Meromorphic Functions and Linear Algebra

By: Olava Nevanlinna

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Hardcover | 1 January 2003

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"Some results appear here for the first time, while the rest are extended from recent papers of the author. The book is intended for researchers in mathematics in general and especially for those working in numerical linear algebra. Much of the book is understandable if the reader has a good background in linear algebra and a first course in complex analysis."--BOOK JACKET. Title Summary field provided by Blackwell North America, Inc. All Rights Reserved

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You Can Find This Book In

Non-Fiction Mathematics Algebra Calculus & Mathematical Analysis Complex Analysis Numerical Analysis

Preface	p. ix
Prologue	p. 1
Tapping away in an evening at Djursholm	p. 1
What does an epsilon weigh?	p. 1
Red wine at the Stock Exchange Club	p. 2
Ice-cream in Madison	p. 2
Exact equality	p. 3
The deficiency of values	p. 5
Zurich	p. 5
Beautiful to look at, but...	p. 6
The unbearable ease of using norms	p. 6
Centenary Colloquium in Joensuu	p. 7
Two basic tasks, stability first	p. 7
And then accelerating the iteration	p. 9
Factoring the resolvent	p. 10
In the Hermann Weyl lecture hall	p. 11
A quiet life in Warsaw	p. 12
Finally, in Kirkkonummi	p. 12
	p. 15
Resolvent	p. 15
Cauchy-integral	p. 20
	p. 23
Entire functions	p. 23
Taylor coefficients	p. 24
Meromorphic functions	p. 25
The first main theorem	p. 30
Cartan's identity	p. 31
Order and type for meromorphic functions	p. 32
Boutroux-Cartan lemma	p. 33
Bound along a circle	p. 34
Representation theorems	p. 36
	p. 37
Analytic vector valued functions	p. 37
Subharmonic functions	p. 37
Meromorphic vector valued functions	p. 38
Rational functions	p. 40
When is the inverse also meromorphic	p. 42
A simple estimate for matrices	p. 44
	p. 47
A product form for matrices	p. 47
Singular value decomposition	p. 50
Basic inequalities for singular values and eigenvalues	p. 51
The total logarithmic size of a matrix	p. 54
Some basic properties of the total logarithmic size	p. 56
Direct sum, Kronecker product and Hadamard product	p. 60
	p. 63
The total logarithmic size is subharmonic	p. 63
Behavior near poles	p. 65
Introducing T[subscript 1] for matrix valued functions	p. 68
Basic identity for inversion	p. 69
Extension to trace class	p. 70
How to work outside the trace class	p. 71
	p. 73
Perturbation results	p. 73
Special results for resolvents	p. 77
Powers and their resolvents	p. 79
Bounded characteristics	p. 83
What if small perturbation means small in norm	p. 85
	p. 87
Combining a scalar function with an operator	p. 87
Representing F as G/g	p. 93
Representations for the resolvent	p. 94
Decay of spectral polynomials	p. 96
Robust bounds for Krylov solvers	p. 98
A bound for spectral projectors	p. 100
	p. 103
Approximate polynomial degree of an analytic function	p. 103
Some properties of the approximate polynomial degree	p. 106
Approximate rational degree of a meromorphic function	p. 108
Spijker's lemma	p. 112
Power bounded operators and bounds for the Laurent coefficients	p. 114
	p. 117
Growth of associated scalar functions	p. 117
Locally algebraic and locally almost algebraic operators	p. 121
	p. 125
Exceptional values	p. 125
Simple asymptotics for resolvents of matrices	p. 126
Eigenvalues and exceptional values	p. 128
Deficient operators	p. 131
Epilogue	p. 133
Lecturing and typing in Toronto	p. 133
Fishing and finishing in Karjalohja	p. 133
Bibliography	p. 135
Table of Contents provided by Ingram. All Rights Reserved.

Meromorphic Functions and Linear Algebra

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