Condition: the geometry of numerical algorithms
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2013
|
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften
349 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XXXI, 554 S. graph. Darst. |
| ISBN: | 9783642388958 |
Internformat
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| 245 | 1 | 0 | |a Condition |b the geometry of numerical algorithms |c Peter Bürgisser ; Felipe Cucker |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2013 | |
| 300 | |a XXXI, 554 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Cucker, Felipe |d 1958- |e Verfasser |0 (DE-588)133832783 |4 aut | |
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Datensatz im Suchindex
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IMAGE 1
CONTENTS
PART I CONDITION IN LINEAR ALGEBRA ( ADAGIO) 1 NORMWISE CONDITION OF
LINEAR EQUATION SOLVING 3
1.1 VECTOR AND MATRIX NORMS 4
1.2 TURING'S CONDITION NUMBER 6
1.3 CONDITION AND DISTANCE TO ILL-POSEDNESS 10
1.4 AN ALTERNATIVE CHARACTERIZATION OF CONDITION 11
1.5 THE SINGULAR VALUE DECOMPOSITION 12
1.6 LEAST SQUARES AND THE MOORE-PENROSE INVERSE 17
2 PROBABILISTIC ANALYSIS 21
2.1 A CRASH COURSE ON INTEGRATION 22
2.2 A CRASH COURSE ON PROBABILITY: I 27
2.2.1 BASIC FACTS 28
2.2.2 GAUSSIAN DISTRIBUTIONS 33
2.2.3 THE X 2 DISTRIBUTION 35
2.2.4 UNIFORM DISTRIBUTIONS ON SPHERES 38
2.2.5 EXPECTATIONS OF NONNEGATIVE RANDOM VARIABLES 39
2.2.6 CAPS AND TUBES IN SPHERES 41
2.2.7 AVERAGE AND SMOOTHED ANALYSES 46
2.3 PROBABILISTIC ANALYSIS OF CW, ( A , X ) 48
2.4 PROBABILISTIC ANALYSIS OF K R S ( A ) 50
2.4.1 PRECONDITIONING 51
2.4.2 AVERAGE ANALYSIS 53
2.4.3 UNIFORM SMOOTHED ANALYSIS 55
2.5 ADDITIONAL CONSIDERATIONS 56
2.5.1 PROBABILISTIC ANALYSIS FOR OTHER NORMS 56
2.5.2 PROBABILISTIC ANALYSIS FOR GAUSSIAN DISTRIBUTIONS 57
3 ERROR ANALYSIS OF TRIANGULAR LINEAR SYSTEMS 59
3.1 RANDOM TRIANGULAR MATRICES ARE ILL-CONDITIONED 60
XI
HTTP://D-NB.INFO/103451539X
IMAGE 2
XII CONTENTS
3.2 BACKWARD ANALYSIS OF TRIANGULAR LINEAR SYSTEMS 64
3.3 COMPONENTWISE CONDITION OF RANDOM SPARSE MATRICES 65
3.3.1 COMPONENTWISE CONDITION NUMBERS 65
3.3.2 DETERMINANT COMPUTATION 67
3.3.3 MATRIX INVERSION 71
3.3.4 SOLVING LINEAR EQUATIONS 72
3.4 ERROR BOUNDS FOR TRIANGULAR LINEAR SYSTEMS 73
3.5 ADDITIONAL CONSIDERATIONS 73
3.5.1 ON NORMS AND MIXED CONDITION NUMBERS 73
3.5.2 ON THE UNDERLYING PROBABILITY MEASURE 74
4 PROBABILISTIC ANALYSIS OF RECTANGULAR MATRICES 77
4.1 A CRASH COURSE ON PROBABILITY: II 78
4.1.1' LARGE DEVIATIONS .79
4.1.2 RANDOM GAUSSIAN MATRICES 81
4.1.3 A BOUND ON THE EXPECTED SPECTRAL NORM 84
4.2 TAIL BOUNDS FOR K (A) 86
4.2.1 TAIL BOUNDS FOR || A'F || 87
4.2.2 PROOF OF THEOREM 4.16 91
4.3 EXPECTATIONS: PROOF OF THEOREM 4.2 92
4.4 COMPLEX MATRICES 93
5 CONDITION NUMBERS AND ITERATIVE ALGORITHMS 101
5.1 THE COST OF COMPUTING: A PRIMER IN COMPLEXITY 102
5.2 THE METHOD OF STEEPEST DESCENT 103
5.3 THE METHOD OF CONJUGATE GRADIENTS 107
5.4 CONJUGATE GRADIENT ON RANDOM DATA 116
INTERMEZZO I: CONDITION OF STRUCTURED DATA 119
PART II CONDITION IN LINEAR OPTIMIZATION (. ANDANTE)
6 A CONDITION NUMBER FOR POLYHEDRAL CONIC SYSTEMS 123
6.1 CONDITION AND CONTINUITY 123
6.2 BASIC FACTS ON CONVEXITY 125
6.2.1 CONVEX SETS 125
6.2.2 POLYHEDRA 128
6.3 THE POLYHEDRAL CONE FEASIBILITY PROBLEM 129
6.4 THE GCC CONDITION NUMBER AND DISTANCE TO ILL-POSEDNESS . 134 6.5
THE GCC CONDITION NUMBER AND SPHERICAL CAPS 136
6.6 THE GCC CONDITION NUMBER AND IMAGES OF BALLS 140
6.7 THE GCC CONDITION NUMBER AND WELL-CONDITIONED SOLUTIONS . . . 142
6.8 CONDITION OF SOLUTIONS AND CONDITION NUMBERS 143
6.9 THE PERCEPTRON ALGORITHM FOR FEASIBLE CONES 144
7 THE ELLIPSOID METHOD 1 47
7.1 A FEW FACTS ABOUT ELLIPSOIDS 147
7.2 THE ELLIPSOID METHOD 150
IMAGE 3
CONTENTS XIII
7.3 POLYHEDRAL CONIC SYSTEMS WITH INTEGER COEFFICIENTS 153
8 LINEAR PROGRAMS AND THEIR SOLUTION SETS 155
8.1 LINEAR PROGRAMS AND DUALITY 155
8.2 THE GEOMETRY OF SOLUTION SETS 160
8.3 THE COMBINATORICS OF SOLUTION SETS 162
8.4 ILL-POSEDNESS AND DEGENERACY 166
8.4.1 DEGENERACY 166
8.4.2 A BRIEF DISCUSSION ON ILL-POSEDNESS 168
9 INTERIOR-POINT METHODS 173
9.1 PRIMAL-DUAL INTERIOR-POINT METHODS: BASIC IDEAS 173
9.2 EXISTENCE AND UNIQUENESS OF THE CENTRAL PATH 177
9.3 ANALYSIS OF IPM FOR LINEAR PROGRAMMING 180
9.4 CONDITION-BASED ANALYSIS OF IPM FOR PCFP 184
9.4.1 REFORMULATION 184
9.4.2 ALGORITHMIC SOLUTION 186
9.4.3 ANALYSIS 188
9.5 FINITE PRECISION FOR DECISION AND COUNTING PROBLEMS 190
10 THE LINEAR PROGRAMMING FEASIBILITY PROBLEM 193
10.1 A CONDITION NUMBER FOR POLYHEDRAL FEASIBILITY 193
10.2 DECIDING FEASIBILITY OF PRIMAL-DUAL PAIRS 195
11 CONDITION AND LINEAR PROGRAMMING OPTIMIZATION 201
11.1 THE CONDITION NUMBER X { D ) 202
11.2 X ( D ) AND OPTIMAL SOLUTIONS 208
11.3 COMPUTING THE OPTIMAL BASIS 211
11.3.1 AN INTERIOR-POINT ALGORITHM 212
11.3.2 A REDUCTION TO POLYHEDRAL FEASIBILITY PROBLEMS 214
11.4 OPTIMIZERS AND OPTIMAL BASES: THE CONDITION VIEWPOINT 219 11.5
APPROXIMATING THE OPTIMAL VALUE 221
12 AVERAGE ANALYSIS OF THE RCC CONDITION NUMBER 223
12.1 PROOF OF THEOREM 12.1 225
12.1.1 THE GROUP 0" AND ITS ACTION 225
12.1.2 PROBABILITIES 229
13 PROBABILISTIC ANALYSES OF THE GCC CONDITION NUMBER 233
13.1 THE PROBABILITY OF PRIMAL AND DUAL FEASIBILITY 235
13.2 SPHERICAL CONVEXITY 238
13.3 A BOUND ON THE VOLUME OF TUBES 240
13.4 TWO ESSENTIAL REDUCTIONS 241
13.5 A CRASH COURSE ON PROBABILITY: III 245
13.6 AVERAGE ANALYSIS 248
13.7 SMOOTHED ANALYSIS : . 252
INTERMEZZO II: THE CONDITION OF THE CONDITION 255
IMAGE 4
XJV CONTENTS
PART III CONDITION IN POLYNOMIAL EQUATION SOLVING {ALLEGRO CON BRIO)
14 A GEOMETRIC FRAMEWORK FOR CONDITION NUMBERS 261
14.1 CONDITION NUMBERS REVISITED 261
14.1.1 COMPLEX ZEROS OF UNIVARIATE POLYNOMIALS 263
14.1.2 A GEOMETRIC FRAMEWORK 265
14.1.3 LINEAR EQUATION SOLVING 267
14.2 COMPLEX PROJECTIVE SPACE 269
14.2.1 PROJECTIVE SPACE AS A COMPLEX MANIFOLD 269
14.2.2 DISTANCES IN PROJECTIVE SPACE 271
14.3 CONDITION MEASURES ON MANIFOLDS 275
14.3.1 EIGENVALUES AND EIGENVECTORS 276
14.3.2 COMPUTATION OF THE KERNEL 280
15 HOMOTOPY CONTINUATION AND NEWTON'S METHOD 283
15.1 HOMOTOPY METHODS 283
15.2 NEWTON'S METHOD 286
16 HOMOGENEOUS POLYNOMIAL SYSTEMS 295
16.1 A UNITARILY INVARIANT INNER PRODUCT 297
16.2 A UNITARILY INVARIANT CONDITION NUMBER 300
16.3 ORTHOGONAL DECOMPOSITIONS OF 304
16.4 A CONDITION NUMBER THEOREM 307
16.5 BEZOUT'S THEOREM 310
16.6 A PROJECTIVE NEWTON'S METHOD 313
16.7 A HIGHER DERIVATIVE ESTIMATE 321
16.8 A LIPSCHITZ ESTIMATE FOR THE CONDITION NUMBER 325
17 SMALE'S 17TH PROBLEM: I 331
17.1 THE ADAPTIVE LINEAR HOMOTOPY FOR HA 332
17.2 INTERLUDE: RANDOMIZATION 340
17.2.1 RANDOMIZED ALGORITHMS 340
17.2.2 A LAS VEGAS HOMOTOPY METHOD 342
17.3 A CRASH COURSE ON PROBABILITY: IV 343
17.4 NORMAL JACOBIANS OF PROJECTIONS 346
17.5 THE STANDARD DISTRIBUTION ON THE SOLUTION VARIETY 350
17.6 BELTRAN-PARDO RANDOMIZATION 353
17.7 ANALYSIS OF ALGORITHM L.V 356
17.8 AVERAGE ANALYSIS OF ^NORM, Z^AV, AND/U, MAX 361
18 SMALE'S 17TH PROBLEM: II 367
18.1 THE MAIN TECHNICAL RESULT 368
18.1.1 OUTLINE OF THE PROOF 368
18.1.2 NORMAL JACOBIANS OF LINEARIZATIONS 371
18.1.3 INDUCED PROBABILITY DISTRIBUTIONS 374
18.2 SMOOTHED ANALYSIS OF LV 377
18.3 CONDITION-BASED ANALYSIS OF LV 378
18.4 A NEAR-SOLUTION TO SMALE'S 17TH PROBLEM 381
IMAGE 5
CONTENTS XV
18.4.1 A DETERMINISTIC HOMOTOPY CONTINUATION 381
18.4.2 AN ELIMINATION PROCEDURE FOR ZERO-FINDING 383
18.4.3 SOME INEQUALITIES OF COMBINATORIAL NUMBERS 387
19 REAL POLYNOMIAL SYSTEMS 391
19.1 HOMOGENEOUS SYSTEMS WITH REAL COEFFICIENTS 392
19.2 ON THE CONDITION FOR REAL ZERO-COUNTING 393
19.3 SMALE'S A-THEORY 396
19.4 AN ALGORITHM FOR REAL ZERO-COUNTING 405
19.4.1 GRIDS AND GRAPHS 405
19.4.2 PROOF OF THEOREM 19.1 408
19.5 ON THE AVERAGE NUMBER OF REAL ZEROS 413
19.6 FEASIBILITY OF UNDERDETERMINED AND SEMIALGEBRAIC SYSTEMS . 414
20 PROBABILISTIC ANALYSIS OF CONIC CONDITION NUMBERS: I. THE COMPLEX
CASE 419
20.1 THE BASIC IDEA 421
20.2 VOLUME OF TUBES AROUND LINEAR SUBSPACES 422
20.3 VOLUME OF ALGEBRAIC VARIETIES 425
20.4 A CRASH COURSE ON PROBABILITY: V 426
20.5 PROOF OF THEOREM 20.1 428
20.6 APPLICATIONS 432
20.6.1 LINEAR EQUATION-SOLVING 432
20.6.2 EIGENVALUE COMPUTATIONS 433
20.6.3 COMPLEX POLYNOMIAL SYSTEMS 436
21 PROBABILISTIC ANALYSIS OF CONIC CONDITION NUMBERS: II. THE REAL CASE
439
21.1 ON THE VOLUME OF TUBES 440
21.1.1 CURVATURE INTEGRALS 441
21.1.2 WEYL'S TUBE FORMULA 443
21.2 A CRASH COURSE ON PROBABILITY: VI 446
21.3 BOUNDING INTEGRALS OF CURVATURE 448
21.4 PROOF OF THEOREM 21.1 450
21.4.1 THE SMOOTH CASE 450
21.4.2 THE GENERAL CASE 452
21.4.3 PROOF OF THEOREM 21.1 454
21.5 AN APPLICATION 455
21.6 TUBES AROUND CONVEX SETS 455
21.6.1 INTEGRALS OF CURVATURE FOR BOUNDARIES OF CONVEX SETS . . . 455
21.6.2 PROOF OF THEOREM 13.18 458
21.7 CONIC CONDITION NUMBERS AND STRUCTURED DATA 459
21.8 SMOOTHED ANALYSIS FOR ADVERSARIAL DISTRIBUTIONS 460
APPENDIX 467
A.L BIG OH, LITTLE OH, AND OTHER COMPARISONS 467
A.2 DIFFERENTIAL GEOMETRY 468
IMAGE 6
XVI CONTENTS
A.2.1 SUBMANIFOLDS OF R" 469
A.2.2 ABSTRACT SMOOTH MANIFOLDS 471
A.2.3 INTEGRATION ON MANIFOLDS 473
A.2.4 SARD'S THEOREM AND TRANSVERSALITY 475
A.2.5 RIEMANNIAN METRICS 477
A.2.6 ORTHOGONAL AND UNITARY GROUPS 479
A.2.7 CURVATURE OF HYPERSURFACES 479
A.3 ALGEBRAIC GEOMETRY 481
A.3.1 VARIETIES 481
A.3.2 DIMENSION AND REGULAR POINTS 483
A.3.3 ELIMINATION THEORY 486
A.3.4 DEGREE 487
A.3.5 RESULTANT AND DISCRIMINANT 490
A.3.6 VOLUMES OF COMPLEX PROJECTIVE VARIETIES 491
A.4 INTEGRAL GEOMETRY 496
A.4.1 POINCARE'S FORMULA 496
A.4.2 THE PRINCIPAL KINEMATIC FORMULA 500
NOTES 503
CODA: OPEN PROBLEMS 521
P.I PROBABILISTIC ANALYSIS OF GROWTH FACTORS 521
P.2 EIGENVALUE PROBLEM 522
P.3 SMALE'S 9TH PROBLEM 524
P.4 SMOOTHED ANALYSIS OF RCC CONDITION NUMBER 524
P.5 IMPROVED AVERAGE ANALYSIS OF GRASSMANN CONDITION 525
P.6 SMOOTHED ANALYSIS OF GRASSMANN CONDITION 525
P.7 ROBUSTNESS OF CONDITION NUMBERS 525
P.8 AVERAGE COMPLEXITY OF IPMS FOR LINEAR PROGRAMMING 526
P.9 SMALE'S 17TH PROBLEM 526
P.10 THE SHUB-SMALE STARTING SYSTEM 526
P. 11 EQUIVARIANT MORSE FUNCTION 527
P. 12 GOOD STARTING PAIRS IN ONE VARIABLE 527
P. 13 APPROXIMATING CONDITION GEODESIES 528
P. 14 SELF-CONVEXITY OF/U-NORM IN HIGHER DEGREES 528
P. 15 STRUCTURED SYSTEMS OF POLYNOMIAL EQUATIONS 529
P. 16 SYSTEMS WITH SINGULARITIES 529
P. 17 CONIC CONDITION NUMBERS OF REAL PROBLEMS WITH HIGH CODIMENSION OF
ILL-POSEDNESS 529
P.18 FEASIBILITY OF REAL POLYNOMIAL SYSTEMS 530
BIBLIOGRAPHY 531
NOTATION 543
. . . C O N C E P T S 5 4 7
. . . A N D T H E P E O P L E W H O C R A F T E D T H E M 553 |
| any_adam_object | 1 |
| author | Bürgisser, Peter 1962- Cucker, Felipe 1958- |
| author_GND | (DE-588)1041607911 (DE-588)133832783 |
| author_facet | Bürgisser, Peter 1962- Cucker, Felipe 1958- |
| author_role | aut aut |
| author_sort | Bürgisser, Peter 1962- |
| author_variant | p b pb f c fc |
| building | Verbundindex |
| bvnumber | BV041108039 |
| classification_rvk | SK 900 |
| ctrlnum | (OCoLC)858051351 (DE-599)DNB103451539X |
| dewey-full | 518.1 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.1 |
| dewey-search | 518.1 |
| dewey-sort | 3518.1 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV041108039 |
| illustrated | Illustrated |
| indexdate | 2024-08-03T00:45:23Z |
| institution | BVB |
| isbn | 9783642388958 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026084306 |
| oclc_num | 858051351 |
| open_access_boolean | |
| owner | DE-11 DE-384 DE-20 DE-634 DE-19 DE-BY-UBM DE-83 DE-29T DE-188 |
| owner_facet | DE-11 DE-384 DE-20 DE-634 DE-19 DE-BY-UBM DE-83 DE-29T DE-188 |
| physical | XXXI, 554 S. graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Bürgisser, Peter 1962- Verfasser (DE-588)1041607911 aut Condition the geometry of numerical algorithms Peter Bürgisser ; Felipe Cucker Berlin [u.a.] Springer 2013 XXXI, 554 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 349 Konditionszahl (DE-588)4598523-6 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Algorithmus (DE-588)4001183-5 gnd rswk-swf Konditionszahl (DE-588)4598523-6 s Numerische Mathematik (DE-588)4042805-9 s Algorithmus (DE-588)4001183-5 s DE-604 Cucker, Felipe 1958- Verfasser (DE-588)133832783 aut Erscheint auch als Online-Ausgabe 978-3-642-38896-5 Grundlehren der mathematischen Wissenschaften 349 (DE-604)BV000000395 349 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4327775&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026084306&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bürgisser, Peter 1962- Cucker, Felipe 1958- Condition the geometry of numerical algorithms Grundlehren der mathematischen Wissenschaften Konditionszahl (DE-588)4598523-6 gnd Numerische Mathematik (DE-588)4042805-9 gnd Algorithmus (DE-588)4001183-5 gnd |
| subject_GND | (DE-588)4598523-6 (DE-588)4042805-9 (DE-588)4001183-5 |
| title | Condition the geometry of numerical algorithms |
| title_auth | Condition the geometry of numerical algorithms |
| title_exact_search | Condition the geometry of numerical algorithms |
| title_full | Condition the geometry of numerical algorithms Peter Bürgisser ; Felipe Cucker |
| title_fullStr | Condition the geometry of numerical algorithms Peter Bürgisser ; Felipe Cucker |
| title_full_unstemmed | Condition the geometry of numerical algorithms Peter Bürgisser ; Felipe Cucker |
| title_short | Condition |
| title_sort | condition the geometry of numerical algorithms |
| title_sub | the geometry of numerical algorithms |
| topic | Konditionszahl (DE-588)4598523-6 gnd Numerische Mathematik (DE-588)4042805-9 gnd Algorithmus (DE-588)4001183-5 gnd |
| topic_facet | Konditionszahl Numerische Mathematik Algorithmus |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4327775&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026084306&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT burgisserpeter conditionthegeometryofnumericalalgorithms AT cuckerfelipe conditionthegeometryofnumericalalgorithms |