Computational Aspects of the Implementation of Disk Inversions
|
|
- Earl Marshall
- 5 years ago
- Views:
Transcription
1 Computatonal Aspects of the Implementaton of Dsk Inversons Ivan Petkovć Faculty of Electronc Engneerng, Department of Computer Scence, Unversty of Nš, Serba Abstract More than three decades the mplementaton of teratve methods for the smultaneous ncluson of polynomal zeros n crcular complex nterval arthmetc s carred out usng the exact nverson of dsks. Based on theoretcal analyss and numercal examples, we show that the centered nverson gves smaller ncluson dsks. Ths surprsng result s the consequence of better convergence of the mdponts of produced dsks when the centered nverson s employed. Some examples of ncluson methods wth the centered and exact nverson, together wth numercal results, are gven. Keywords: Zeros of polynomals; smultaneous methods; ncluson methods; dsk nverson; crcular nterval arthmetc; convergence. AMS subject classfcatons: 65H05, 65G20, 30C5. Introducton A great mportance of numercal methods for determnng polynomal zeros n the theory and practce (for example, n solvng many problems of appled and fnance mathematcs, control theory, sgnal processng, nonlnear crcuts, boscence, and other dscplnes) has led to the development of a great number of zero-fndng methods n ths feld, see, e.g., the books [7], [2], [4]. These numercal methods have become practcally applcable together wth the rapd growth of dgtal computers some ffty years ago. However, the computed soluton of an algebrac equaton s only an approxmaton to the exact soluton due to the errors orgnatng from dscretzaton, truncaton and from roundng. Ths naturally leads to the queston what s the error n the result? Solvng polynomal equatons, apart from the work engaged n the procedure appled to mprove the approxmate result, a consderable amount of work s nvolved n determnng error bounds of the mproved approxmatons to the polynomal zeros. An Submtted: January 8, 2009; Accepted: January 4, 200. Ths work was supported by the Serban Mnstry of Scence under grant
2 82 Ivan Petkovć, Computatonal Aspects of Dsk Inversons effcent approach that overcomes the aforementoned problem and gves satsfactory results s based on the use of nterval arthmetc. Partcularly, t turns out that teratve nterval methods for the smultaneous ncluson of polynomal zeros, realzed n crcular complex nterval arthmetc, are effcent n the case of complex zeros. These methods produce dsks that contan the wanted zeros n each teraton. For ths reason, such methods can be regarded as a self-valdated numercal tool that provdes automatc computaton of rgorous error bounds (gven by rad of resultng ncluson dsks) to the approxmate solutons. Ths very useful (ncluson) property s the man advantage of nterval methods. The am of ths note s to pont to the effcent use of a proper nverson of a dsk n the mplementaton of a class of smultaneous ncluson root-fndng methods based on fxed pont relatons. Although nterval methods started beng developed snce the 970 s, they were realzed usng the nverson based on Möbus s transformaton of a dsk Z by the functon z /z, the so-called exact nverson, denoted by Z. In crcular nterval arthmetc (arthmetc whch deals wth dsks) ths operaton s the exact operaton snce the mage Z completely concdes wth the exact range {/z : z Z}. Note that only a couple of authors tred to deal wth some other type of nversons n order to obtan smaller ncluson dsks. The reason probably les n the fact that the exact nverson gves the smallest dsks compared wth other nversons so that t seemed that ts applcaton s qute reasonable. In ths paper we show that the sze of ncluson dsks depend heavly on some other (extra-arthmetcal) features, not only of the employed arthmetc. 2 Crcular complex nterval arthmetc We start wth a short revew of the basc operatons n crcular nterval arthmetcs. For more detals see the books [2] and [2]. A crcular closed regon (dsk) Z := {z : z c r} wth center c := md Z and radus r := rad Z we wll denote n ths paper by parametrc notaton Z := {c; r}. Usng the Möbus transformaton we ntroduce the exact nverson { {c; r} = c c 2 r 2 ; r c 2 r 2 } = {/z : z {c; r} (0 / {c; r}). () As we wll see n ths paper, n some applcatons t s more convenent to take an nverse dsk {c; r} Ic := {/c; ρ} whose center s just /c, where c s the center of the orgnal dsk Z = {c; r}. Denote the crcumference of such nverse dsk wth D c, and let D e = { z : z c c 2 r 2 = r c 2 r 2 be the crcumference of the exact nverse dsk {c; r} gven by (). Snce {c; r} s the exact range, t has to be {c; r} Ic := D c ntd c {c; r}. Accordng to ths and Fg., the radus ρ = rad {c; r} Ic s equal to ρ = max w {c;r} c w = max θ [0,2π) c + r exp(θ) c c 2 r 2 = r max c θ [0,2π) r + cexp(θ) c 2 r 2 = r max r + c exp(α) c α [0,2π) c 2 r 2 = r c ( c r). }
3 Relable Computng 5, Ths formula, often used by Rokne, Wu, Ratschek, Rump and others, can be also derved usng a general approach to crcular centered forms of elementary complex functons, see [] and [3]. Fg. The exact and centered nverson It s necessary to check that the dsk {c; r} Ic = {/c; ρ} completely contans the exact range {/z : z C} = {c; r}, n other words, we have to prove the nequalty md D c md D e rad Dc radd e, that s, c c c 2 r 2 r c ( c r) r c 2 r 2, whch reduces to the equalty. Ths means that the crcle D e touches (nsde) D c (see Fg. ). Therefore, the so-called centered nverson s gven by { } {c; r} Ic = c ; r {c; r} (0 / {c; r}). (2) c ( c r) Actually, the nverson defned n ths way concdes wth the Taylor form of nverson derved n []. One observes that the centered nverson always produces larger dsks than the exact nverson (). If Z k = {c k ; r k }(k =,2), then Z ± Z 2 = {c ± c 2; r + r 2}, w Z = {w md Z; w rad Z} (w C), Z Z 2 = {c c 2; c r 2 + c 2 r + r r 2}, Z : Z 2 = Z INVZ 2 (0 / Z 2, INV {(),() Ic }). The addton, subtracton and nverson Z are exact operatons, that s, Z Z 2 = {z z 2 : z Z, z 2 Z 2}, {+,,() }. We wll use the abbrevaton INV to denote the nverson of a dsk. 3 Smultaneous ncluson of polynomal complex zeros Let P be a monc polynomal of degree n wth smple real or complex zeros ζ,..., ζ n and assume that we have found an array of n dsks Z = (Z,..., Z n) such that
4 84 Ivan Petkovć, Computatonal Aspects of Dsk Inversons ζ Z ( I n := {,..., n}). Denote by ζ = (ζ,..., ζ n) and z = (z,..., z n) the vectors of the exact zeros of P and the centers of dsks, z = md Z, and let us represent a fxed pont relaton n a general form ζ = F (z, ζ) ( I n). (3) Let N(z) = P(z)/P (z) denote Newton s correcton. Our study wll be carred out n partcular cases of the followng two examples of the fxed pont relatons ζ = z ζ = z P(z ) n (z ζ j) j= j N(z ) n j= j ( I n), (4) z ζ j whch can be easly obtaned from the factorzaton P(z) = n (z ζ j) j= ( I n), (5) applyng the logarthmc dervatve n the case of (5). Substtutng the zeros on the rght sde of (3) by ther ncluson dsks and usng the ncluson property, we obtan the ncluson ζ Ẑ := F (z, Z) ( I n). (6) Under sutable ntal condtons (takng nto account the sze of ntal dsks and ther dstrbuton), the set Ẑ s a new contracted dsk contanng the zero ζ. In general, we wll use the symbol to denote quanttes n the subsequent teraton. Settng (Z,..., Z n) =: (Z (0),..., Z(0) n ), from (6) we can construct the followng teratve methods for the smultaneous ncluson of all smple zeros of the polynomal P : Z (m+) = F (z (m), Z (m) ) (m = 0,,... ; I n). (7) Let Z (m) such that r (m) := {c (m) ; r (m) } be ncluson dsks produced by the teratve method (7) 0 ( I n) when m, and let r (m) = max. If there exsts a real number k and a nonzero constant γ such that r (m+) (r (m) ) k γ, n r(m) then k s called the order of convergence of the teratve nterval method (7). In practce, for small enough r (m) t s suffcent to show that r (m+) = O ) k ) ((r (m), where O s the Landau symbol. Ths defnton of the order of convergence s satsfactory for the class of nterval methods consdered n ths paper. A more general defnton of the convergence speed, expressed by the so-called R-order, can be found n [2]. Havng n mnd (6) and (7), we start from the fxed pont relatons (4) and (5) and construct the followng partcular methods for the smultaneous ncluson of all smple zeros of the polynomal P :
5 Relable Computng 5, Weerstrass-lke method [2], [5], the convergence order 2: Ẑ = z P(z ) n INV(z Z j) ( I n). (8) j= j Gargantn-Henrc s method [5], the convergence order 3: Ẑ = z INV 2 /N(z ) j)) n INV (z Z ( I n). (9) j= j Here we assume that INV,INV,INV 2 {(),() Ic }. The subscrpt ndces and 2 n (9) pont to the order of applcaton of nversons. The nterval method (9) (wth the exact nversons) was proposed n [5] so that t s often refereed to as Gargantn- Henrc s method. Let us note that orgnal methods (8) and (9) presented n the papers cted above, as many other smlar methods based on fxed pont relatons, used only the exact nverson, that s, INV = INV = INV 2 = (). We could consder some other fxed pont relatons and correspondng nterval methods, but the conclusons are entrely the same as n the case of nterval methods (8) and (9). Remark The man advantage of nterval methods (8) and (9) s the ncluson property; namely, n each teraton these nterval methods produce the array of dsks Z (m),..., Z n (m) such that ζ Z (m) (m = 0,, 2,... ; I n). In ths way the automatc control of error s provded snce md Z (m) ζ rad Z (m), takng the mdponts of dsks to be approxmatons to the zeros. From the convergence analyss of nterval methods (8) and (9) we can fnd that ˆr = rad Ẑ = O( P(z) r), r = max r (0) n for the Weerstrass-lke method (8) and ˆr = rad Ẑ = O( P(z) 2 r) () for the Gargantn-Henrc method (9), see the book [2] for detals. Obvously, snce P(z ) = z ζ z ζ j = O(r), j from (0) and () we conclude that the convergence order of the methods (8) and (9) s two and three, respectvely. Besdes the study of smultaneous ncluson methods, let us consder teratve methods for the smultaneous determnaton of complex zeros realzed n ordnary complex arthmetc. Wthout loss of generalty, we wll restrct out attenton to the methods correspondng to the ncluson methods (8) and (9). If we start from the fxed pont relatons (4) and (5) and substtute the exact zeros ζ,..., ζ n by ther ( pont ) approxmatons z,..., z n, then we obtan the followng two methods for the smultaneous approxmaton of polynomal zeros:
6 86 Ivan Petkovć, Computatonal Aspects of Dsk Inversons Weerstrass-Durand-Kerner method [6], the convergence order 2: ẑ = z P(z ) n (z z j) j= j Ehrlch-Aberth s method [], [4], the convergence order 3: ẑ = z N(z ) n j= j z z j ( I n); (2) ( I n). (3) For more detals on these methods see the recent book [7]. From (0) and () we nfer that the convergence of rad strongly depends on the centers of ncluson dsks; when the centers are closer to the zeros, the convergence of rad s faster. Let us examne now the convergence behavor of the centers of dsks Ẑ produced by the ncluson methods (8) and (9) dstngushng two cases: () the exact nverson () s appled; () the centered nverson (2) s appled. Ths behavor can be smply examned consderng the resultng dsks obtaned by the nversons () and (2). Startng from () we fnd (assumng that r s suffcently small and c > r). { } {c; r} c = c 2 r ; r 2 c 2 r 2 { ( ) } = c + (r/ c )2 + (r/ c ) 2 (r/ c ) 4 r + ;. (4) c c 2 r 2 }{{} Bearng n mnd (2) and the mappng functon z /z, we note that the centered nverson preserves the property of centerng, whle the exact nverson does not. Ths means that the centers of dsks produced by the ncluson methods (8) and (9) concde wth the teratve methods (2) and (3), respectvely, when the centered nverson s appled. On the other hand, applyng the exact nverson, we observe by comparng () and (4) that the centers of dsks obtaned by the methods (8) and (9) are removed (for underlned part) and wll not concde wth (2) and (3). Therefore, the convergence of centers wll be spoled when the exact nverson s employed. Consequently, takng nto account the estmaton (0) and (), the ncluson methods (8) and (9) show faster convergence when the centered nverson s appled. At frst sght ths s a paradox snce the centered nverson always produces larger dsks than the centered nverson (see (2)). However, we have shown that the convergence speed of nterval methods depends not only of the dsk sze but also of the convergence behavor of centers of dsks. Accordng to the presented analyss t follows that the better convergence of centers of resultng ncluson dsks provdes the faster convergence. Hence, the followng natural queston could be posed: can the mprovement of convergence of centers accelerate the convergence speed of nterval methods? The answer s yes, whch was demonstrated for the frst tme n [3] where the followng ncluson method of Gargantn-Henrc s type wth Newton s correctons N(z j) = P(z j)/p (z j) was stated: Ẑ = z INV 2 (/N(z ) j)) n INV (z Z j + N(z ( I n). (5) j= j
7 Relable Computng 5, We note that the centers md (Z j N(z j)) = z j N(z j) behave as the approxmatons obtaned by Newton s method, { that} eventually provdes the acceleraton of convergence of the sequences of rad rad Z (m). The followng statement was proved n [3]: Theorem If ntal ncluson dsks Z (0),..., Z(0) n are reasonable small, then the R-order of convergence O R(5) of the nterval method (5) s gven by O R(5) { (3 + 7)/2 = f INV = (), 4 f INV = () Ic. In essence, the ncrease of the convergence rate s the result of the accelerated convergence of the centers of the dsks Ẑ calculated by (5). { In partcular, } when the centered nverson s appled n (5), then the sequences md Ẑ(m) behave as the sequences of approxmatons defned by the fourth-order Nouren s method [9] ẑ = z N(z ) n j j= z z j + N(z j) ( I n). In general, t s desrable to accelerate the convergence of centers of dsks appearng n teratve nterval formulas. The applcaton of the centered nverson moves the center of the mproved dsk Ẑ very close to the zero ζ. Further mprovement of the convergence rate of the nterval methods (9) and (5) can be acheved by applyng more rapd method nstead of Newton s method. The followng teratve method for solvng algebrac equaton P(z) = 0, proposed by Ostrowsk [0], s convenent n the acceleraton of convergence of nterval methods: P(z N(z)) P(z) ẑ = z N(z) 2P(z N(z)) P(z) = z g(z), P(z) N(z) = P (z). (6) The order of convergence of the Ostrowsk method (6) s four. The term g(z) n (6) s called Ostrowsk s correcton. Let us note that the teratve method (6) can be also appled to arbtrary (real or complex) functon. In a smlar way as n the constructon of the nterval method (5), we can derve the Gargantn-Henrc method wth Ostrowsk s correctons g(z j) n crcular complex arthmetc: j)) n Ẑ = z INV 2 /u(z ) INV (z Z j + g(z ( I n). (7) j= j Applyng crcular arthmetc operatons, t can be proved that the choce INV = INV 2 = () Ic n (7) produces the dsks Ẑ whose centers behave as the approxmatons obtaned by the smultaneous method ẑ = z N(z ) n j= j z z j + g(z j) ( I n). (8)
8 88 Ivan Petkovć, Computatonal Aspects of Dsk Inversons The convergence order of the method (8) s sx so that we can expect very fast convergence of the nterval method (7) snce Ostrowsk s approxmaton z j g(z j) s very close to the exact zero ζ j. More precsely, we can state the followng result: Theorem 2 Let (Z (0),..., Z(0) n ) := (Z,..., Z n) be an array of dsjont ntal dsks contanng the zeros ζ,..., ζ n of P. If the mdponts of ntal dsks are close enough to the zeros of P, then the R-order of convergence of the teratve method (7) s gven by { (3 + 7)/2 = f INV = (), O R(7) 6 f INV = () Ic. We note that the R-order of the nterval method (3) s not ncreased when INV = () although the correctons of the method of hgher order s appled. Numercal examples confrms ths fact, see, e.g., Table 2. Detaled theoretcal explanaton of ths phenomenon s gven n [8] and [2]. 4 Numercal examples The presented analyss wll be llustrated by the followng examples. Example We have appled Weerstrass-lke method (8) wth INV = () and INV = () Ic for the ncluson of zeros of the polynomal P(z) = z 7 + z 5 0z 4 z 3 z + 0. We have started wth the ntal dsks Z (0) = {z (0) ;0.3} that contan the exact zeros 2, ±, ±, ±2. The maxmal rad of the obtaned dsks are gven n Table, where A( q) means A 0 q. Methods max r () max r (2) max r (3) max r (4) max r (5) max r (6) (8) wth () ( 2) 3.2( 5) 6.8( ).7( 22) (8) wth () Ic ( 3).7( 6) 8.84( 5) 3.77( 3) Table : Weerstrass-lke method (8) wth exact and centered nverson Example 2 We have appled two versons of the Gargantn-Henrc method (9) for the ncluson of zeros of the followng polynomal of the 25th degree wth ntal dsks wth the same rad r (0) = 0.3, P(z) = (z 4)(z 4 )(z 4 8)(z 2 8z + 7)(z 2 6z + 3)(z 2 4z + 5)(z 2 2z + 5) (z 2 4z + 3)(z 2 + 2z + 5)(z 2 + 4z + 5)(z 2 + 4z + 3). The maxmal rad are dsplayed n Table 2. Methods max r () max r (2) max r (3) max r (4) (9) wth () 9.27( 2) 6.96( 4).99( 2).42( 39) (9) wth () Ic.70( ).08( 4).8( 5) 8.99( 50) Table 2: Gargantn-Henrc method (9) wth exact and centered nverson
9 Relable Computng 5, Example 3 Apart from the Gargantn-Henrc method (9), we have also appled the accelerated methods (5) and (7) wth INV = INV 2 = () and INV = INV 2 = () Ic for ncluson of the zeros of the polynomal P(z) = z 9 + 3z 8 3z 7 9z 6 + 3x 5 + 9z z z 2 00z 300. We have started wth the ntal dsks Z (0) = {z (0) ;0.3} that contan the zeros 3, ±, ±2, ±2 ±. The maxmal rad of dsks are gven n Table 3. Methods max r () max r (2) max r (3) max r (4) (9) wth () 6.20( 2) 8.3( 5) 4.45( 5).47( 46) (9) wth () Ic.0( ) 5.73( 5) 6.2( 6).52( 50) (5) wth () 6.20( 2) 5.65( 5).2( 7) 5.05( 62) (5) wth () Ic.0( ) 4.57( 5) 2.6( 9) 3.0( 76) (7) wth () 6.0( 2).78( 5) 2.0( 8) 3.90( 64) (7) wth () Ic.0( ) 6.40( 6).70( 3) 6.9( 89) Table 3: The methods (9), (5) and (7) wth exact and centered nverson From Tables, 2 and 3 we observe that the centered nverson (2) gves smaller dsks n the case of all tested methods (8), (9), (5) and (7). The mprovement s especally stressed when the methods (5) and (7) wth correcton s appled. Slghtly larger dsks n the frst teraton, when the centered nverson s appled, s the results of relatvely slow convergence of the centers at the begnnng of teratve process. References [] O. Aberth, Iteraton methods for fndng all zeros of a polynomal smultaneously, Math. Comp., vol. 27, pp , 973. [2] G. Alefeld, J. Herzberger, Introducton to Interval Computaton, Academc Press, New York, 983. [3] C. Carstensen, M. S. Petkovć, An mprovement of Gargantn s smultaneous ncluson method for polynomal roots by Schroeder s correcton, Appl. Numer. Math., vol. 25, pp , 993. [4] L.W. Ehrlch, A modfed Newton method for polynomals, Comm. ACM, vol. 0, pp , 967. [5] I. Gargantn, P. Henrc, Crcular arthmetc and the determnaton of polynomal zeros, Numer. Math., vol. 8, pp , 972. [6] I. O. Kerner, En Gesamtschrttverfahren zur Berechnung der Nullstellen von Polynomen, Numer. Math., vol. 8, pp , 966. [7] J. M. McNamee, Numercal Methods for Roots of Polynomals, Part I, Elsever, Amsterdam, [8] D. Mloševć, Iteratve methods for the smultaneous nlcuson of polynomal zeros, Ph. D. Theses, Unversty of Nš, Nš, [9] A. W. M. Nouren, An mprovement on two teraton methods for smultanneously determnaton of the zeros of a polynomal, Internat. J. Comput. Math., vol. 6, pp , 977.
10 90 Ivan Petkovć, Computatonal Aspects of Dsk Inversons [0] A. M. Ostrowsk, Solutons of Equatons and System of Equatons, Academc Press, New York, 966. [] Lj. D. Petkovć, M. S. Petkovć, The representaton of complex crcular functons usng Taylor seres, Z. Angew. Math. Mech., vol. 6, pp , 98. [2] M. S. Petkovć, Iteratve Methods for Smultaneous Incluson of Polynomal Zeros, Sprnger-Verlag, Berln-Hedelberg-New York, 989. [3] H. Ratschek, J. Rokne, Computer Methods for the Range of Functons, Ells Horwood, Chchester, 984. [4] B. Sendov, A. Andreev, N. Kyurkchev, Numercal Soluton of Polynomal Equatons (Handbook of Numercal Analyss), Vol. VIII, Elsever Scence, New York, 994. [5] X. Wang, S. Zheng, The quas-newton method n parallel crcular teraton, J. Comput. Math., vol. 4, pp , 984.
Applied Mathematics Letters
Appled Mathematcs Letters 28 (204) 60 65 Contents lsts avalable at ScenceDrect Appled Mathematcs Letters journal homepage: wwwelsevercom/locate/aml On an effcent smultaneous method for fndng polynomal
More informationOn the Interval Zoro Symmetric Single-step Procedure for Simultaneous Finding of Polynomial Zeros
Appled Mathematcal Scences, Vol. 5, 2011, no. 75, 3693-3706 On the Interval Zoro Symmetrc Sngle-step Procedure for Smultaneous Fndng of Polynomal Zeros S. F. M. Rusl, M. Mons, M. A. Hassan and W. J. Leong
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationFixed point method and its improvement for the system of Volterra-Fredholm integral equations of the second kind
MATEMATIKA, 217, Volume 33, Number 2, 191 26 c Penerbt UTM Press. All rghts reserved Fxed pont method and ts mprovement for the system of Volterra-Fredholm ntegral equatons of the second knd 1 Talaat I.
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationGlobal Sensitivity. Tuesday 20 th February, 2018
Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values
More informationFundamental loop-current method using virtual voltage sources technique for special cases
Fundamental loop-current method usng vrtual voltage sources technque for specal cases George E. Chatzaraks, 1 Marna D. Tortorel 1 and Anastasos D. Tzolas 1 Electrcal and Electroncs Engneerng Departments,
More informationSolving Nonlinear Differential Equations by a Neural Network Method
Solvng Nonlnear Dfferental Equatons by a Neural Network Method Luce P. Aarts and Peter Van der Veer Delft Unversty of Technology, Faculty of Cvlengneerng and Geoscences, Secton of Cvlengneerng Informatcs,
More informationThe Jacobsthal and Jacobsthal-Lucas Numbers via Square Roots of Matrices
Internatonal Mathematcal Forum, Vol 11, 2016, no 11, 513-520 HIKARI Ltd, wwwm-hkarcom http://dxdoorg/1012988/mf20166442 The Jacobsthal and Jacobsthal-Lucas Numbers va Square Roots of Matrces Saadet Arslan
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationApproximate Smallest Enclosing Balls
Chapter 5 Approxmate Smallest Enclosng Balls 5. Boundng Volumes A boundng volume for a set S R d s a superset of S wth a smple shape, for example a box, a ball, or an ellpsod. Fgure 5.: Boundng boxes Q(P
More information8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before
.1 Arc Length hat s the length of a curve? How can we approxmate t? e could do t followng the pattern we ve used before Use a sequence of ncreasngly short segments to approxmate the curve: As the segments
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationA New Refinement of Jacobi Method for Solution of Linear System Equations AX=b
Int J Contemp Math Scences, Vol 3, 28, no 17, 819-827 A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan,
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationHigh resolution entropy stable scheme for shallow water equations
Internatonal Symposum on Computers & Informatcs (ISCI 05) Hgh resoluton entropy stable scheme for shallow water equatons Xaohan Cheng,a, Yufeng Ne,b, Department of Appled Mathematcs, Northwestern Polytechncal
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationChapter 3 Differentiation and Integration
MEE07 Computer Modelng Technques n Engneerng Chapter Derentaton and Integraton Reerence: An Introducton to Numercal Computatons, nd edton, S. yakowtz and F. zdarovsky, Mawell/Macmllan, 990. Derentaton
More informationTHE SUMMATION NOTATION Ʃ
Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the
More informationThe Expectation-Maximization Algorithm
The Expectaton-Maxmaton Algorthm Charles Elan elan@cs.ucsd.edu November 16, 2007 Ths chapter explans the EM algorthm at multple levels of generalty. Secton 1 gves the standard hgh-level verson of the algorthm.
More informationOn the Multicriteria Integer Network Flow Problem
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 5, No 2 Sofa 2005 On the Multcrtera Integer Network Flow Problem Vassl Vasslev, Marana Nkolova, Maryana Vassleva Insttute of
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationP A = (P P + P )A = P (I P T (P P ))A = P (A P T (P P )A) Hence if we let E = P T (P P A), We have that
Backward Error Analyss for House holder Reectors We want to show that multplcaton by householder reectors s backward stable. In partcular we wsh to show fl(p A) = P (A) = P (A + E where P = I 2vv T s the
More informationInexact Newton Methods for Inverse Eigenvalue Problems
Inexact Newton Methods for Inverse Egenvalue Problems Zheng-jan Ba Abstract In ths paper, we survey some of the latest development n usng nexact Newton-lke methods for solvng nverse egenvalue problems.
More informationThe Second Eigenvalue of Planar Graphs
Spectral Graph Theory Lecture 20 The Second Egenvalue of Planar Graphs Danel A. Spelman November 11, 2015 Dsclamer These notes are not necessarly an accurate representaton of what happened n class. The
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More informationExample: (13320, 22140) =? Solution #1: The divisors of are 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 41,
The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no confuson
More informationSpectral Graph Theory and its Applications September 16, Lecture 5
Spectral Graph Theory and ts Applcatons September 16, 2004 Lecturer: Danel A. Spelman Lecture 5 5.1 Introducton In ths lecture, we wll prove the followng theorem: Theorem 5.1.1. Let G be a planar graph
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
More informationHongyi Miao, College of Science, Nanjing Forestry University, Nanjing ,China. (Received 20 June 2013, accepted 11 March 2014) I)ϕ (k)
ISSN 1749-3889 (prnt), 1749-3897 (onlne) Internatonal Journal of Nonlnear Scence Vol.17(2014) No.2,pp.188-192 Modfed Block Jacob-Davdson Method for Solvng Large Sparse Egenproblems Hongy Mao, College of
More informationGrover s Algorithm + Quantum Zeno Effect + Vaidman
Grover s Algorthm + Quantum Zeno Effect + Vadman CS 294-2 Bomb 10/12/04 Fall 2004 Lecture 11 Grover s algorthm Recall that Grover s algorthm for searchng over a space of sze wors as follows: consder the
More informationSpeeding up Computation of Scalar Multiplication in Elliptic Curve Cryptosystem
H.K. Pathak et. al. / (IJCSE) Internatonal Journal on Computer Scence and Engneerng Speedng up Computaton of Scalar Multplcaton n Ellptc Curve Cryptosystem H. K. Pathak Manju Sangh S.o.S n Computer scence
More informationSolution of Linear System of Equations and Matrix Inversion Gauss Seidel Iteration Method
Soluton of Lnear System of Equatons and Matr Inverson Gauss Sedel Iteraton Method It s another well-known teratve method for solvng a system of lnear equatons of the form a + a22 + + ann = b a2 + a222
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationModelli Clamfim Equazione del Calore Lezione ottobre 2014
CLAMFIM Bologna Modell 1 @ Clamfm Equazone del Calore Lezone 17 15 ottobre 2014 professor Danele Rtell danele.rtell@unbo.t 1/24? Convoluton The convoluton of two functons g(t) and f(t) s the functon (g
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationChapter - 2. Distribution System Power Flow Analysis
Chapter - 2 Dstrbuton System Power Flow Analyss CHAPTER - 2 Radal Dstrbuton System Load Flow 2.1 Introducton Load flow s an mportant tool [66] for analyzng electrcal power system network performance. Load
More informationErratum: A Generalized Path Integral Control Approach to Reinforcement Learning
Journal of Machne Learnng Research 00-9 Submtted /0; Publshed 7/ Erratum: A Generalzed Path Integral Control Approach to Renforcement Learnng Evangelos ATheodorou Jonas Buchl Stefan Schaal Department of
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationThe Quadratic Trigonometric Bézier Curve with Single Shape Parameter
J. Basc. Appl. Sc. Res., (3541-546, 01 01, TextRoad Publcaton ISSN 090-4304 Journal of Basc and Appled Scentfc Research www.textroad.com The Quadratc Trgonometrc Bézer Curve wth Sngle Shape Parameter Uzma
More informationUncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
More informationThe internal structure of natural numbers and one method for the definition of large prime numbers
The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract
More informationRemarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence
Remarks on the Propertes of a Quas-Fbonacc-lke Polynomal Sequence Brce Merwne LIU Brooklyn Ilan Wenschelbaum Wesleyan Unversty Abstract Consder the Quas-Fbonacc-lke Polynomal Sequence gven by F 0 = 1,
More informationModule 14: THE INTEGRAL Exploring Calculus
Module 14: THE INTEGRAL Explorng Calculus Part I Approxmatons and the Defnte Integral It was known n the 1600s before the calculus was developed that the area of an rregularly shaped regon could be approxmated
More informationTHE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION
THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationA MODIFIED METHOD FOR SOLVING SYSTEM OF NONLINEAR EQUATIONS
Journal of Mathematcs and Statstcs 9 (1): 4-8, 1 ISSN 1549-644 1 Scence Publcatons do:1.844/jmssp.1.4.8 Publshed Onlne 9 (1) 1 (http://www.thescpub.com/jmss.toc) A MODIFIED METHOD FOR SOLVING SYSTEM OF
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationNorms, Condition Numbers, Eigenvalues and Eigenvectors
Norms, Condton Numbers, Egenvalues and Egenvectors 1 Norms A norm s a measure of the sze of a matrx or a vector For vectors the common norms are: N a 2 = ( x 2 1/2 the Eucldean Norm (1a b 1 = =1 N x (1b
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationACTM State Calculus Competition Saturday April 30, 2011
ACTM State Calculus Competton Saturday Aprl 30, 2011 ACTM State Calculus Competton Sprng 2011 Page 1 Instructons: For questons 1 through 25, mark the best answer choce on the answer sheet provde Afterward
More informationfor Linear Systems With Strictly Diagonally Dominant Matrix
MATHEMATICS OF COMPUTATION, VOLUME 35, NUMBER 152 OCTOBER 1980, PAGES 1269-1273 On an Accelerated Overrelaxaton Iteratve Method for Lnear Systems Wth Strctly Dagonally Domnant Matrx By M. Madalena Martns*
More informationarxiv: v1 [math.co] 12 Sep 2014
arxv:1409.3707v1 [math.co] 12 Sep 2014 On the bnomal sums of Horadam sequence Nazmye Ylmaz and Necat Taskara Department of Mathematcs, Scence Faculty, Selcuk Unversty, 42075, Campus, Konya, Turkey March
More informationMEASUREMENT OF MOMENT OF INERTIA
1. measurement MESUREMENT OF MOMENT OF INERTI The am of ths measurement s to determne the moment of nerta of the rotor of an electrc motor. 1. General relatons Rotatng moton and moment of nerta Let us
More informationLecture 3. Ax x i a i. i i
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationParametric fractional imputation for missing data analysis. Jae Kwang Kim Survey Working Group Seminar March 29, 2010
Parametrc fractonal mputaton for mssng data analyss Jae Kwang Km Survey Workng Group Semnar March 29, 2010 1 Outlne Introducton Proposed method Fractonal mputaton Approxmaton Varance estmaton Multple mputaton
More informationTHERE ARE INFINITELY MANY FIBONACCI COMPOSITES WITH PRIME SUBSCRIPTS
Research and Communcatons n Mathematcs and Mathematcal Scences Vol 10, Issue 2, 2018, Pages 123-140 ISSN 2319-6939 Publshed Onlne on November 19, 2018 2018 Jyot Academc Press http://jyotacademcpressorg
More informationSome Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)
Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998
More informationSolving Fractional Nonlinear Fredholm Integro-differential Equations via Hybrid of Rationalized Haar Functions
ISSN 746-7659 England UK Journal of Informaton and Computng Scence Vol. 9 No. 3 4 pp. 69-8 Solvng Fractonal Nonlnear Fredholm Integro-dfferental Equatons va Hybrd of Ratonalzed Haar Functons Yadollah Ordokhan
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationGeneral viscosity iterative method for a sequence of quasi-nonexpansive mappings
Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 9 (2016), 5672 5682 Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence,
More informationarxiv: v1 [math.co] 1 Mar 2014
Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationJournal of Universal Computer Science, vol. 1, no. 7 (1995), submitted: 15/12/94, accepted: 26/6/95, appeared: 28/7/95 Springer Pub. Co.
Journal of Unversal Computer Scence, vol. 1, no. 7 (1995), 469-483 submtted: 15/12/94, accepted: 26/6/95, appeared: 28/7/95 Sprnger Pub. Co. Round-o error propagaton n the soluton of the heat equaton by
More informationComplete subgraphs in multipartite graphs
Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G
More informationSnce h( q^; q) = hq ~ and h( p^ ; p) = hp, one can wrte ~ h hq hp = hq ~hp ~ (7) the uncertanty relaton for an arbtrary state. The states that mnmze t
8.5: Many-body phenomena n condensed matter and atomc physcs Last moded: September, 003 Lecture. Squeezed States In ths lecture we shall contnue the dscusson of coherent states, focusng on ther propertes
More informationWorkshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department
More informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
More informationSingular Value Decomposition: Theory and Applications
Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real
More informationU.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016
U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and
More informationMATH 829: Introduction to Data Mining and Analysis The EM algorithm (part 2)
1/16 MATH 829: Introducton to Data Mnng and Analyss The EM algorthm (part 2) Domnque Gullot Departments of Mathematcal Scences Unversty of Delaware Aprl 20, 2016 Recall 2/16 We are gven ndependent observatons
More informationSpin-rotation coupling of the angularly accelerated rigid body
Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationOn a direct solver for linear least squares problems
ISSN 2066-6594 Ann. Acad. Rom. Sc. Ser. Math. Appl. Vol. 8, No. 2/2016 On a drect solver for lnear least squares problems Constantn Popa Abstract The Null Space (NS) algorthm s a drect solver for lnear
More informationDETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM
Ganj, Z. Z., et al.: Determnaton of Temperature Dstrbuton for S111 DETERMINATION OF TEMPERATURE DISTRIBUTION FOR ANNULAR FINS WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY BY HPM by Davood Domr GANJI
More informationTHE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
More information1 Rabi oscillations. Physical Chemistry V Solution II 8 March 2013
Physcal Chemstry V Soluton II 8 March 013 1 Rab oscllatons a The key to ths part of the exercse s correctly substtutng c = b e ωt. You wll need the followng equatons: b = c e ωt 1 db dc = dt dt ωc e ωt.
More informationSuppose that there s a measured wndow of data fff k () ; :::; ff k g of a sze w, measured dscretely wth varable dscretzaton step. It s convenent to pl
RECURSIVE SPLINE INTERPOLATION METHOD FOR REAL TIME ENGINE CONTROL APPLICATIONS A. Stotsky Volvo Car Corporaton Engne Desgn and Development Dept. 97542, HA1N, SE- 405 31 Gothenburg Sweden. Emal: astotsky@volvocars.com
More informationMin Cut, Fast Cut, Polynomial Identities
Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a mult-graph.
More informationA Bayes Algorithm for the Multitask Pattern Recognition Problem Direct Approach
A Bayes Algorthm for the Multtask Pattern Recognton Problem Drect Approach Edward Puchala Wroclaw Unversty of Technology, Char of Systems and Computer etworks, Wybrzeze Wyspanskego 7, 50-370 Wroclaw, Poland
More information