Computer solution of large linear systems:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Amsterdam [u.a.]
Elsevier
1999
|
| Ausgabe: | 1. ed |
| Schriftenreihe: | Studies in mathematics and its applications
28 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008758570&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XXII, 753 S. graph. Darst. |
| ISBN: | 044450169X |
Internformat
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| 245 | 1 | 0 | |a Computer solution of large linear systems |c G. Meurant |
| 250 | |a 1. ed | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 1999 | |
| 300 | |a XXII, 753 S. |b graph. Darst. | ||
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| 490 | 1 | |a Studies in mathematics and its applications |v 28 | |
| 650 | 4 | |a Großes System - Lineares System - Numerisches Verfahren | |
| 650 | 0 | 7 | |a Lineares Gleichungssystem |0 (DE-588)4035826-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |D s |
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Datensatz im Suchindex
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| adam_text | COMPUTER SOLUTION OF LARGE LINEAR SYSTEMS G. MEURANT DCSA/EC CEA
BRUYERES LE CHATEL, BP 12 91680 BRUYERES LE CHATEL FRANCE 1999 -
ELSEVIER AMSTERDAM - LAUSANNE - NEW YORK - OXFORD - SHANNON - SINGAPORE
- TOKYO XI CONTENTS 1. INTRODUCTORY MATERIAL 1 1.1. VECTOR AND MATRICES
NORMS 2 1.2. EIGENVALUES 9 1.3. IRREDUCIBILITY AND DIAGONAL DOMINANCE 17
1.4. M*MATRICES AND GENERALIZATIONS 23 1.5. SPLITTINGS 29 1.6. POSITIVE
DEFINITE MATRICES 30 1.7. THE GRAPH OF A MATRIX 35 1.8. CHEBYSHEV
POLYNOMIALS 38 1.9. DISCRETIZATION METHODS FOR PARTIAL DIFFERENTIAL
EQUATIONS . . . . 41 1.10. EIGENVALUES AND FOURIER ANALYSIS 50 1.11.
FLOATINGPOINT ARITHMETIC 58 1.12. VECTOR AND PARALLEL COMPUTERS 62 1.13.
BLAS AND LAPACK 65 1.14. BIBLIOGRAPHICAL COMMENTS 67 2. GAUSSIAN
ELIMINATION FOR GENERAL LINEAR SYSTEMS 69 2.1. INTRODUCTION TO GAUSSIAN
ELIMINATION 69 2.1.1. GAUSSIAN ELIMINATION WITHOUT PERMUTATIONS 70
2.1.2. GAUSSIAN ELIMINATION WITH PERMUTATIONS (PARTIAL PIVOT- ING) 78
XII 2.1.3. GAUSSIAN ELIMINATION WITH OTHER PIVOTING STRATEGIES . . 80
2.1.4. OPERATION COUNTS 81 2.2. GAUSSIAN ELIMINATION FOR SYMMETRIC
SYSTEMS 82 2.2.1. THE OUTER PRODUCT ALGORITHM 82 2.2.2. THE BORDERING
ALGORITHM 84 2.2.3. THE INNER PRODUCT ALGORITHM 84 2.2.4. CODING THE
THREE FACTORIZATION ALGORITHMS 85 2.2.5. POSITIVE DEFINITE SYSTEMS 93
2.2.6. INDEFINITE SYSTEMS 94 2.3. GAUSSIAN ELIMINATION FOR H*MATRICES 95
2.4. BLOCK METHODS 99 2.5. TRIDIAGONAL AND BLOCK TRIDIAGONAL SYSTEMS 99
2.6. ROUNDOFF ERROR ANALYSIS ILL 2.7. PERTURBATION ANALYSIS 117 2.8.
SCALING 121 2.9. ITERATIVE REFINEMENT 122 2.10. PARALLEL SOLUTION OF
GENERAL LINEAR SYSTEMS 123 2.11. BIBLIOGRAPHICAL COMMENTS 128 3.
GAUSSIAN ELIMINATION FOR SPARSE LINEAR SYSTEMS 131 3.1. INTRODUCTION 131
3.2. THE FILL-IN PHENOMENON 132 3.3. GRAPHS AND FILL-IN FOR SYMMETRIC
MATRICES 134 3.4. CHARACTERIZATION OF THE FILL-IN 137 3.5. BAND AND
ENVELOPE NUMBERING SCHEMES FOR SYMMETRIC MATRICES . 140 3.5.1. THE
CUTHILL-MCKEE AND REVERSE CUTHILL-MCKEE ORDER- INGS 141 XIII 3.5.2.
SLOAN S ALGORITHM 146 3.6. SPECTRAL SCHEMES 148 3.6.1. THE BASIC IDEA
148 3.6.2. THE MULTILEVEL SPECTRAL ALGORITHM 150 3.6.3. THE KUMFERT AND
POTHEN HYBRID ALGORITHM . . . . 151 3.6.4. THE BOMAN-HENDRICKSON
MULTILEVEL ALGORITHM . . . 152 3.7. THE MINIMUM DEGREE ORDERING 152 3.8.
THE NESTED DISSECTION ORDERING 155 3.9. GENERALIZATION OF DISSECTION
ALGORITHMS 159 3.9.1. GENERAL DISSECTION ALGORITHMS 159 3.9.2. GRAPH
BISECTION IMPROVEMENT TECHNIQUES 161 3.9.3. THE MULTISECTION ALGORITHM
163 3.10. THE MULTIFRONTAL METHOD 163 3.11. NON-SYMMETRIC SPARSE
MATRICES 166 3.12. NUMERICAL STABILITY FOR SPARSE MATRICES 169 3.13.
PARALLEL ALGORITHMS FOR SPARSE MATRICES 170 3.14. BIBLIOGRAPHICAL
COMMENTS 176 4. FAST SOLVERS FOR SEPARABLE PDES 177 4.1. INTRODUCTION
177 4.2. FAST FOURIER TRANSFORM 179 4.2.1. THE BASICS OF THE FFT 179
4.2.2. THE COMPLEX FFT 180 4.2.3. THE REAL TRANSFORMS 183 4.2.4. FFT ON
VECTOR AND PARALLEL COMPUTERS 186 4.2.5. STABILITY OF THE FFT 186 4.2.6.
OTHER ALGORITHMS 187 XIV 4.2.7. DOUBLE FOURIER ANALYSIS 187 4.3. THE
FOURIER/TRIDIAGONAL METHOD 187 4.4. THE CYCLIC REDUCTION METHOD 191 4.5.
THE FACR(L) METHOD 202 4.6. THE CAPACITANCE MATRIX METHOD 203 4.7.
BIBLIOGRAPHICAL COMMENTS 206 5. CLASSICAL ITERATIVE METHODS 209 5.1.
INTRODUCTION 209 5.2. THE JACOBI METHOD 210 5.3. THE GAUSS-SEIDEL METHOD
218 5.4. THE SOR METHOD 223 5.5. THE SSOR METHOD 229 5.6. ALTERNATING
DIRECTION METHODS 233 5.7. RICHARDSON METHODS 241 5.8. ACCELERATION
TECHNIQUES 246 5.9. STABILITY OF CLASSICAL ITERATIVE METHODS 248 5.10.
BIBLIOGRAPHICAL COMMENTS 250 6. THE CONJUGATE GRADIENT AND RELATED
METHODS 251 6.1. DERIVATION OF THE METHOD 251 6.2. GENERALIZATION AND
SECOND FORM OF PCG . 256 6.3. OPTIMALITY OF PCG 260 6.4. THE CONVERGENCE
RATE OF PCG 265 6.5. THE LANCZOS ALGORITHM 281 6.6. A POSTERIORI ERROR
BOUNDS 284 6.7. THE EISENSTAT S TRICK 302 6.8. THE CONJUGATE RESIDUAL
METHOD 303 XV 6.9. SYMMLQ . 304 6.10. THE MINIMUM RESIDUAL METHOD 307
6.11. HYBRID ALGORITHMS 308 6.12. ROUNDOFF ERRORS OF CG AND LANCZOS 310
6.13. SOLVING FOR SEVERAL RIGHT HAND SIDES 315 6.14. BLOCK CG AND
LANCZOS 319 6.14.1. THE BLOCK LANCZOS ALGORITHM 319 6.14.2. THE BLOCK CG
ALGORITHM 322 6.15. INNER AND OUTER ITERATIONS 323 6.1^. CONSTRAINED CG
324 6.17. VECTOR AND PARALLEL PCG 325 6.18. BIBLIOGRAPHICAL COMMENTS 329
7. KRYLOV METHODS FOR NON-SYMMETRIC SYSTEMS 331 7.1. THE NORMAL
EQUATIONS 332 7.2. THE CONCUS AND GOLUB NON-SYMMETRIC CG 335 7.3.
CONSTRUCTION OF BASIS FOR KRYLOV SPACES 337 7.3.1. THE ARNOLDI ALGORITHM
337 7.3.2. THE HESSENBERG ALGORITHM 339 7.3.3. THE GENERALIZED
HESSENBERG PROCESS 341 7.4. FOM AND GMRES 342 7.4.1. DEFINITION OF FOM
AND GMRES 342 7.4.2. CONVERGENCE RESULTS 346 7.4.3. TRUNCATED AND
RESTARTED VERSIONS 351 7.4.4. METHODS EQUIVALENT TO GMRES 351 7.4.5.
METHODS EQUIVALENT TO FOM 357 7.5. ROUNDOFF ERROR ANALYSIS OF GMRES 357
XVI 7.6. EXTENSIONS TO GMRES 360 7.6.1. FLEXIBLE GMRES 360 7.6.2. GMRES*
361 7.7. HYBRID GMRES ALGORITHMS 363 7.8. THE NON-SYMMETRIC LANCZOS
ALGORITHM 364 7.8.1. DEFINITION OF THE NON-SYMMETRIC LANCZOS ALGORITHM .
365 7.8.2. VARIANTS OF THE NON-SYMMETRIC LANCZOS ALGORITHM . . 367
7.8.3. MAINTAINING SEMI BI-ORTHOGONALITY 368 7.9. THE BICONJUGATE
GRADIENT ALGORITHM 370 7.10. ROUNDOFF ERROR ANALYSIS OF BICG 373 7.11.
HANDLING OF BREAKDOWNS 374 7.11.1. FOP 374 7.11.2. PADE APPROXIMATION
376 7.11.3. BLOCK BI-ORTHOGONALITY 378 7.11.4. MODIFIED KRYLOV SPACES
379 7.12. THE CONJUGATE GRADIENT SQUARED ALGORITHM 379 7.13. EXTENSIONS
OF BICG 382 7.14. THE QUASI MINIMAL RESIDUAL ALGORITHM 387 7.15. CMRH .
390 7.16. WHICH METHOD TO USE? 391 7.17. COMPLEX LINEAR SYSTEMS 392
7.18. KRYLOV METHODS ON PARALLEL COMPUTERS 394 7.19. BIBLIOGRAPHICAL
COMMENTS 394 8. PRECONDITIONING 397 8.1. INTRODUCTION 397 8.2. THE
DIAGONAL PRECONDITIONER 399 XVII 8.3. THE SSOR PRECONDITIONER 401 8.3.1.
DEFINITION OF SSOR 401 8.3.2. CONVERGENCE RESULTS FOR SSOR 402 8.3.3.
FOURIER ANALYSIS OF SSOR 405 8.4. THE BLOCK SSOR PRECONDITIONER 407
8.4.1. DEFINITION OF BSSOR 407 8.4.2. ANALYSIS OF BSSOR 407 8.4.3.
FOURIER ANALYSIS OF BSSOR 409 8.5. THE INCOMPLETE CHOLESKY DECOMPOSITION
416 8.5.1. THE GENERAL DECOMPOSITION 416 8.5.2. INCOMPLETE DECOMPOSITION
OF H*MATRICES 418 8.5.3. INCOMPLETE DECOMPOSITION OF NON-SYMMETRIC
MATRICES . 422 8.5.4. DIFFERENT INCOMPLETE DECOMPOSITION STRATEGIES . .
. 423 8.5.5. FINITE DIFFERENCE MATRICES 424 8.5.6. FOURIER ANALYSIS OF
IC(1,1) 427 8.5.7. COMPARISON OF PERIODIC AND DIRICHLET BOUNDARY CONDI-
TIONS 430 8.5.8. AXELSSON S RESULTS 441 8.6. THE MODIFIED INCOMPLETE
CHOLESKY DECOMPOSITION 442 8.6.1. THE DKR PRECONDITIONER 442 8.6.2.
ANALYSIS OF DKR 443 8.6.3. FOURIER ANALYSIS OF DKR 446 8.6.4. EXTENSIONS
OF DKR 448 8.7. THE RELAXED INCOMPLETE CHOLESKY DECOMPOSITION 448 8.8.
MORE ON THE INCOMPLETE DECOMPOSITIONS FOR THE MODEL PROBLEM . 449 8.9.
STABILITY OF INCOMPLETE DECOMPOSITION 452 XVIII 8.10. THE GENERALIZED
SSOR PRECONDITIONER 454 8.11. INCOMPLETE DECOMPOSITION OF POSITIVE
DEFINITE MATRICES . . . 458 8.12. DIFFERENT ORDERINGS FOR IC 461 8.12.1.
EXPERIMENTAL RESULTS 461 8.12.2. THEORY FOR MODEL PROBLEMS 467 8.12.3.
VALUE DEPENDENT ORDERINGS 470 8.12.4. MULTICOLOR ORDERINGS 471 8.13. THE
REPEATED RED-BLACK DECOMPOSITION 472 8.13.1. DESCRIPTION OF THE METHODS
472 8.13.2. ANALYSIS OF RRB 475 8.14. THE BLOCK INCOMPLETE CHOLESKY
DECOMPOSITION 479 8.14.1. BLOCK TRIDIAGONAL MATRICES 479 8.14.2.
POINTWISE EQUIVALENT DECOMPOSITION 480 8.14.3. THE MODIFIED INCOMPLETE
BLOCK DECOMPOSITION . . . 481 8.14.4. BLOCK INCOMPLETE DECOMPOSITION FOR
H-MATRICES . . . 482 8.14.5. GENERALIZATION OF THE BLOCK INCOMPLETE
DECOMPOSITION . 482 8.14.6. FOURIER ANALYSIS OF INV AND MINV 482 8.14.7.
AXELSSON S RESULTS 493 8.14.8. BLOCK*SIZE REDUCTION 494 8.15. THE BLOCK
CHOLESKY DECOMPOSITION FOR 3D PROBLEMS . . . . 494 8.15.1. POINT
PRECONDITIONERS 495 8.15.2. ID BLOCK PRECONDITIONERS 496 8.15.3. 2D
POINT PRECONDITIONERS 497 8.15.4. 2D BLOCK PRECONDITIONERS 499 8.16.
NESTED FACTORIZATION 501 8.16.1. THE ACP PRECONDITIONER FOR 3D PROBLEMS
. . . . 502 XIX 8.16.2. THE PRECONDITIONER OF APPLEYARD, CHESHIRE AND
POLLARD 502 8.16.3. IMPROVEMENTS OF ACP 503 5.17. SPARSE APPROXIMATE
INVERSES 504 8.17.1. THE SPARSE INVERSES OF HUCKLE AND GROTE 505 8.17.2.
THE SPARSE INVERSES OF GOULD AND SCOTT 506 8.17.3. THE SPARSE INVERSES
OF CHOW AND SAAD 506 8.17.4. SPARSE APPROXIMATE INVERSES FOR SYMMETRIC
MATRICES . 507 8.17.5. THE SPARSE INVERSES OF BENZI, MEYER AND TUMA . .
. 507 !.18. POLYNOMIAL PRECONDITIONERS 509 8.18.1. TRUNCATED NEUMANN
SERIES 510 8.18.2. THE MINMAX POLYNOMIAL 511 8.18.3. LEAST SQUARES
POLYNOMIALS 514 8.18.4. STABLE EVALUATION OF POLYNOMIALS 520 8.18.5. A
POLYNOMIAL INDEPENDENT OF EIGENVALUE ESTIMATES . . 525 8.18.6. ADAPTIVE
ALGORITHMS FOR SPD MATRICES 526 8.18.7. POLYNOMIALS FOR SYMMETRIC
INDEFINITE PROBLEMS . . . 527 8.18.8. POLYNOMIALS FOR NON-SYMMETRIC
PROBLEMS 528 1.19. DOUBLE PRECONDITIONERS 528 !.20. OTHER IDEAS 529
8.20.1. ADI PRECONDITIONER 529 8.20.2. ADDKR PRECONDITIONER 530 8.20.3.
ELEMENT BY ELEMENT PRECONDITIONER 530 8.20.4. FAST SOLVERS 531 8.20.5.
WAVELETS 531 1.21. VECTOR AND PARALLEL COMPUTING 532 8.21.1.
VECTORIZATION OF IC(1,1) 533 XX 8.21.2. PARALLEL ORDERINGS 534 8.21.3.
VECTORIZATION OF INV 535 8.21.4. TWISTED INCOMPLETE BLOCK FACTORIZATIONS
535 8.21.5. INCOMPLETE BLOCK CYCLIC REDUCTION 537 8.21.6. A MASSIVELY
PARALLEL PRECONDITIONER 538 8.22. BIBLIOGRAPHICAL COMMENTS 539 9.
MULTIGRID METHODS 541 9.1. INTRODUCTION 541 9.2. THE TWO-GRID METHOD 542
9.3. A ONE DIMENSIONAL EXAMPLE 546 9.3.1. THE CHOICE OF THE SMOOTHING
547 9.3.2. THE CHOICE OF THE RESTRICTION 549 9.3.3. THE CHOICE OF
PROLONGATION 549 9.3.4. THE CHOICE OF THE COARSE GRID MATRIX 550 9.4.
THE CHOICES OF COMPONENTS 557 9.4.1. THE SMOOTHING 557 9.4.2. THE
COARSENING 560 9.4.3. GRID TRANSFERS 561 9.4.4. THE COARSE GRID OPERATOR
563 9.5. THE MULTIGRID METHOD 563 9.6. CONVERGENCE THEORY 567 9.7.
COMPLEXITY OF MULTIGRID 571 9.8. THE FULL MULTIGRID METHOD 573 9.9.
VECTOR AND PARALLEL MULTIGRID 576 9.10. ALGEBRAIC MULTIGRID 579 9.11.
BIBLIOGRAPHICAL COMMENTS 583 XXI 10. DOMAIN DECOMPOSITION AND MULTILEVEL
METHODS . . . . . 585 10.1. INTRODUCTION TO DOMAIN DECOMPOSITION 585
10.2. SCHWARZ METHODS 587 10.2.1. THE CLASSICAL SCHWARZ ALTERNATING
METHOD . . . . 587 10.2.2. THE MATRIX FORM OF THE SCHWARZ ALTERNATING
METHOD . 589 10.2.3. THE RATE OF CONVERGENCE 592 10.2.4. OTHER BOUNDARY
CONDITIONS 595 10.2.5. PARALLELIZING MULTIPLICATIVE SCHWARZ 595 10.2.6.
THE ADDITIVE SCHWARZ METHOD 596 10.2.7. ADDING A COARSE MESH CORRECTION
597 10.3. AN ADDITIVE SCHWARZ PRECONDITIONER FOR PARABOLIC PROBLEMS . .
597 10.4. ALGEBRAIC DOMAIN DECOMPOSITION METHODS WITHOUT OVERLAPPING .
600 10.4.1. EXACT SOLVERS FOR THE SUBDOMAINS 601 10.4.2. APPROXIMATE
SOLVERS FOR THE SUBDOMAINS 604 10.5. APPROXIMATE SCHUR COMPLEMENTS IN
THE TWO SUBDOMAINS CASE . 606 10.5.1. THE SCHUR COMPLEMENT FOR BLOCK
TRIDIAGONAL MATRICES . 607 10.5.2. EIGENVALUES OF THE SCHUR COMPLEMENT
FOR SEPARABLE PROB- LEMS 608 10.5.3. DRYJA S PRECONDITIONER 613 10.5.4.
GOLUB AND MAYERS PRECONDITIONER 614 10.5.5. THE NEUMANN-DIRICHLET
PRECONDITIONER 615 10.5.6. THE NEUMANN-NEUMANN PRECONDITIONER 617
10.5.7. DEPENDENCE ON THE ASPECT RATIO 618 10.5.8. DEPENDENCE ON THE
COEFFICIENTS 620 10.5.9. PROBING 621 10.5.10. INV AND MINV
APPROXIMATIONS 623 XXII 10.5.11. THE SCHUR COMPLEMENT FOR MORE GENERAL
PROBLEMS . . 625 10.6. APPROXIMATIONS OF SCHUR COMPLEMENTS WITH MANY
SUBDOMAINS . 627 10.7. INEXACT SUBDOMAIN SOLVERS 632 10.8. DOMAIN
DECOMPOSITION WITH BOXES 637 10.8.1. THE BRAMBLE, PASCIAK AND SCHATZ
PRECONDITIONER . . 638 10.8.2. VERTEX SPACE PRECONDITIONERS 642 10.9. A
BLOCK RED-BLACK DD PRECONDITIONER 643 10.10. MULTILEVEL PRECONDITIONERS
646 10.10.1. ADDITIVE MULTILEVEL SCHWARZ PRECONDITIONERS . . . . 647
10.10.2. MULTILEVEL ILU PRECONDITIONERS 648 10.11. BIBLIOGRAPHICAL
COMMENTS 655 REFERENCES 657 INDEX 749
|
| any_adam_object | 1 |
| author | Meurant, Gérard A. 1948- |
| author_GND | (DE-588)138962545 |
| author_facet | Meurant, Gérard A. 1948- |
| author_role | aut |
| author_sort | Meurant, Gérard A. 1948- |
| author_variant | g a m ga gam |
| building | Verbundindex |
| bvnumber | BV012869503 |
| classification_rvk | SK 915 |
| ctrlnum | (OCoLC)247060538 (DE-599)BVBBV012869503 |
| discipline | Mathematik |
| edition | 1. ed |
| format | Book |
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| id | DE-604.BV012869503 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T10:37:36Z |
| institution | BVB |
| isbn | 044450169X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008758570 |
| oclc_num | 247060538 |
| open_access_boolean | |
| owner | DE-29T DE-20 DE-83 |
| owner_facet | DE-29T DE-20 DE-83 |
| physical | XXII, 753 S. graph. Darst. |
| publishDate | 1999 |
| publishDateSearch | 1999 |
| publishDateSort | 1999 |
| publisher | Elsevier |
| record_format | marc |
| series | Studies in mathematics and its applications |
| series2 | Studies in mathematics and its applications |
| spellingShingle | Meurant, Gérard A. 1948- Computer solution of large linear systems Studies in mathematics and its applications Großes System - Lineares System - Numerisches Verfahren Lineares Gleichungssystem (DE-588)4035826-4 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4035826-4 (DE-588)4128130-5 |
| title | Computer solution of large linear systems |
| title_auth | Computer solution of large linear systems |
| title_exact_search | Computer solution of large linear systems |
| title_full | Computer solution of large linear systems G. Meurant |
| title_fullStr | Computer solution of large linear systems G. Meurant |
| title_full_unstemmed | Computer solution of large linear systems G. Meurant |
| title_short | Computer solution of large linear systems |
| title_sort | computer solution of large linear systems |
| topic | Großes System - Lineares System - Numerisches Verfahren Lineares Gleichungssystem (DE-588)4035826-4 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Großes System - Lineares System - Numerisches Verfahren Lineares Gleichungssystem Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008758570&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000646 |
| work_keys_str_mv | AT meurantgerarda computersolutionoflargelinearsystems |