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Stetter H. J. — Numerical polynomial algebra

Stetter H. J. — Numerical polynomial algebra

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Название: Numerical polynomial algebra

Автор: Stetter H. J.

Аннотация:

In many important areas of scientific computing, polynomials in one or more variables are employed in the mathematical modeling of real-life phenomena; yet most of classical computer algebra assumes exact rational data. This book is the first comprehensive treatment of the emerging area of numerical polynomial algebra, an area that falls between classical numerical analysis and classical computer algebra but, surprisingly, has received little attention so far.

The author introduces a conceptual framework that permits the meaningful solution of various algebraic problems with multivariate polynomial equations whose coefficients have some indeterminacy; for this purpose, he combines approaches of both numerical linear algebra and commutative algebra. For the application scientist, Numerical Polynomial Algebra provides both a survey of polynomial problems in scientific computing that may be solved numerically and a guide to their numerical treatment. In addition, the book provides both introductory sections and novel extensions of numerical analysis and computer algebra, making it accessible to the reader with expertise in either one of these areas.

Graduate students or academic and industrial research scientists in numerical analysis and computer algebra can use Numerical Polynomial Algebra as a textbook or reference book. The book is clearly written and standard numerical linear algebra notation is used consistently throughout. Principles and their application are explained through numerical examples and exercises avoiding excessive technical detail. Numerous open-ended problems invite further investigation and research.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 472

Добавлена в каталог: 31.10.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$({\bar p}, e)$       71
$({\bar \alpha}, \epsilon)$       68
$({\mathbb F}(D), \Box)$       116
$S({p}_{1},{p}_{2})$       186
$\Box$       115
$\cal Z$       73
$\cal {N}$ -index      287
$\cal {N}$ -leading term      287
$\cal {N}[\cal I]$       56
$\cal {N}_{\prec}$       317
$\cal{A}$       73
$\delta({\tilde z})$       87
$\langle ... \rangle$       14
$\mathbb C$       7
$\mathbb T$       173
$\partial{\cal N}$       290
$\prec$ -leading term      315
${A}_{\omega}$       34
${A}_{{b}_{\mu}}$       33
${bb}_{k}$       59
${BW}_{\cal N}$       291
${c}^{T}$       40
${c}^{T}(b)$       40
${c}^{T}_{0}(b)$       47
${e}_{\mu{\omega}_{1}{\omega}_{2}}$       290
${M}_{0}$       47
${NF}_{\cal I}[p]$       58
${N}_{\delta}({\bar P}, E)}$       350
${N}_{\delta}({\bar \alpha}, \epsilon)}$       69
${Z}^{s}(m)$       275
${z}^{\dag}$       394
${Z}_{\delta}({\bar a},e)$       73
${[p]}_{\cal I}$       29
${\cal B}_{\cal N}[{\cal I}]$       60
${\cal C}[\langle P \rangle]$       319
${\cal D[{N}_{\delta}({\bar P}, E)]}$       351
${\cal D}[\cal I]$       42 45
${\cal G}[\langle P \rangle]$       207
${\cal I}[\cal D]$       44
${\cal M}_{\tilde z}$       86
${\cal P}^{s} / {\cal I}$       31
${\cal P}^{s}$       4
${\cal P}^{s}(m)$       275
${\cal P}^{s}_{d}$       4
${\cal P}_{n}$       135
${\cal R}[\cal I]$       29 31 34
${\cal R}[{N}_{\delta}({\bar P}, E)}]$       351
${\cal T}^{s}$       4
${\cal T}^{s}(m)$       56
${\cal T}^{s}_{d}$       4
${\hat b}, {\check a}$       449
${\hat {\cal N}}, {\check {\cal N}}$       449
${\mathbb F}(D)$       114
${\partial}_{j}$       8
${\|..\|}^{*}$       7o
Aberth method      148
Admissible-data manifold      295
Affine variety      17
Algebraic predicate      173
Algebraic predicate, backward error      175
Algebraic predicate, critical manifold      175
Algebraic predicate, critical, noncritical      174
Algebraic predicate, extension to empirical polynomial      175
Algebraic predicate, truth domain      173
Algorithmic scheme      102
Anti-differentiation operators      330
Asymptotic cluster ideal asymptotic basis      377
b(x)      35
B6zout number      309
Backward analysis      21
Backward error      87
Backward error, common pseudozero      212
Backward error, complex zero      151
Backward error, determination      88
Backward error, interpolation      169
Backward error, membership in $\langle {N}_{\delta}({\bar P}, E) \rangle$       353
Backward error, polynomial division      160
Backward error, polynomial zero      89
Backward error, pseudodivisor      184
Backward error, simultaneous zeros      151 346
Bases, conjugate      40
Bases, dual space      40
Bases, minimal      26
Bases, of ideal      25
Bases, of quotient ring      31
Bases, transformations      46
Basis theorem (Hilbert)      27
BKK(P)      310
BKK-bound      309
BKK-deficient      400
Border basis      60 280
Border basis, comprehensive      389
Border basis, expansion in terms of      289
Border basis, order-compatible      317
Border basis, syzygies      290
Border Groebner basis      318
Border set      59
Border web      291
Boundary set      290
Cancellation, leading digits      111
Central theorem, general version      52
Central theorem, positive-dimensional ideals      452
Central theorem, preliminary version      49
Central theorem, univariate      144
Chebyshev polynomials      138
Closed, monomial set      55
Closed, space of linear functional      44
Closed, space of linear functional, criterion      330
Cluster ideal      374
Cluster ideal, asymptotic      377
Cluster polynomial      198
Cluster polynomial, information about zeros      201
Commuting matrix family      35 36
Companion matrix      144
Complete intersection      18 300
Complete intersection, expansion in terms of      306
Complete intersection, number of zeros      309
Complete intersection, system, ideal      300
Comprehensive border basis      389
Comprehensive Groebner basis      386
Computational error, generated      106
Computational error, propagated      107
Computational steps      106
Condition      79
Condition number      79
Condition number, of matrix      80
Condition, manifold      242
Condition, multiple zero      204
Condition, multivariate factorization      266
Condition, polynomial division      157
Condition, polynomial interpolation      166
Condition, zero of univariate polynomial      84
Conjugate bases      40
Corner basis      319
Corner set      58
D-point, multiple      394
D-point, simple      387
Data $\to$ result mapping      73
Data $\to$ result mapping, implicit      83
Data space $\cal A$       73
Data, empirical, intrinsic      68
Degeneracy, degree of      325
Differential functional      43
Differential functional, internal, external      395
Differential operators      7
Differential operators, order      8
Divided differences      164
Dual norm      70
Dual space, of ideal      42
Dual space, of vector space      40

Durand — Kerner method      147
Empirical data, specified value, tolerance      68
Empirical data, valid instance      68
Empirical manifolds      241
Empirical multivariate polynomials      237
Empirical multivariate polynomials, backward error      238
Empirical multivariate polynomials, empirical support      237
Empirical multivariate polynomials, pseudofactors      255
Empirical multivariate polynomials, pseudozero sets      240
Empirical multivariate polynomials, singular zero      246
Empirical multivariate polynomials, valid instance      238
Empirical multivariate polynomials, valid results      238
Empirical multivariate systems      343
Empirical multivariate systems, backward error      344
Empirical multivariate systems, cluster ideal      374
Empirical multivariate systems, feasible normal set      349
Empirical multivariate systems, ideal membership      351
Empirical multivariate systems, m-cluster domain      347
Empirical multivariate systems, multiple zeros, zero clusters      366
Empirical multivariate systems, pseudozero domains      347
Empirical multivariate systems, regular      344
Empirical multivariate systems, singular      381
Empirical polynomial      71
Empirical polynomial, transformation      92
Empirical system of polynomials      72
Empirical univariate polynomials      148
Empirical univariate polynomials, common zeros, divisors      211
Empirical univariate polynomials, divisors      183
Empirical univariate polynomials, multiple zeros      202
Empirical univariate polynomials, multiples      192
Empirical univariate polynomials, zero cluster      196
Equivalent-data set, manifold      86
Error propagation      107
Euclidean algorithm      207
Evaluation basis      43
Expansion in terms of complete intersection      306
Extended Groebner basis      415
f      73
Floating-point, arithmetic      116
Floating-point, macheps      118
Floating-point, mantissa, exponent      114
Floating-point, numbers      114
Floating-point, operations      115
Floating-point, round-off error      118
Floating-point, rounding      115
Frechet derivative      8 80
Frobenius matrix      144
Gcd(P)      207
Gradient      9
Greatest common divisor      207
Greatest common divisor, determination      214
Greatest common divisor, refinement      218
Groebner basis      60 318
Groebner basis, algorithmic computation      411
Groebner basis, discontinuous dependence      323
Groebner basis, extended      326
Groebner basis, for floating-point system      416
Groebner basis, reduced      320
hessian      10
Hull sets      59 287
Ideal      14
Ideal, basis      14 25
Ideal, border basis      60
Ideal, d-dimensional      19
Ideal, Groebner basis      60
Ideal, normal set representation      60 286
Ideal, zero, zero set of      15
Ill-conditioned problem      79
Ill-posed problem      76
Indetermination of results      84
Infinite multiplication matrix finite section      449
Infinite normal set      448
Infinite normal set, bounded parts, niche monomials      448
Infinite normal set, finite border basis      451
Infinite normal set, inner, outer normal set      450
Integral domain      13
Interpolation, polynomial      61
Interpolation, polynomial, multivariate      405
Interval arithmetic      123
Interval validation      125
Intervals in scientific computing      128
Inverse function theorem      10
Iterative improvement      95
Jacobian      9
Kharitonov's Theorem      181
Lagrange basis      30
Lagrange basis, determination      48
Lagrange basis, of dual space      141
Leading term      315
Leibniz rule      8 330
Lifting      258
linear combination      14
Linear functional closed space of      44
Linear program      90
Local linearization      96
Manifold      19
Manifold, condition      242
Manifold, irreducible component      255
Manifold, nearest point on      233
Manifold, singular point      246
Manifold, structure at singular point      251
Mantissa      114
Maximal independent subset      463
Mixed volume      312
Monomial      4
Monomial, degree      4
Monomial, exponent      4
Multiple d-point      394
Multiple d-point, dual space      394
Multiple d-point, multiplicity      394
Multiple zero, algorithmic approach      330
Multiple zero, dual space      328
Multiple zero, multiplicity structure      329
Multiple zero, normal set representation      334
Multiple zero, of multivariate polynomial system      328
Multiplication matrices      34
Multiplication matrices, (non)trivial rows      34
Multiplication matrices, constraints      279
Multiplication matrices, family of      36
Multiplication matrices, overdetermination      282
Multiplication matrices, univariate      143
Multiplication matrix computation in Maple      416
Multivariate factorization      254
Multivariate factorization, algorithmic approach      258
Multivariate factorization, condition      266
Multivariate factorization, germ      261
Multivariate factorization, target values, $\beta$ -test      262
Multivariate interpolation      405
Multivariate interpolation, smoothing      407
Multivariate polynomial systems, complete intersection      300
Multivariate polynomial systems, regular systems      273 300
Multivariate polynomial systems, zeros      304
Multivariate polynomials      227
Multivariate polynomials, expansion in terms of border basis      289
Multivariate polynomials, expansions      232
Multivariate polynomials, factorization      254
Multivariate polynomials, singular zero      246
Multivariate polynomials, Taylor approximations      232
Multivariate polynomials, Taylor expansion      231
Near-singular polynomial system      387
Neighbor positive, negative      56
Neighborhoods, family of      69
Newton poly tope      311
Newton's method      96
Nonderogatory      37
Nonderogatory, matrix family      37
Normal form      58
Normal form, computation      286
Normal set      56
Normal set representation      60 286

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