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Linear Search 137 the variable metric method, since Davidon (and later Fletcher and PowelI) publicized the idea of updating an estimate of the inverse Hessian matrix. Finally, conjugate directions were defined in comparison to orthogonal directions. The Davidon-Fletcher-Powell variable metric search directions are A-conjugate, but the symmetric, inverse-Hessian matrix estimate requires a substantial amount of memory to store. The Fletcher-Reeves method requires memory for only 3N vectors and works nearly as well, except for final convergence. The Fletcher-Reeves search algorithm works well for engineering accuracy in the memory space provided in desktop computers. The mechanics of a linear search by Fletcher are discussed next, because of the important assumption that each linear search is stopped at the exact minimum in that direction. 5.3. Linear Search Nearly all gradient search methods require linear searches, i.e., line searches to minima in a sequence of directions. The single, real variable in such searches has been defined as IX in (5.39). A value for IX may be calculated according to (5.41) when the surface is known to be a quadratic function in the general form of (5.6). However important a quadratic model may be in formulating search strategies, the usual surface is not at alI quadratic except in the immediate vicinity of local minima, so that linear searches must find the minimum as a function of IX by a comprehensive procedure. Figure 5.20 illustrates a typical linear search profile. There are three stages in the linear search for the optimum value a: (I) estimate the order of magnitude of a; (2) establish bounds on the vicinity of the minimum; (3) interpolate the value of IX within those bounds. First, the slope in the search direction (directional derivative) will be defined, and an order of magnitude of IX will be determined based on the expected quadratic behavior of IX near the minimum. The classical cubic interpolation using two function values and two derivatives will be explained, F(~) '\ "\. F'(o:);(gi)TSi FIOI . 1 1 I 1 1 • d a Figure S.20. Profile of linear search function F(a).
138 Gradient Optimization and terminating criteria will be discussed. Finally, the linear search procedure programmed by Fletcher and used in the following Fletcher-Reeves optimizer will be described and illustrated by an example. 5.3.1. Slope in tire Linear Search Direction. The components of the gradient vector at any point p, namely g(p), indicate the rate of change of function value in each coordinate direction. During linear searches, Figure 5.20 indicates the need to have the function's rate of change in some arbitrary direction s. This may be obtained by recalling the Taylor series expansion about point p, as given in (5.30). In this case, the
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Wile,.· Interactenee ..A. .. 1 18
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Circuit Design U sing Personal Comp
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To my late parents, Tommy and Brown
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viii Preface computer. Excessive th
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Contents l. Introduction 1 2. Some
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Contetlts xiii 4.3.4. Transmission
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~----- ---- Contents xv 6.6. Pseudo
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-------------------------- Contents
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-- ------- 2 Introduction viewpoint
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4 Introduction Chapters Four and Fi
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6 Introduction the selectivity effe
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8 Some Fundamenurl Numerical Method
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10 Some Fundamental Numerical Metho
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--------------------- Romberg Integ
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and it is convenient to rearrange (
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Romberg Integranon 17 Truncation Er
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---~-- - - -- ----- - -------------
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Polynomial Minimox Approximation of
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Polynomial Minimax Approximation of
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Rational Polynomial LSE Approximati
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---~~--_._--- ------ ID(w)£(w)1 2
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Rational Polynomial LSE Approximati
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Problems 31 2.6. If a,Z+a2 w = -a'-
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sense by the sum of first-kind Cheb
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Complex Zeros of Complex Polynomial
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Complex Zeros ofComplex Polynomials
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which are equal to the polynomial C
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Complex Zeros of Complex Polynomial
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Table 3.3. Complex Zeros of Complex
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Polynomials From Complex Zeros and
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Polynomials From Complex Zeros and
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Polynomial Addition and Subtraction
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Continued Fraction Expansion 51 Not
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Case 1 (row ~t (al N" 3 -1 '" 25 +
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---- ------------------------- Cont
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----~------- Input ImpedafJce $ytrt
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Input Impedance Synthesis From Its
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------- - ------------- ---- Input
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Long Division and Partial Fraction
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Problems 6S magnitude. The program
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- Problems 67 3.12. Given the resis
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.Chapter Four Ladder Network Analys
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- -- - -------- Recun;"" ludder Met
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Recursive Ladder Method 73 This alw
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--------------- Example a: 3,325,20
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Recursive Ladder Method 77 This las
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- - - -------------- Embedded T>vo-
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- - - - - --- - - - ---- Unifonn Tr
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--- - .._----- Unifonn T/'Qnsmissio
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--------------- Uniform Transmissio
Page 102 and 103: ------- --- -----------_. - ------
Page 104 and 105: Table 4.4. 6.d L1 6.dc, 6.d L2 6.d
Page 106 and 107: ------- - _. - - Input and Transfer
Page 108 and 109: Input and Transfer Network Response
Page 110 and 111: Input and Transfer Network Response
Page 112 and 113: ----------------------------- Time
Page 114 and 115: ----- - - h(-rJ h(2.6.Tl o I II ---
Page 116 and 117: -------- Sensitivities 101 the corr
Page 118 and 119: formula: Sensitivities 103 dZ""A,Z
Page 120 and 121: Sensitivities 105 obeying at least
Page 122 and 123: and its derivative with respect to
Page 124 and 125: Problems 109 branch immittance; the
Page 126 and 127: Problems 111 Solve by using (4.40)
Page 128 and 129: Chapter Five Gradient Optimization
Page 130 and 131: Gradient Optimization 115 Figure 5.
Page 132 and 133: -- ------------------ Quadratic Fon
Page 134 and 135: Quadratic Fonns and Ellipsoids 119
Page 136 and 137: Quadratic Forms and EUipsoids 121 F
Page 138 and 139: Quadratic Fonm and Ellipsoids 123 x
Page 140 and 141: Quadratic Forms and Ellipsoids 115
Page 142 and 143: Conjugate Gradient Search 127 choic
Page 144 and 145: -_.. ---------- Conjugate Gradient
Page 146 and 147: Conjugate Gradient Search 131 x, Fi
Page 148 and 149: -- -~--- Conjugate Gradient Search
Page 150 and 151: Conjugate Gradient Search 135 Hxl =
Page 154 and 155: Then, the minimum function value in
Page 156 and 157: - ------- Linear Search 141 subtrac
Page 158 and 159: The F/etcher- Reeves Optimizer 143
Page 160 and 161: The netcher- Reeves Optimizer 145 c
Page 162 and 163: Table 5.4. Typical Output for the R
Page 164 and 165: The Fletcher- Reeves Optimizer 149
Page 166 and 167: Network Objective Functions 151 and
Page 168 and 169: -------- - - Network Objecti"" Func
Page 170 and 171: Network Objective Functions 155 Tab
Page 172 and 173: -- ------------ Constraints 157 tra
Page 174 and 175: - - - - - -------------- Constraint
Page 176 and 177: Table 5.7. Objective Subroutine lor
Page 178 and 179: -70. Constrainis 163 315-325). For
Page 180 and 181: Constraints 165 In fact, one applic
Page 182 and 183: --------~ - - ---------------------
Page 184 and 185: (b) Problems 169 Find the diagonal
Page 186 and 187: Impedance Matching 171 I, z, ---+__
Page 188 and 189: Narrow·Band L, T, and Pi Networks
Page 190 and 191: - - - --------------- + '" R, . " N
Page 192 and 193: NalTO...Band L, T, and Pi Networks
Page 194 and 195: Narrow-Bond L, T, and Pi Networks ]
Page 196 and 197: Narrow-BaruJ L, T, aruJ Pi Networks
Page 198 and 199: Loss/ess Unifonn Transmission Lines
Page 200 and 201: ------- - -------------------------
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----------- and (6.34) can be writt
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Fano's Broodband-Matching Limitatio
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Fano's Broadband-Matching Limitatio
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----------------- Fano's Broadband-
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Fono's Broadband·Matching Limitati
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Green's recursive element formula i
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f 9, Figure 6.24. 92
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Bandpass Network Transformations 20
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Bandpass Network Trans/onnotions 20
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50 II 2.6548 16.594 Bandpass Networ
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BaruJpass Network TransJof1lUltions
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Pseudobandpass Matching Networks 21
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I InlPJ =In Pseudobandpass Matching
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Pseudobandpass Matching Networks 21
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----- -- Carlin's Broodband-Matchin
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---- -------------------- expressio
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Car/in's Broadband.Matching Method
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Carlin's Broadband-Matching Methad
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Prohle"" 227 Third, an objective fu
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Problems 229 6.16. Program Equation
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Bilinear Tronsfonnatl'ons 231 I, ~-
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Bi/inear Transjormations 233 (w" w"
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Bilinear Transfonnations 235 accomp
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Bilinear Transfonnations 237 circle
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Impedance Mapping 239 1 3 al~ ~a3 [
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Impedance Mapping 241 r-- Z, s" I,
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---------- Impedan£e Mapping 243 T
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Impedance Mapping 245 in Figure 7.7
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Two-Port Impedance and Power Models
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---------------- This may be put in
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----~-------~--- - - --------------
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-------- -- - -------------- and th
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Two-Port Impedance and Power Models
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Billlteral Scattering Stability and
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Bilateral Scattering Stability and
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--- -------- Bilateral Scattering S
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Unilateral Scattering Gain 267 the
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Unilateral Scattering Gain 269 The
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Prohle"" 271 _merit is u=0.08; by (
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Chapter Eight Direct-Coupled Filter
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- -- ~-------~- -------------- Dire
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Prototype Network 277 and Wo is the
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Prototype Network 279 element value
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Prototype Network 281 8./.4. Protot
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Designing with Lande Inverters 283
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Designing with L and C Inverters 28
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Designilrg with Lande Inverters Sup
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Designing with L and C Inverters 28
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General Inverters, Resonators, and
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Genera/Inverters, Resonators, and E
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conductance of the Kth resonator is
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Table 8.2. General Inverters, Reson
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General Inveners, Resonators, and E
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Four Importunt Pussband Shapes 305
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Four Important Passband Shapes 307
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FOIl' Important Possband Shopes 309
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Fou, Important Passband Shapes 317
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Four lmporlunt Pussband Shupes 319
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Comments on a Design Procedure 321
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-----~------------ A Complete Desig
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A Complete Design Example 329 DB3 =
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A Complete Design Example 331 R,,(l
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------- Problems 333 For Z=O+jX, fi
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------------- Chapter Nine Other Di
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, I II r L;.. I I . C, ~ ~ Wo ell :
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• e/2 e/2 Figure 9.3. A typical e
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Another trigonometric identity can
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o---iS~-----, I '--------;::==-----
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Introduction to Cauer Elliptic Filt
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Introduction to Cauer Elliptic Filt
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40 20 140 12 11 130 10 120 110 9 10
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00 Introduction to Cauer Elliptic F
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Doubly Termi/lllted Elliptic Filter
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Doubly TermillQted Elliptic Filters
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Doubly Terminated Elliptic Filters
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Doubly Terminated Elliptic FilteN 3
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Doubly TerrnilUlted Elliptic Filten
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Some Lumped.Element Transformations
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-lin '" 1 +jOn : c, 3L, L, I ) r So
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1 1CT (T Some Lumped.Element Tronsj
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Load Effects on Passive Networks 36
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Invulnerable Filters 377 9.6. Invul
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Invulnerable Filters 379 20 t lmin
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Invulnerable Filters 381 40 t L mil
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Problems. 383' corresponding number
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Problems 385 9.11. A passive networ
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Program AS-I. Swain's Snrface See E
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------- -------- - _. _. - Program
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----- _. - ------------- - Program
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- - - _._------------------ &61 TAN
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Program A6-4. Norton Transformation
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861 862 863 86' 865 866 B6? 868 869
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Program A7·2. Three-Port to Two·P
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15B .LBU Cue & 5to 3x3 Elements J5J
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e61 STOB 32 degrees 115 RCL9 a3 deg
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Program A7-4. Maximally Efficient G
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169 CHS Register Assignments: 17e e
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84e 841 842 843 844 845 846 847 848
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Program A8-2. Doubly Termiuated Min
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----- - - - _. - _.--------- - -- _
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-------_._--- ------- Program A9-1.
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Appendix B PET BASIC Programs Progr
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Program B2-2A. Gauss-Jordan Solutio
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- - -------------------------------
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Program B2-4B. Polynomial Minimax A
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Program B2-5. Levy's Matrix Coeffic
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----------- _. - - - Flowchart for
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Program B3-2. Polynomials From Comp
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--------- - - Program 83-4. Polynom
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Program 83-7. Partial Fraction Expa
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Flowchart for Ladder Analysis Progr
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Program B5-1. The Fletcher-Reeves O
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Program B5-2. L-Seetion Optimizatio
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Flowchart for Matching Program B6-1
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Program B6-3. Levy Matching to Resi
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---------- -- Program 86-5. Hilbert
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--~-- - ---------- Program B9-1. El
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---------- - -_.-------------------
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3260 TO~(TO+W)/(l-TO'N) 3270 B(l,=B
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Appendix D Linear Search Flowchart*
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Appendix E Defined Complex Constan
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Appendix F _ Doubly Terminated Mini
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Appendix G _ Direct-Coupled Filter
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461 (G.22) (G.23) (G.24) A,= (G.25)
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G.4.2. Resonator Asymptote Slopes D
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Appendix G 465 except for overcoupl
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----------_._-_.- ---- - - -_.. - A
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Appendix H _ Zverev's Tables ofEqui
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------ 6(0) 6(b) L _ (L,C,- L,C,)'
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L, L, c. " 12(a) C\=W C 2 =X L\=Y L
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Appendix H 475 Simplified Notations
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References 477 Box, M. J., D. Davie
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--------- Johnson, L. W" References
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References 481 Van Valkenburg, M. E
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_ 484 Author Index Noble, B.. 120.
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486 Subject Index second kind, 32.
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488 Subject Index proper, 62 quadra
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490 Subject Index equivalent: bandp
Page 506 and 507:
1 1 _ 492 Subject Index I I Reflect
Page 508 and 509:
494 Subject Index length. electrica
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