•klv lhqhd Z1, w2, w3 • \•^hf• \•^ih\•^g• pbf lhqdZf agZq_ggy nmgdp•€ I1, f2 lZ I3 lh fh`gZ \bagZqblb ihkl•cg• \_ebqbgb D0, a1 lZ Z2 lZdbf qbghf sh agZq_ggy d\Z^jZlbqgh€ nmgdp•€ q(w) = a0+a1(w - w1)+a2(w - w1) (w - w2 a[•]Zxlvky a• agZq_ggyfb I Z m ljvho aZagZq_gbo lhqdZo I_j_c^_fh ^h h[qbke_ggy T Z m dh`g•c •a ljvho aZ^Zgbo lhqhd

f3 = f(w3) = q(w3) = f1+(f2–f 1)/(w2–w 1)(w3-w1)+a2(w3–w 1)(w3–w 2).

<bj•rmxqb hklZgg} j•\gyggy \•^ghkgh Z2 h^_j`m}fh

Ydsh lhqg•klv ZijhdkbfZp•€ ^hke•^`m\Zgh€ nmgdp•€ \ •gl_j\Ze•

\•^ Z1 ^h Z3 aZ ^hihfh]hx d\Z^jZlbqgh]h ihe•ghfZ \by\ey}lvky ^hkblv \bkhdhx lh \•^ih\•^gh ^h aZijhihgh\Zgh€ kljZl_]•€ ihrmdm ih[m^h\Zgbc ihe•ghf fh`gZ \bdhjbklh\m\Zlb ^ey hp•gx\Zggy dhhj^bgZlb lhqdb hilbfmfm

GZ]Z^Z}fh sh klZp•hgZjg• lhqdb nmgdp•€ h^gh€ af•ggh€ \bagZqZxlvky reyohf ijbj•\gx\Zggy ^h gmey €€ i_jrh€ iho•^gh€ • gZklmigh]h agZoh^`_ggy dhj_g•\ hljbfZgh]h lZdbf qbghf j•\gyggy

M ^Zghfm \biZ^dm a j•\gyggy

dq = a1 + a 2 + (w − w 2 )+ a 2 (w − w1 ) = 0 dw

Hkd•evdb nmgdp•y I Z gZ jha]eygmlhfm •gl_j\Ze• fZ} \eZklb\•klv mg•fh^Zevghkl• Z Zijhdkbfmxqbc d\Z^jZlbqgbc ihe•ghf lZdh` } mg•fh^Zevghx nmgdp•}x lh fh`gZ hq•dm\Zlb sh \_ebqbgZ w

\by\blvky ijbcgylghx hp•gdhx dhhj^bgZlb lhqdb •klbgh]h hilbfmfm Z*.

F_lh^ ihke•^h\gh]h hp•gx\Zggy a \bdhjbklZggyf d\Z^jZlbqgh€ ZijhdkbfZp•€ f_lh^ IZm_eeZ

F_lh^ IZm_eeZ aZkgh\Zgbc gZ ihke•^h\ghfm aZklhkm\Zgg• ijhp_^mjb hp•gx\Zggy a \bdhjbklZggyf d\Z^jZlbqgh€ ZijhdkbfZp•€

Ko_fm Ze]hjblfm fh`gZ hibkZlb lZdbf qbghf G_oZc Z1 ihqZldh\Z w h[jZgZ \_ebqbgZ djhdm ih hk• Z

Djhd H[qbkeblb I Z3 • agZclb Fmin = min {f1, f2, f3}, wmin lhqdZ wi sh \•^ih\•^Z} )min.

Djhd Ih ljvho lhqdZo Z1, w2, w3 h[qbkeblb x \bdhjbklh\mxqb nhjfmem ^ey hp•gx\Zggy aZ ^hihfh]hx d\Z^jZlbqgh€ ZijhdkbfZp•€

Djhd I_j_\•jdZ gZ aZd•gq_ggy ihrmdm

- f( w ^hklZlgvh fZehx"

- w ^hklZlgvh fZehx"

Ydsh h[b^\• mfh\b \bdhgmxlvky aZd•gqblb ihrmd M ijhlbe_`ghfm \biZ^dm i_j_clb ^h djhdm

Djhd <b[jZlb ©gZcdjZsmª lhqdm Zmin Z[h w • ^\• lhqdb ih

h[b^\• klhjhgb \•^ g_€ IhagZqblb p• lhqdb \ ijbjh^ghfm ihjy^dm • i_j_clb ^h djhdm

AZagZqbfh sh ijb i_jr•c j_Ze•aZp•€ djhdm f_`• •gl_j\Zem sh f•klblv lhqdm f•g•fmfm g_ h[h\ yadh\h \by\eyxlvky \klZgh\e_gbfb Ijb pvhfm hljbfZgZ lhqdZ w

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lh]h sh[ \bdexqblb fh`eb\•klv aZgZ^lh \_ebdh]h _dkljZiheyp•cgh]h i_j_f•s_ggy \Zjlh ijh\_klb i•key djhdm ^h^Zldh\m i_j_\•jdm c m \biZ^dm dheb lhqdZ o agZoh^blvky aZgZ^lh ^Ze_dh \•^ Z3 aZf•gblb w lhqdhx dhhj^bgZlZ ydh€ h[qbkex}lvky a

jhajZomgdhf aZa^Ze_]•^v mklZgh\e_gh€ ^h\`bgb djhdm

F_lh^b ihrmdm a \bdhjbklZggyf iho•^gbo

F_lh^b ijyfh]h ihrmdm • lhqdh\h]h hp•gx\Zggy ‰jmglmxlvky gZ ijbims_ggyo ijh mg•fh^Zevg•klv c m jy^• \biZ^d•\ ijh [_ai_j_j\g•klv ^hke•^`m\Zgh€ p•evh\h€ nmgdp•€ >hp•evgh ijbimklblb sh ydsh gZ ^h^Zlhd ^h mfh\b [_ai_j_j\ghkl• \\_klb \bfh]m ^bn_j_gp•x\Zggy nmgdp•€ lh _n_dlb\g•klv ihrmdh\bo ijhp_^mj fh`gZ •klhlgh i•^\bsblb

G_h[o•^ghx mfh\hx •kgm\Zggy ehdZevgh]h f•g•fmfm nmgdp•€ \ ^_yd•c lhqp• ] } h[_jlZggy \ gmev i_jrh€ iho•^gh€ nmgdp•€ \ lhqp• ] lh[lh I ] GI GZ_w=z = 0.

Ydsh nmgdp•y I Z f•klblv qe_gb sh \dexqZxlv Z m lj_lvhfm • [•evr \bkhdbo klmi_gyo lh [_aihk_j_^g} h^_j`Zggy ZgZe•lbqgh]h j•r_ggy j•\gyggy I Z fh`_ \by\blbky kdeZ^gbf M lZdbo \biZ^dZo \bdhjbklh\mxlvky gZ[eb`_g• f_lh^b ihke•^h\gh]h ihrmdm klZp•hgZjgh€ lhqdb nmgdp•€ I

3.3. F_lh^ GvxlhgZ JZnkhgZ

>ey nmgdp•c h^g•}€ af•ggh€ deZkbqgbc i•^o•^ ijb ihrmdm agZq_gv Z m lhqdZo i_j_]bgm nmgdp•€ I Z ihey]Z} m \bj•r_gg• j•\gyggy f'(w) = 0.

<bj•rblb lZd_ j•\gyggy g_ aZ\`^b ijhklh Lhfm dhjhldh jha]eyg_fh qbkeh\bc f_lh^ ch]h j•r_ggy Ydsh fh`gZ agZclb ^\Z agZq_ggy Z lZ E lZdbo sh I D lZ I E fZxlv ijhlbe_`g• agZdb lh^• m kbem hq_\b^gbo ijbims_gv ijh [_ai_j_j\g•klv [m^_ •kgm\Zlb dhj•gv pvh]h j•\gyggy Q ijbqhfm Z Q E

F_lh^ GvxlhgZ ^ha\hey} ihe•irblb \•^ghkgh ]jm[m ZijhdkbfZp•x sh[ h^_j`Zlb dhj•gv j•\gyggy I Z

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M aZ]Zevghfm \biZ^dm ihe•ir_g_ agZq_ggy ^ey

1l_jZp•€ fh`mlv [mlb ijh^h\`_g• ^hlb ^hdb ^ey ^\ho gZklmigbo ZijhdkbfZp•c g_ [m^_ ^hky]gmlZ g_h[o•^gZ lhqg•klv

F_lh^ k_j_^gvh€ lhqdb f_lh^ ;hevpZgh

Ydsh nmgdp•y I Z mg•fh^ZevgZ \ aZ^Zghfm •gl_j\Ze• ihrmdm lh lhqdhx hilbfmfm } lhqdZ m yd•c I Z Ydsh ijb pvhfm } fh`eb\•klv h[qbkex\Zlb yd agZq_ggy nmgdp•€ lZd • €€ iho•^gh€ lh ^ey agZoh^`_ggy dhj_gy j•\gyggy I Z fh`gZ kdhjbklZlbky _n_dlb\gbf Ze]hjblfhf \bdexq_ggy •gl_j\Ze•\ gZ dh`g•c •l_jZp•€ ydh]h jha]ey^Z}lvky ebr_ h^gZ ijh[gZ lhqdZ GZijbdeZ^ ydsh \ lhqp• ] \bdhgm}lvky g_j•\g•klv I ] lh a mjZom\Zggyf ijbims_ggy ijh mg•fh^Zevg•klv ijbjh^gh kl\_j^`m\Zlb sh lhqdZ

f•g•fmfm g_ fh`_ agZoh^blbky e•\•r_ lhqdb ] 1grbfb keh\Zfb •gl_j\Ze Z ≤ ] i•^ey]Z} \bdexq_ggx A •grh]h [hdm ydsh I ] ! lh

lhqdZ f•g•fmfm g_ fh`_ agZoh^blbky ijZ\•r_ ] lZ •gl_j\Ze Z ≤ z

fh`gZ \bdexqblb

<bagZqbfh ^\• lhqdb / lZ 5 lZdbf qbghf sh I / lZ I 5 ! KlZp•hgZjgZ lhqdZ jhalZrh\ZgZ f•` / lZ 5 H[qbkebfh agZq_ggy iho•^gh€ nmgdp•€ \ k_j_^g•c lhqp• jha]eygmlh]h •gl_j\Zem ] / 5 Ydsh I ] ! lh •gl_j\Ze ] 5 fh`gZ \bdexqblb a •gl_j\Zem ihrmdm A •grh]h [hdm ydsh I ] lh fh`gZ \bdexqblb •gl_j\Ze / ] Gb`q_ ^Z}lvky nhjfZe•ah\Zgbc hibk djhd•\

Ze]hjblfm

G_oZc } h[f_`_gbc •gl_j\Ze D ≤ w ≤ E lZ aZ^Zgbc iZjZf_lj a[•`ghkl• ε .

Djhd Ydsh _I ]_ ε aZd•gqblb ihrmd M ijhlbe_`ghfm \biZ^dm ydsh I ] ihdeZklb / ] lZ i_j_clb ^h djhdm Ydsh

34

I ] ! ihdeZklb 5 ] lZ i_j_clb ^h djhdm

Ke•^ aZagZqblb sh eh]•qgZ kljmdlmjZ ihrmdm \•^ih\•^gh ^h \bdeZ^_gh]h f_lh^m \bdexq_ggy •gl_j\Ze•\ aZkgh\ZgZ ebr_ gZ ^hke•^`_gg• agZdZ iho•^gh€ g_aZe_`gh \•^ agZq_gv ydL py iho•^gZ ijbcfZ}.

F_lh^ k•qgbo f_lh^ ohj^

F_lh^ k•qgbo sh } dhf[•gZp•}x f_lh^m GvxlhgZ • aZ]Zevgh€ ko_fb \bdexq_ggy •gl_j\Ze•\ hj•}glh\Zgbc gZ agZoh^`_ggy dhj_gy j•\gyggy I Z \ •gl_j\Ze• Z E ydsh ajhamf•eh lZdbc dhj•gv •kgm}

Ijbimklbfh sh \ ijhp_k• ihrmdm klZp•hgZjgh€ lhqdb nmgdp•€ Io \ •gl_j\Ze• Z E \by\e_g• ^\• lhqdb / lZ 5 m ydbo agZdb iho•^gh€ j•ag• M pvhfm \biZ^dm Ze]hjblf f_lh^m k•qgbo ^ha\hey} Zijhdkbfm\Zlb nmgdp•x I Z ©k•qghx ijyfhxª ijyfhx e•g•}x sh a }^gm} ^\• lhqdb • agZclb lhqdm m yd•c k•qgZ ]jZn•dZ I Z i_j_lbgZ}

\•kv Z[kpbk LZdbf qbghf \bieb\Z} sh gZ[eb`_ggy ^h klZp•hgZjgh€ lhqdb o* \bagZqZ}lvky aZ nhjfmehx

z = R − f ′(R )(R − L) . f ′(R )− f ′(L)

Ydsh _I ]_ ≤ ε ihrmd \Zjlh aZd•gqblb M ijhlbe_`ghfm \biZ^dm

g_h[o•^gh \b[jZlb h^gm a lhqhd / Z[h 5 lZdbf qbghf sh[ agZdb iho•^gh€ \ p•c lhqp• • lhqp• [meb j•ag• Z ihl•f ih\lhjblb hkgh\gbc djhd Ze]hjblfm

E_]dh [Zqblb sh gZ \•^f•gm \•^ f_lh^m k_j_^gvh€ lhqdb f_lh^ k•qgbo aZkgh\Zgbc gZ ^hke•^`_gg• g_ l•evdb agZdZ Ze_ • agZq_gv iho•^gh€ \ ijh[gbo lhqdZo • lhfm \ jy^• \biZ^d•\ ^ha\hey} \bdexqblb [•evr_ iheh\bgb •gl_j\Zem ihrmdm

3.3. F_lh^ ihrmdm a \bdhjbklZggyf dm[•qgh€ ZijhdkbfZp•€

<•^ih\•^gh ^h f_lh^m sh jha]ey^Z}lvky nmgdp•y I sh i•^ey]Z} f•g•f•aZp•€ Zijhdkbfm}lvky ihe•ghfhf lj_lvh]h ihjy^dm Eh]•qgZ ko_fZ f_lh^m ZgZeh]•qgZ ko_f• f_lh^•\ a \bdhjbklZggyf

35

d\Z^jZlbqgh€ ZijhdkbfZp•€ H^gZd m ^Zghfm \biZ^dm ih[m^h\Z Zijhdkbfmxqh]h ihe•ghfZ ijh\h^blvky gZ hkgh\• f_grh]h qbkeZ lhqhd hkd•evdb \ dh`g•c lhqp• fh`gZ h[qbkex\Zlb agZq_ggy yd nmgdp•€ lZd • €€ iho•^gh€

Jh[hlZ Ze]hjblfm ihqbgZ}lvky \ ^h\•evgh h[jZg•c lhqp• Z1:

agZoh^blvky •grZ lhqdZ Z2 lZdZ sh iho•^g• I Z1 lZ I Z2 fZxlv j•ag• agZdb 1grbfb keh\Zfb g_h[o•^gh ihdeZklb klZp•hgZjgm lhqdm

:ijhdkbfmxqZ dm[•qgZ nmgdp•y aZibkm}lvky \ gZklmighfm \b]ey^•:

f*(w1) = a0+a1(w - w1 Z2(w - w1)(w – w 2 Z3(w – w 1)2(w – w 2).

I Z • I Z m pbo lhqdZo

Ijb\_^_fh nhjfZe•ah\Zgbc hibk Ze]hjblfm G_oZc aZ^Zg•

Z` ^h lhqdb Z m yd•c I ZM-1I Z Ihl•f ihdeZklb Z1 = wM-1, w2 =

Z H[qbkeblb agZq_ggy I Z1), f(w2), f'(w1 lZ I Z2).

Djhd AgZclb klZp•hgZjgm lhqdm Z* Zijhdkbfmxqh]h dm[•qgh]h ihe•ghfZ

z = 3(f ¢(w1 )- f ¢(w 2 )) + f ¢(w1 )+ f ¢(w 2 ), w2 - w1

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NhjfmeZ ^ey β aZ[_ai_qm} gZe_`gbc \b[•j h^gh]h a ^\ho dhj_g•\ d\Z^jZlgh]h j•\gyggy ^ey agZq_gv μ ihdeZ^_gbo \ •gl_j\Ze• \•^

^h nhjfmeZ ]ZjZglm} sh h^_j`m\ZgZ lhqdZ Z jhalZrh\ZgZ f•` Z1

lZ Z2.

Djhd Ydsh I Z*)<f(w1 i_j_clb ^h djhdm M ijhlbe_`ghfm

\biZ^dm h[qbkex\Zlb Z* ih nhjfme• Z* = w* + 0,5(w* - w1 ^hlb ^hdb g_ [m^_ \bdhgm\Zlbky g_j•\g•klv I Z*) ≤ f(w1).

Djhd I_j_\•jdZ gZ aZd•gq_ggy ihrmdm

Ydsh _I Z*)| ≤ ε 1 lZ _I Z*)| ≤ ε 2 ihrmd aZd•gqblb M ijhlbe_`ghfm \biZ^dm ihdeZklb Z[h

(a) w2=w1 • Z1=w* ydsh I Z*)f'(w1)<0,

Z[h

[ Z1=w* ydsh I Z*)f'(w2)<0.

Ihl•f i_j_clb ^h djhdm

AZagZqbfh sh djhdb lZ j_Ze•amxlv ijhp_^mjm ihrmdm f_` •gl_j\Zem _\jbklbqgbf f_lh^hf ijbqhfm af•gZ agZdZ iho•^gh€ \bdhjbklh\m}lvky yd djbl_j•c i_j_oh^m q_j_a lhqdm hilbfmfm GZ djhp• ijh\h^ylvky h[qbke_ggy dhhj^bgZlb lhqdb hilbfmfm Zijhdkbfmxqh]h ihe•ghfZ Djhd Zkhp•ch\Zgbc a i_j_\•jdhx lh]h nZdlm sh hljbfZgZ hp•gdZ ^•ckgh } ihe•ir_gbf gZ[eb`_ggyf ^h lhqdb hilbfmfm

37

JHA>1E ;:=:LH<BF1JG1 ;?A=J:>1/GLG1 F?LH>B G?E1G1CGH2 HILBF1A:P12

4.1 F_lh^ G_e^_jZ F•^Z

F_lh^ G_e^_jZ F•^Z ihrmd ih kbfie_dkm ihrmd ih ^_nhjfh\Zghfm [Z]Zlh]jZggbdm 62 f_lh^ } jha\bldhf

kbfie_dkgh]h f_lh^m Ki_g^e• O_dklZ • O•fk\hjlZ Fgh`bgZ

Qh€ j•\gh\•^^Ze_gh€ lhqdb \ Qf•jghfm ijhklhj• gZab\Z}lvky j_]meyjgbf kbfie_dkhf Py dhgn•]mjZp•y jha]ey^Z}lvky \ f_lh^• Ki_g^e• O_dklZ • O•fk\hjlZ Hl`_ m ^\h\bf•jghfm ijhklhj• kbfie_dkhf } j•\ghklhjhgg•c ljbdmlgbd Z \ ljb\bf•jghfm ijhklhj• ² ijZ\bevgbc l_ljZ_^j

1^_y f_lh^m ihey]Z} \ ihj•\gygg• agZq_gv nmgdp•€ \ Q \_jrbgZo kbfie_dkm • i_j_f•s_gg• kbfie_dkm \ gZijyfdm hilbfZevgh€ lhqdb aZ ^hihfh]hx •l_jZp•cgh€ ijhp_^mjb M kbfie_dkghfm f_lh^• aZijhihgh\Zghfm kihqZldm j_]meyjgbc kbfie_dk \bdhjbklh\m\Z\ky gZ dh`ghfm _lZi• G_e^_j • F•^ aZijhihgm\Zeb d•evdZ fh^bn•dZp•c pvh]h f_lh^m sh ^himkdZxlv sh[ kbfie_dkb [meb g_ijZ\bevgbfb M j_amevlZl• \bcrh\ ^m`_

gZ^•cgbc f_lh^ ijyfh]h ihrmdm sh } h^gbf a gZc_n_dlb\g•rbo ydsh Q ≤ 6.

M f_lh^• Ki_g^e• O_dklZ • O•fk\hjlZ kbfie_dk i_j_f•sm}lvky aZ ^hihfh]hx ljvho hkgh\gbo hi_jZp•c \•^h[jZ`_ggy, jhaly]Zggy •

klbkdm Af•kl pbo hi_jZp•c klZg_ ajhamf•ebf ijb jha]ey^• djhd•\ ijhp_^mjb G_e^_jZ F•^Z

Djhd AgZc^_fh agZq_ggy nmgdp•€ I1 = f(w1), f2 = f(w2), ..., fn+1 = f(wn+1 m \_jrbgZo kbfie_dkm

Djhd AgZc^_fh gZc[•evr_ agZq_ggy nmgdp•€ Ih sh \bieb\Z} a

gZc[•evrh]h agZq_ggy nmgdp•€ Ig gZcf_gr_ agZq_ggy nmgdp•€ Il lZ \•^ih\•^g• €f lhqdb Zh, wg lZ Zl.

Djhd AgZc^_fh p_glj \Z]b \k•o lhqhd aZ \bgyldhf lhqdb Zh.

G_oZc p_gljhf \Z]b [m^_ w0 = 1 åwi H[qbkebfh I Z0) = f0.

n i¹h

Djhd Ajmqg•r_ aZ \k_ ihqZlb i_j_f•s_ggy \•^ lhqdb Zh.

<•^h[jZab\rb lhqdm Zh sh^h lhqdb Z0 h^_j`bfh lhqdm Zr lZ

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agZc^_fh I Zr) = fr.

Hi_jZp•y \•^h[jZ`_ggy •exkljm}lvky jbk

Jbk Hi_jZp•y \•^h[jZ`_ggy kbfie_dkm

Ydsh Z ! dh_n•p•}gl \•^h[jZ`_ggy lh iheh`_ggy lhqdb Zr \bagZqZ}lvky lZdbf qbghf

wr-w0 = a(w0-wh),

^_ D _Zr – w 0| / |w0 – w h|.

Djhd Ihj•\gy}fh agZq_ggy nmgdp•c Ir lZ Il.

1. Ydsh Ir < fl lh fb h^_j`Zeb gZcf_gr_ agZq_ggy nmgdp•€

GZijyfhd •a lhqdb Z0 m lhqdm Zr gZc[•evr ajmqgbc ^ey i_j_f•s_ggy LZdbf qbghf fb jh[bfh jhaly]Zggy \ pvhfm gZijyfdm • agZoh^bfh lhqdm Ze • agZq_ggy nmgdp•€ Ie = f(we).

Jbk •exkljm} hi_jZp•x jhaly]Zggy kbfie_dkm. Dh_n•p•}gl jhaly]Zggy Y ! fh`gZ agZclb a \bjZam

Y_oe ± o0_ _or ± o0|.

Z Ydsh Ie < fl lh aZf•gy}fh lhqdm Zh gZ lhqdm Ze • i_j_\•jy}fh Qm lhqdm kbfie_dkm gZ a[•`g•klv ^h f•g•fmfm ^b\ djhd

Ydsh a[•`g•klv ^hky]gmlZ lh ijhp_k amibgy}lvky m ijhlbe_`ghfm

\biZ^dm ih\_jlZ}fhky gZ djhd

[ Ydsh Ie ³ fl lh \•^db^Z}fh lhqdm Ze Hq_\b^gh fb

39

i_j_f•klbebky aZgZ^lh ^Ze_dh \•^ lhqdb Z0 ^h lhqdb Zr Lhfm \Zjlh

aZf•gblb lhqdm Zh gZ lhqdm Zr m yd•c [meh hljbfZgh ihe•ir_ggy djhd

i_j_\•jblb a[•`g•klv • ydsh \hgZ g_ ^hky]gmlZ ih\_jgmlbky gZ djhd

Jbk Hi_jZp•y jhaly]Zggy kbfie_dkm

Ydsh Ir > fl Ze_ Ir ≤ fg lh Zr } djZshx lhqdhx \ ihj•\gygg• a

•grbfb ^\hfZ lhqdZfb kbfie_dkm • fb aZf•gy}fh lhqdm Zh gZ lhqdm wr • ydsh a[•`g•klv g_ ^hky]gmlZ ih\_jlZ}fhky gZ djhd lh[lh \bdhgm}fh imgdl [ hibkZgbc \bs_

Ydsh Ir > fl • Ir > fg lh i_j_c^_fh gZ djhd Djhd Ihj•\gy}fh agZq_ggy nmgdp•c Ir • Ih .

Ydsh Ir > fh lh i_j_oh^bfh [_aihk_j_^gvh ^h djhdm klbkdm

m ijhlbe_`ghfm \biZ^dm ydsh Ir < fh lh aZf•gy}fh lhqdm oh gZ

lhqdm or • agZq_ggy nmgdp•€ Ih gZ agZq_ggy nmgdp•€ Ir aZiZf ylh\m}fh agZq_ggy Ir > fg •a djhdm ijb\_^_gh]h \bs_ ihl•f i_j_oh^bfh gZ djhd

2. M pvhfm \biZ^dm Ir > fh lhfm ykgh sh fb i_j_f•klbebky

aZgZ^lh ^Ze_dh \•^ lhqdb oh ^h lhqdb Z0 Kijh[m}fh \bijZ\blb p_ agZcrh\rb lhqdm Zc Z ihl•f Ic aZ ^hihfh]hx djhdm klbkdm, ihdZaZgh]h gZ jbk

Ydsh Ir > fh lh \•^jZam i_j_oh^bfh ^h djhdm klbkdm • agZoh^bfh lhqdm Zc a• ki•\\•^ghr_ggy Zc – w 0 = b(wh – w 0 ^_ E E

dh_n•p•}gl klbkdm Lh^• Zk = bwh + (1-b)w0.

Ydsh Ir < fh lh kihqZldm aZf•gbfh lhqdm Zh gZ lhqdm Zr Z ihl•f

ajh[bfh klbkd Lh^• lhqdm Zc ^b\ jbk agZc^_fh a• ki•\\•^ghr_ggy Zk = bwr + (1-b)w0.