Critical Points and Error Rank in Best $H_2$ Matrix Rational Approximation of Fixed McMillan Degree
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Critical points and error rank matrix rational approximation in best of fixed McMillan degree. . Laurent Baratchart, Martine Olivi. N˚ 2970 September 1996 THE`ME 4. ISSN 0249-6399. apport de recherche.
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(57) R "bcmZi r " r gtl u |bhmZiwkmpE|qbc\] lE]rjrngt|q] u ]× lf] u bhlOgrvz\\]ij|qbcw¡ g=vm·ÃY ZE] lEktijbckmlkt§ l r l bcr¤dErqd gIe¢ecvgIijisgtwZE] u ijkisZE]CrqdEE\k u dfeh]Ct] lE]|sgIij] u v gtl u bhlÄ c# |jgIisZE]| isZ gIl£Ä ¶# e mVf À l gt\]e¢vijk¥ijZE]*|j] pf|j] rq] ligGijbckml Ä c# y Ä c' Ê* Ä c# . . . za dEwZng`|j]pE|j]rj] lisgIijbckmlnZEkte u r bc§ gtl u kmlfe¢v3b¢§ijZE]¤ec]§Xi Z gIl u rqb u ]¤bcl isZE]*gtJkDm]/]¿LdEgG¬ ijbckml.bcr¥g¦¡³] gt²t¼w-ehkmrq] u rjdEErqp gtw]£kt§Ä c# QVrq] ]ge I^EÀ Ö³ZEgtpfis]|jÀ}Y ZE]km|j]\ z·HjSf ¡"ZEkmrj]\¥gIij|qb«y m]|jrqbckml2bcr]=gtrbce¢v u ]|qb¢m] u R-À³bcl ¡"ZfbhwZ°w=gtrq] Ä ¶# y Ä ¶' bcrweckI¬ rj] u bcl Ķ`gtl u isZE]¥i ¡³k u ]× lfb¢ijbckmlErkt§"g r l wktbclEwb u ]t·¸ÕZE] l Ébhr|sgIiqbhktl gIeVÀ ¡"ZfbhwZTbhr*kmdf| ktlfecv¦w kmlEw] |ql ZE]|j]IÀijZfbhr/¡»]=gt²t¼ weckmrj] u lE]rjr`gGec¡³g=vzr ZEkte u r ]w=gtdfrj] Ä c# y Ä c ' Ä c# < 2 Ä c# Z gtr× lzbcij] w k u bh\]lErqbckml#ÀNrqbclEw ] < 2 bcrg × lfb¢is] »ehgtrjwZf²t]pf|jk u dEwi ·Nkm|q] t] lE]|sgIe r l " r \dfrqio]/wkmlErb u ]|j] u b¢§ijZE]*|sgtlfm] bhr"lEkti §¯dzecze e m fVÀEEdfi"¡»]rqZ gIe¢e+lEkti"lE]] u isZfbcr · Yk¥gtlvnÄ 4 Ä ! c#&%&' ÀE¡»]gtrqrjkLwbhgIis]/ijZE]*weckmrj] u rqZfbc§Xi¬®bcltgI|qbhgtlixrqdEErqp gtw ]kt§ @ Ä ¶# A -RÄ 1¾Ê W 4 Ä ¶# A > Ä 4 Ä ¶ ' [ QV R @ ¡"ZE] |q]isZE]rqdEpJ] |jrqw |bhpf
(58) i ¨\¥] gtlEr`ij|sgtlErqpktrj]t· oktis]3bcl p gtrqrqbclEisZ gGi -RÄ 1*bcr ~dErqi"isZf] ²I] |qlE]ekt§isZE] ±xgtlE²t]-ekmp]|sgIijkm| Ä # A  s Ä ! @ ¶  s 5 - ¶' Ä ? 1 Y ZE]ngtl gIe¢v?ijbcwnrb u ]kt§´ dEZE|q\ªgIlEl rª|q]=gIe¢bh½ gIijbckml isZf] km|v gtrqbcw=gGece¢v w ktlErqbcrqijrªbcl°gIpz¬ pfecv?bclEijZE] ³] df|qe¢bhlEG¬ $+gGyÍisZf] km|q] \ iskrqdEwZrjdfErjp gIw ] r`gtl u isZE]C§¯kte¢ehkD¡ bclE£|q] rqdfeci À ¡"ZfbhwZÍbcr pf|jkDm] u §¯km|¤bclErigtlEw]Cbhl e mF fVÀ#bhr/\] |q]e¢vTg rjpJ] wbhgIe¢bh½ gIijbckmlÍisk |jgIijbckml gIecr/kt§ isZE] ktdEtehgtr¬azZ gtpfbc|jkG¬azZfbc]e u r §Rgtw-iskm|bh½ gIijbckm]l e
(59) f+gIr m] lE]|sgIe¢bc½ ] u iskC\¥gIij|qb«y?¬®IgIecdE] u rqis|bhw-ije¢vlEkmlEw-vLwe¢bhw*§¯dElEw-ijbckmlEr bhl e m fV· \+ÏLÓ Ï l Ä Ä ! ¶#& %&? ' l n l n k ÄÌÊ Qq= R . . . . . |}| íd~VF.
(60) Q. n = l V V n K. n 0 k = n
(61) n n J k 3Ä c#&%&' =n 2 Z < 2 ³Ä n k n 0 =n _^ kKn n n < l V n =n n l ^ k , k =n = n PU l $ , n T V 6 . l X k_l kKn k_l
(62) l n n =n l -RÄ 1¾Ê Ä ¶# A . k @ l V n n k k l 0 k = < 2 »Ä Ê < 2 . ¸ÕZE] l < 2 »Ä Ê < 2 ¥ À:¡³]w=gIe¢ezqQ D R ijZE]. n m k n k l K 0 kI§}Ä gtl u ¡»]`] lErqdE|j]dElfbc¿Ldf] lE]rjr/vn|j] ¿?dfbc|qbclE .- »1 Ê # ·oY ZE]bc\pkm|igtlfw ] kt§ ijZfbcr"§RgIwiskt|qbc½=gIiqbhkml bclktdE| rqijd u vrqij] \r"§¯|jkm\ isZE]/i ¡»kª§¯kte¢ehkD¡ bclEC§Rgtwijr
(63) /Y ZE]*§Rgtw-iskm|qr gtl u bhlgqQ =»R ¡ bce¢egIe¢ehkD¡ dEroijkp gt|sgI\¥]-is|bh½] #? ' R- Æb 1 gtl u ¡ bce¢e pE|jkD?b u ]gClfbcw ]*\dzeciqbhpfe¢bcw=gIiqbct] u ]rjw |bhpzijbckml£kt§isZf]*isgtlEm]lixrqp gtw]t·
(64) $¹]§Xi"\dfe¢ijbcpfecbcw=gGijbckml£vijZE]/bhlElf] | Ä #&%F# bhrogtl bcrjkm\]ij|qvnÄ ¶# s Ä ¶# · /df|mkgGezbcl`isZfbcr¨rq] wiqbhktl`bhrijk/gtww kmdEli¾§¯km|+4 §Rgtw-i c
(65) Nvtb¢?bhlf*gxiskmpJkteckmtbcw=gIezw ktl?ij] li iskY ZE]km|j]\ m·Cnkm|j]CpE|q] w-bhrq]e¢vmÀ+¡³]rjZEgIece¨] l u kD¡ isZE]Crq]i*kt§»p gIbc|jr - o1 ¡ b¢isZTg m] w-iskm|EdEl u eh]»rqij|jdEw-isdE|q]³gtl u |q] pE|q] rj]li #? ' ¯- Æh¹1 gtr gtl kmp]lrqdEErq]i=· ¸º] u ] lfktis]Nv # isZE]"rq]iNkt§lEkm|j\¥gIe¢bc½ ] u bhlflE] |¾§¯dElEw-ijbckmlEr}kt§¹wDnb¢e¢e©gtl u ]m|j]]xÆ La ijZLdfr À 4 Ä c#&%F# ]-ehkmlfmr isk # b¢§¨gtl u kmlz§Xvbc§ <.2 -0D E G 1 .ÊË-0DÆEHG 1 I Ê # N 476 8 qQ tSR .- 4 N1 Ê # ®§ 4 # À³ijZE] l .- 4K` /C1 ]-ehkmlfmrªijk Ä ! c#&%F# gtl u ijZE] |q]§¯km|q]£pJkmrjrq] rq] r gT\bclfbc\¥gIe |j]=gGecbc½=gIiqbhktl .- 4K` /}1 Ê .-0 / Â
(66) 1 y qQ DR $+]-iÊ 2 K-0 / C 1 J] ijZE]pJkte¢vzlfkm\Cb©gIekt§¨pktec] ro¡"ZEkmrq] u ]m|j]]bcrÆ gtl u ¡"ZEktrj] |jkLktisrCe¢bh]< bhl isZf]dElzbci u bcrj² ZE] |q]=gI§Xij] | u ] lfktis] u vT· .¾¿LdEgIijbckml Q Q`R ]-yzpE|q] rqrj] r $- 4S` /¥1 gtrªijZE]£¿?dEktiqbh]livÍkt§¤rjkm\]$( * + pkIecvLlEkm\CbhgIex\ªgGis|qb«yÀ kt§/pJkte¢vzlfkI ¬ \bhgIe u ]m|j]]£gGiª\kmri3Æ ·ÇazdfErqiqbcijdfijbclEºEgtw² Í/ §¯km| º/ gtl u \dzeciqbhpfe¢v?bhlfv / isZE|qkmdEmZEktdfi=Àf¡»] kmfigGbhlng`§¯|sgtw-ijbckml gIe+|q] pE|q] rq] ligI4Siq` bhktl
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(77) . bhrisZE] |q] w-bhpE|qkzw gIeEpJkte¢vLlEkm\Cb©gGefkt§ L·¾aLdEwZg|j]pE|j]rj]l?isgIijbckmlZEkte u r¡"Zf] lE]-m] | .- 4K` /1 bhr}gtlEgIecv?iqbhw gGi¨bhlz× lfb¢i v a?¡"Z gIi¾bcr¨pJ] wdfecbhgt|¨ijk¤isZE] §Rgtw-i¨isZEgIi Obcr¨bclElE] |¨bcrijZE] b u ] lijb¢i v Ê Qq r^ R # ¡"ZE] |q] ÉÊ / - 4S` /1v u ]-× lfb¢ijbckml#· |sgtlis] u .- 4 1 Ê # À¤ijZfbhrbc\¥pzecbc] r 2 Ê Ue«tmÀ¹pE|jktp#·c farqbclEw ]¥lEkw=gIlEw ]-ecehgIijbckmlTw=gtlÍkLw wdE|*]-i ¡³]] l gtl u L À¹b¢i §¯< ktececkD¡"r bhln` p gt|ijbcw dfehgt| isZEgIi < 2 )- ¶ 1NÊ < 2 y < 2 ¶ ¡"ZE] lE]-m]| gtl u ¶xgI|j]¤bclElE] |okt§isZE]*rjgt\] rbh½]t· \+ÏLÓ Ï =n kKn # k k $ K = P n k nÆ ( n n n Ä c#&%F#
(78) l =n^l n ol n = k ] n QqP Rq nzk $ K l k 7 K »b¢i"§¯ktececkD¡"ro§¯|qkm\ ep fÀ isZE\ L·çgtl u pE|jktp#·"z·ñ f¹isZEgIi # bhr"bhl u ] ] u g¥\¥gtlzbc§¯kte u kt§xijZE]|qbcmZi u bh\]lErqbckmlÈrq\¥kLktijZfecv] \J] ufu ] u bcl°Ä #&%F# À³gIl u ijZ gIiisZf]bc\¥gtm]kt§ # dEl u ]|isZf]3rjdfErqiqbcijdfijbckml / s 4K` /nbcr`gnrjdEE\¥gtlzbc§¯c kte u kt§ # ' -RbÆ 1-·aLbclEw ]¥ehkLw=gGe w kLkm| u bcl gIij] rktlÈisZE]¦ehgIijij] |\¥g=vÈJ]¦kmzigIbclE] u §¯|qkm\ w=gtlEktlfbhw gIe/|q]=gIe¢bh½ gIijbckmlErÀoijZE] \ªgIp2gtrqrjkLwbhgIijbclEÍiskTg¦§¯dElEw-ijbckml2gIl gIe¢vLiqbhwgIiCbclf× lfb¢i v b¢isrpkIecvLlEkm\CbhgIe"kt§xpkIeh]rbcr rj\kzkIisZ#· azdEfrqijb¢isdzijbclE gtw² /§¯km| 4S` /fÀ"¡»]rj] ]¦ijZ gIiijZE]¦wkzV] 2ªw-bh]lisrkt§ gt|q] rj\kzkIisZ §¯dElEw-ijbckmlErokt§ 4 # ·N´ bcl gIe¢e¢vmÀ s £Ê bhrogIecrjkrj\kLktisZgtr"\dzeciqbhpfe¢bcw=gIiqbhkmlnbhr"fb¢e¢bhlE] gt|ogtl u w ktl?iqbhl?dEktdErxbclnÄ c · ´:kt| 4 # Àfeh]-i - 1}Ê W 4 Ä ¶#&%&' 4 Ä ! ¶#& %&? ' [ ¸º] u ]-× lE]*kmdE|"EdEl u eh]*bclgrj]-iq¬øisZE]km|j]-ijbcw \¥gtlflE] |ogtr Ê W - 1 4 # 4 - 1 [ ¸ÇbcijZnijZE] gIb u kt§Y ZE]km|j]\ zÀf¡»]rqZ gIe¢e+] l u kD¡ C¡ b¢isZg`× J] |j] u ris|jdfwisdf|j] . .
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(82) =^. n = l V V n K. \+ÏLÓ Ï =n n n n ºÄ c#&%F# * Ä ¶#&%&' =n$kKn k k $ K n l n
(83) kKn kKn k l n k #
(84) kKn n X V n k - 1F =n n - 1 k n k ,+ Æ n 6 8 l k k k 0 V n . l k 7 K N´ bh|qrqi"lEkIijbcw ]/isZ gGi 4 - 1»b¢§¾gtl u kmlzecvb¢§¨] gtwZnw ktecdE\l£J]eckmlEmr ijk g- 1}ÊËÄ ¶ # Ä ! ¶# ? ¡"ZfbhwZnbhr ijZE]*w km\pfec] \] lixkt§ Ä ¶# bclnijZE]*km|qijZEkmmkmlEgIe u ]w km\pJkmrqb¢ijbckml Ä ¶# Ê Ä ¶# g- 1 ´:|qkm\ |q]=gIe¢bhrjgIijbckml2ijZE] kt|qv úe tSVf À ¡³]²LlfkQ¡ÉijZ gIi¥isZE] u bc\¥]lErqbckmlÈkI§g- 1 kDt] | bhr] ¿?d gGe iskÍisZE]w=nbce¢ehgtl u ]m|j]]kt§ lEgt\]e¢v Æ·°Ö³kmlErq] ¿?dE] liqecvtÀ - 1 bhr¥6 8 g + ÆJ ¬ u bh\] lfrqbckml gIe*|q]=gIexm] w-iskm|rqp gtw ]I· kt|j] kDt] | Àxbc§ 4 - 1 gtl u b¢§ ¡³]rq]i Ê Ä 4 Ä ! ¶#& %&? ' Àf¡³] m]-io§¯|qkm\xQ K R Ê Ä qQ KjR Y ZE]`eh]-§Xi*Z gtl u ¬ rqb u ]Ckt§^Q Sj"R e¢bc] r/bhlÍÄ ¶#%&' ÀA¡"Zzbcec]CisZE]C|bhmZi/ZEgtl u ¬®rqb u ]`bhr¤ijZE]Crjdf\ kt§}g3pJkte¢vzlfkm\Cb©gIe \¥gIis|b¢ykt§¨pkte¢vLlEkm\CbhgIe u ]m|j]]`gIi\kmrqiÆ 4 gtl u kt§}g3\]\J] | kt§ Ä.! ¶# %&? ' ·N±o] lEw] bcr bhl§RgtwixgCpJktecvLlEkt\bhgIe+\¥gIij|qb«yAÀEijZ gIi ¡»] u ] lEktij]*?v . . . . . . %. . )Ê rjZEkD¡ bclE¥bclnp gt|qiqbhwdfehgt| isZ gIi »vQqP R gtl u Qq ^rR-Àf¡»] Z g=m]. . . . < 2
(85) ÇÆÂ. . . 4.
(86). 1 wkmlErbhrisrokt§|jgIijbckml gGe¹]ec] \] lijr · Ê ` `rqk¥ijZ gIi 4 - ³1 bc§gtl u kmlfe¢vb¢§ . 4 Ä ! ¶#& %&? ' xpJkmlwZEgtlEtbclE /bhlijk 4S` /3gtl unu b¢?b u bclEª?vBf/ À ¡»]rq] ]*isZ gGixijZfbhr bcrx]¿?dfbcIgIec] lixisk. isZE]*|j]¿?dfbh|q] \] li. - . ). Ä ¶#&%&' 4 . . }| ¯û. .
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(89) 1 rjdEwZnisZ gIi §¯kt|"gtlv 4 ÀfisZf] pEgt|qiqb©gIe+\¥gtp 6 8 '  ss $ -- $ - 1 1
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