Numerical polynomial algebra:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
SIAM
2004
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 472 S. graph. Darst. |
| ISBN: | 0898715571 |
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MARC
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| 100 | 1 | |a Stetter, Hans J. |d 1930- |e Verfasser |0 (DE-588)108696219 |4 aut | |
| 245 | 1 | 0 | |a Numerical polynomial algebra |c Hans J. Stetter |
| 264 | 1 | |a Philadelphia |b SIAM |c 2004 | |
| 300 | |a XV, 472 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 7 | |a Algebra |2 gtt | |
| 650 | 7 | |a Numerieke methoden |2 gtt | |
| 650 | 7 | |a Polynomen |2 gtt | |
| 650 | 4 | |a Numerical analysis | |
| 650 | 4 | |a Polynomials | |
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Record in the Search Index
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| adam_text | Contents
Preface
xi
Acknowledgments
xv
I Polynomials and Numerical Analysis
1
1
Polynomials
3
1.1
Linear Spaces of Polynomials
........................ 4
1.2
Polynomials as Functions
.......................... 7
1.3
Rings and Ideals of Polynomials
...................... 12
1.3.1
Polynomial Rings
. ...................... 12
1.3.2
Polynomial Ideals
....................... 13
1.4
Polynomials and
Affine
Varieties
...................... 16
1.5
Polynomials in Scientific Computing
.................... 20
1.5.1
Polynomials in Scientific and Industrial Applications
..... 21
2
Representations of Polynomial Ideals
25
2.1
Ideal Bases
.................................. 25
2.2
Quotient Rings of Polynomial Ideals
.................... 29
2.2.1
Linear Spaces of Residue Classes and Their Multiplicative
Structure
............................ 29
2.2.2
Commuting Families of Matrices
............... 36
2.3
Dual Spaces of Polynomial Ideals
...................... 39
2.3.1
Dual Vector Spaces
...................... 39
2.3.2
Dual Spaces of Quotient Rings
................ 42
2.4
The Central Theorem of Polynomial Systems Solving
........... 46
2.4.1
Basis Transformations in Tt and T>
. . ............ 46
2.4.2
A Preliminary Version
..................... 49
2.4.3
The General Case
....................... 50
2.5
Normal Sets and Border Bases
....................... 54
2.5.1
Monomial Bases of a Quotient Ring
.............. 54
2.5.2
Border Bases of Polynomial Ideals
.............. 58
2.5.3
Groebner Bases
........................ 60
2.5.4
Polynomial Interpolation
................... 61
Polynomials with Coefficients of Limited Accuracy
67
3.1
Data of Limited Accuracy
.......................... 67
3.1.1
Empirical Data
......................... 67
3.1.2
Empirical Polynomials
..................... 71
3.1.3
Valid Approximate Results
................... 73
3.2
Estimation of the Result
Indétermination
.................. 76
3.2.1
Well-Posed and Ill-Posed Problems
.............. 76
3.2.2
Condition of an Empirical Algebraic Problem
......... 79
3.2.3
Linearized Estimation of the Result
Indétermination
..... 84
3.3
Backward Error of Approximate Results
.................. 86
3.3.1
Determination of the Backward Error
............. 88
3.3.2
Transformations of an Empirical Polynomial
......... 92
3.4
Refinement of Approximate Results
..................... 95
Approximate Numerical Computation
101
4.1
Solution Algorithms for Numerical Algebraic Problems
.......... 101
4.2
Numerical Stability of Computational Algorithms
............. 105
4.2.1
Generation and Propagation of Computational Errors
..... 105
4.2.2
Numerical Stability
...................... 108
4.2.3
Causes for Numerical Instability
............... 110
4.3
Floating-Point Arithmetic
.......................... 113
4.3.1 Floating-Point
Numbers
.................... 114
4.3.2
Arithmetic with
Floating-Point
Numbers
........... 115
4.3.3 Floating-Point
Errors
..................... 118
4.3.4
Local Use of Higher Precision
................. 120
4.4
Use of Intervals
. .............................. 123
4.4.1
Interval Arithmetic
....................... 123
4.4.2
Validation Within Intervals
................... 125
4.4.3
Interval Mathematics and Scientific Computing
........ 128
II Univariate Polynomial Problems
133
5
Univariate Polynomials
135
5.1
Intrinsic Polynomials
............................ 135
5.1.1
Some Analytic Properties
................... 135
5.1.2
Spaces of Polynomials
.................... . 137
5.1.3
Some Algebraic Properties
................... 140
5.1.4
The Multiplicative Structure
.................. 143
5.1.5
Numerical Determination of Zeros of Intrinsic Polynomials
. 146
5.2
Zeros of Empirical Univariate Polynomials
................. 148
5.2.1
Backward Error of Polynomial Zeros
............. 149
5.2.2
Pseudozero
Domains for Univariate Polynomials
....... 152
5.2.3
Zeros with Large Modulus
.......... ......... 154
5.3
Polynomial Division
............................. 157
5.3.1
Sensitivity Analysis of Polynomial Division
......... 157
5.3.2
Division of Empirical Polynomials
.............. 160
5.4
Polynomial Interpolation
.......................... 163
5.4.1
Classical Representations of Interpolation Polynomials
. . . . 163
5.4.2
Sensitivity Analysis of Univariate Polynomial Interpolation
. 166
5.4.3
Interpolation Polynomials for Empirical Data
......... 168
Various Tasks with Empirical Univariate Polynomials
173
6.1
Algebraic Predicates
............................. 173
6.1.1
Algebraic Predicates for Empirical Data
............ 173
6.1.2
Real Polynomials with Real Zeros
............... 178
6.1.3
Stable Polynomials
...................... 180
6.2
Divisors of Empirical Polynomials
..................... 183
6.2.1
Divisors and Zeros
....................... 183
6.2.2
Sylvester Matrices
....................... 185
6.2.3
Refinement of an Approximate Factorization
......... 188
6.2.4
Multiples of Empirical Polynomials
.............. 192
6.3
Multiple Zeros and Zero Clusters
...................... 194
6.3.1
Intuitive Approach
....................... 194
6.3.2
Zero Clusters of Empirical Polynomials
............ 196
6.3.3
Cluster Polynomials
...................... 198
6.3.4
Multiple Zeros of Empirical Polynomials
........... 202
6.3.5
Zero Clusters about Infinity
.................. 204
6.4
Greatest Common Divisors
......................... 206
6.4.1
Intrinsic Polynomial Systems in One Variable
........ 206
6.4.2
Empirical Polynomial Systems in One Variable
........ 211
6.4.3
Algorithmic Determination of Approximate Common
Divisors
............................ 214
6.4.4
Refinement of Approximate Common Zeros and Divisors
. .218
6.4.5
Example
............................ 219
III Multivariate Polynomial Problems
225
7
One Multivariate Polynomial
229
7.1
Analytic Aspects
...............................229
7.1.1
Intuitional Difficulties with Real and Complex Data
.....229
7.1.2
Taylor Approximations
....................230
7.1.3
Nearest Points on a Manifold
.................233
7.2
Empirical Multivariate Polynomials
.....................237
7.2.1
Valid Results for Empirical Polynomials
...........237
7.2.2
Pseudozero
Sets of Empirical Multivariate Polynomials
. . . 240
7.2.3
Condition of Zero Manifolds
.................242
7.3
Singular Points on Algebraic Manifolds
...................246
7.3.1
Singular Zeros of Empirical Polynomials
...........246
7.3.2 Determination
of
Singular
Zeros
............... 249
7.3.3
Manifold Structure at
a
Singular Point ............ 251
7.4
Numerical Factorization of a Multivariate Polynomial
........... 254
7.4.1
Analysis of the Problem
.................... 254
7.4.2
An Algorithmic Approach
................... 258
7.4.3
Algorithmic Details
...................... 261
7.4.4
Condition of a Multivariate Factorization
........... 266
Zero-Dimensional Systems of Multivariate Polynomials
273
8.1
Quotient Rings and Border Bases of
О
-Dimensional Ideals
.........274
8.1.1
The Quotient Ring of a Specified Dual Space
.........274
8.1.2
The Ideal Generated by a Normal Set Ring
..........278
8.1.3
Quasi-Univariate Normal Sets
.................282
8.2
Normal Set Representations of
О
-Dimensional Ideals
............286
8.2.1
Computation of Normal Forms and Border Basis Expansions
286
8.2.2
The Syzygies of a Border Basis
................290
8.2.3
Admissible Data for a Normal Set Representation
......295
8.3
Regular Systems of Polynomials
......................300
8.3.1
Complete Intersections
.....................300
8.3.2
Continuity of Polynomial Zeros
................304
8.3.3
Expansion by a Complete Intersection System
........306
8.3.4
Number of Zeros of a Complete Intersection System
.....309
8.4
Groebner Bases
...............................314
8.4.1
Term Order and Order-Based Reduction
...........314
8.4.2
Groebner Bases
........................317
8.4.3
Direct Characterization of Reduced Groebner Bases
.....321
8.4.4
Discontinuous Dependence of Groebner Bases on
Ρ
.....323
8.5
Multiple Zeros of Intrinsic Polynomial Systems
..............328
8.5.1
Dual Space of a Multiple Zero
.................328
8.5.2
Normal Set Representation for a Multiple Zero
........334
8.5.3
From Multiplication Matrices to Dual Space
.........335
Systems of Empirical Multivariate Polynomials
343
9.1
Regular Systems of Empirical Polynomials
................. 344
9.1.1
Backward Error of Polynomial Zeros
............. 344
9.1.2
Pseudozero
Domains for Multivariate Empirical Systems
. . 347
9.1.3
Feasible Normal Sets for Regular Empirical Systems
..... 349
9.1.4
Sets of Ideals of System Neighborhoods
........... 350
9.2
Approximate Representations of Polynomial Ideals
............. 355
9.2.1
Approximate Normal Set Representations
........... 355
9.2.2
Refinement of an Approximate Normal Set Representation
. .358
9.2.3
Refinement Towards the Exact Representation
........ 363
9.3
Multiple Zeros and Zero Clusters
...................... 366
9.3.1
Approximate Dual Space for a Zero Cluster
.......... 367
9.3.2
Further Refinement
...................... 371
9.3.3
Cluster Ideals
......................... 373
9.3.4
Asymptotic Analysis of Zero Clusters
.............376
9.4
Singular Systems of Empirical Polynomials
................381
9.4.1
Singular Systems of Linear Polynomials
...........382
9.4.2
Singular Polynomial Systems; Simple d-Points
........386
9.4.3
A
Nontrivial
Example
.....................391
9.4.4
Multiple d-Points
.......................394
9.5
Singular Polynomial Systems with Diverging Zeros
............398
9.5.1
Inconsistent Linear Systems
..................398
9.5.2
BKK-Deficient Polynomial Systems
.............400
9.6
Multivariate Interpolation
..........................404
9.6.1
Principal Approach
.......................404
9.6.2
Special Situations
.......................406
9.6.3
Smoothing Interpolation
....................407
10
Numerical Basis Computation
411
10.1
Algorithmic Computation of Groebner Bases
................411
10.1.1
Principles of Groebner Basis Algorithms
...........411
10.1.2
Avoiding Ill-Conditioned Representations
...........414
10.1.3
Groebner Basis Computation for
Floating-Point
Systems
. . . 416
10.2
Algorithmic Computation of Normal Set Representations
.........417
10.2.1
An Intuitive Approach
.....................418
10.2.2
Determination of a Normal Set for a Complete Intersection
. 421
10.2.3
Basis Computation with Specified Normal Set
........423
10.3
Numerical Aspects of Basis Computation
..................429
10.3.1
Two Fundamental Difficulties
.................429
10.3.2
Pivoting
............................431
10.3.3
Basis Computation with Empirical Data
...........433
10.3.4
A Numerical Example
.....................434
IV Positive-Dimensional Polynomial Systems
443
11
Matrix
Eigenproblems
for Positive-Dimensional Systems
447
11.1
Multiplicative Structure of oo-Dimensional Quotient Rings
........447
11.1.1
Quotient Rings and Normal Sets of Positive-Dimensional
Ideals
..............................447
11.1.2
Finite Sections of Infinite Multiplication Matrices
......449
11.1.3
Extension of the Central Theorem
...............451
11.2
Singular Matrix
Eigenproblems.......................452
11.2.1
The Solution Space of a Singular Matrix
Eigenproblem . . . 452
11.2.2
Algorithmic Determination of Parametric
Eigensolutions . . . 454
11.2.3
Algorithmic Determination of Regular
Eigensolutions . . . .456
11.3
Zero Sets from Finite Multiplication Matrices
...............457
11.3.1
One-Dimensional Zero Sets
..................457
11.3.2
Multi-Dimensional Zero Sets
.................460
11.3.3
Direct Computation of Two-Dimensional
Eigensolutions . . . 460
11.4
A Quasi-O-Dimensional Approach
......................462
11.4.1
Quotient Rings with Parameters
................463
11.4.2
A Modified Approach
.....................464
Index
467
|
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| id | DE-604.BV019287427 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:43:36Z |
| institution | BVB |
| isbn | 0898715571 |
| language | English |
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| publisher | SIAM |
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| spelling | Stetter, Hans J. 1930- Verfasser (DE-588)108696219 aut Numerical polynomial algebra Hans J. Stetter Philadelphia SIAM 2004 XV, 472 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algebra gtt Numerieke methoden gtt Polynomen gtt Numerical analysis Polynomials Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Polynomalgebra (DE-588)4297306-5 gnd rswk-swf Polynomalgebra (DE-588)4297306-5 s Numerische Mathematik (DE-588)4042805-9 s DE-604 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012039378&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Stetter, Hans J. 1930- Numerical polynomial algebra Algebra gtt Numerieke methoden gtt Polynomen gtt Numerical analysis Polynomials Numerische Mathematik (DE-588)4042805-9 gnd Polynomalgebra (DE-588)4297306-5 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4297306-5 |
| title | Numerical polynomial algebra |
| title_auth | Numerical polynomial algebra |
| title_exact_search | Numerical polynomial algebra |
| title_full | Numerical polynomial algebra Hans J. Stetter |
| title_fullStr | Numerical polynomial algebra Hans J. Stetter |
| title_full_unstemmed | Numerical polynomial algebra Hans J. Stetter |
| title_short | Numerical polynomial algebra |
| title_sort | numerical polynomial algebra |
| topic | Algebra gtt Numerieke methoden gtt Polynomen gtt Numerical analysis Polynomials Numerische Mathematik (DE-588)4042805-9 gnd Polynomalgebra (DE-588)4297306-5 gnd |
| topic_facet | Algebra Numerieke methoden Polynomen Numerical analysis Polynomials Numerische Mathematik Polynomalgebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012039378&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT stetterhansj numericalpolynomialalgebra |