viii CONTENTS 20 Symbolic Calculations 461 20.1 Introduction....... 461 20.2 Strategy.········· 462 20.3 Examples.······· 465 20.4 Summary and Open Problems... 472 21 The Tau-Method 473 473 21.2 T-Approximation for a Rational Function.................... 474 21.3 Differential Equations 476 21.4 Canonical Polynomials.·.···,························ 476 2l.5 Nomenclature................。..· 478 22 Domain Decomposition Methods 479 22.1 ntroduction..············· 479 22.2 Notation 480 22.3 Connecting the Subdomains:Patching 480 22.4 Weak Coupling of Elemental Solutions 481 22.5 Variational Principles........... 484 22.6 Choice of Basis&Grid.···.····· 485 22.7 Patching versus Variational Formalism...... 4 486 22.8 Matrix Inversion......·.·.·.·.··············· 487 22.9 The Influence Matrix Method 488 22.10Two-Dimensional Mappings Sectorial Elements 491 2211 Prospectus.···························· 492 23 Books and Reviews 494 AA Bestiary of Basis Functions 495 A.1 Trigonometric Basis Functions:Fourier Series............ 495 A.2 Chebyshev Polynomials:Tn() 497 A.3 Chebyshev Polynomials of the Second Kind:Un(z) 499 A.4 Legendre Polynomials:Pn(e).···.·:·:··········· 500 A.5 Gegenbauer Polynomials.................... 4 502 A.6 Hermite Polynomials:H(x) 505 A.7 Rational Chebyshev Functions:TBn(y) 507 A.8 Laguerre Polynomials:In(x)................. 508 A.9 Rational Chebyshev Functions:TLn(y) 509 A.10 Graphs of Convergence Domains in the Complex Plane............ 511 B Direct Matrix-Solvers 514 B.1 Matrix Factorizations..··..·.······················ 514 B2 Banded Matrix........…·.,.······················ 518 B.3 Matrix-of-Matrices Theorem.........·...··············· 520 B.4 Block-Banded Elimination:the "Lindzen-Kuo"Algorithm ·。··…···· 520 B.5 Block and“Bordered"Matrices.·, 522 B.6 Cyclic Banded Matrices (Periodic Boundary Conditions) 524 B.7 Parting shots.。。························· 524viii CONTENTS 20 Symbolic Calculations 461 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 20.2 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462 20.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 20.4 Summary and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 21 The Tau-Method 473 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 21.2 τ -Approximation for a Rational Function . . . . . . . . . . . . . . . . . . . . 474 21.3 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 21.4 Canonical Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 21.5 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478 22 Domain Decomposition Methods 479 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 22.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480 22.3 Connecting the Subdomains: Patching . . . . . . . . . . . . . . . . . . . . . . 480 22.4 Weak Coupling of Elemental Solutions . . . . . . . . . . . . . . . . . . . . . . 481 22.5 Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 22.6 Choice of Basis & Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 22.7 Patching versus Variational Formalism . . . . . . . . . . . . . . . . . . . . . . 486 22.8 Matrix Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 22.9 The Influence Matrix Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 22.10Two-Dimensional Mappings & Sectorial Elements . . . . . . . . . . . . . . . 491 22.11Prospectus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492 23 Books and Reviews 494 A A Bestiary of Basis Functions 495 A.1 Trigonometric Basis Functions: Fourier Series . . . . . . . . . . . . . . . . . . 495 A.2 Chebyshev Polynomials: Tn(x) . . . . . . . . . . . . . . . . . . . . . . . . . . 497 A.3 Chebyshev Polynomials of the Second Kind: Un(x) . . . . . . . . . . . . . . 499 A.4 Legendre Polynomials: Pn(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 A.5 Gegenbauer Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 A.6 Hermite Polynomials: Hn(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 A.7 Rational Chebyshev Functions: T Bn(y) . . . . . . . . . . . . . . . . . . . . . 507 A.8 Laguerre Polynomials: Ln(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . 508 A.9 Rational Chebyshev Functions: T Ln(y) . . . . . . . . . . . . . . . . . . . . . 509 A.10 Graphs of Convergence Domains in the Complex Plane . . . . . . . . . . . . 511 B Direct Matrix-Solvers 514 B.1 Matrix Factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 B.2 Banded Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 B.3 Matrix-of-Matrices Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 B.4 Block-Banded Elimination: the “Lindzen-Kuo” Algorithm . . . . . . . . . . 520 B.5 Block and “Bordered” Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 522 B.6 Cyclic Banded Matrices (Periodic Boundary Conditions) . . . . . . . . . . . 524 B.7 Parting shots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524