Introduction to computational linear algebra:
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
2015
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028168996&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028168996&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | "A CRC title." |
| Umfang: | XXVI, 235 S. |
| ISBN: | 9781482258691 |
Internformat
MARC
| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV042738088 | ||
| 003 | DE-604 | ||
| 005 | 20150821 | ||
| 007 | t| | ||
| 008 | 150805s2015 xxu |||| 00||| eng d | ||
| 010 | |a 015011621 | ||
| 020 | |a 9781482258691 |9 978-1-4822-5869-1 | ||
| 035 | |a (OCoLC)920396761 | ||
| 035 | |a (DE-599)BVBBV042738088 | ||
| 040 | |a DE-604 |b ger |e aacr | ||
| 041 | 0 | |a eng | |
| 044 | |a xxu |c US | ||
| 049 | |a DE-739 | ||
| 084 | |a SK 220 |0 (DE-625)143224: |2 rvk | ||
| 100 | 1 | |a Nassif, Nabil R. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Introduction to computational linear algebra |c Nabil Nassif ; Jocelyne Erhel ;XXIV, Bernard Philippe |
| 246 | 1 | 3 | |a Computational linear algebra |
| 264 | 1 | |c 2015 | |
| 300 | |a XXVI, 235 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a "A CRC title." | ||
| 650 | 4 | |a Algebras, Linear |v Textbooks | |
| 650 | 0 | 7 | |a Lineare Algebra |0 (DE-588)4035811-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Numerische Mathematik |0 (DE-588)4042805-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Computeralgebra |0 (DE-588)4010449-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Computeralgebra |0 (DE-588)4010449-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Numerische Mathematik |0 (DE-588)4042805-9 |D s |
| 689 | 1 | 1 | |a Lineare Algebra |0 (DE-588)4035811-2 |D s |
| 689 | 1 | |5 DE-604 | |
| 700 | 1 | |a Erhel, Jocelyne |e Verfasser |4 aut | |
| 700 | 1 | |a Philippe, Bernard |e Verfasser |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028168996&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028168996&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-028168996 | |
Datensatz im Suchindex
| _version_ | 1819257176824217600 |
|---|---|
| adam_text | Introduction to Computational Linear Algebra presents class-
room-tested material on computational linear algebra and its appli-
cation to numerical solutions of partial and ordinary differential equa-
tions.
The text first introduces BLAS operations of types 1,2, and 3 adapt-
ed to a scientific computer environment, specifically MATLAB®. It
next covers the basic mathematical tools needed in numerical linear
algebra and discusses classical material on Gauss decompositions
as well as LU and Cholesky’s factorizations of matrices. The text
then shows how to solve linear least squares problems, provides a
detailed numerical treatment of the algebraic eigenvalue problem,
and discusses (indirect) iterative methods to solve a system of lin-
ear equations. The final chapter illustrates how to solve discret^
sparse systems of linear equations.
Features
® Discusses the fundamentals needed in numerical linear algebra,
including eigenvalues, vector and matrix norms, orthogonal
matrices, the Gram-Schmidt process, and singular value
decomposition
• Illustrates algorithms for eigenvalue problems with examples
from population dynamics and Google matrices
• Covers iterative methods for solving a system of linear
equations, including stationary methods based on matrix
splitting and Krylov methods
• Explains the implementation of algorithms using MATLAB’s
syntax
«· Expresses the numerical methods using pseudo-code or a
detailed MATLAB programj^^p^
® Includes numerous e^^^p^nd computer projects that test
your understandi^^^Hrthe mathematics of numerical
methods and ttiÊ^^Hromputer programming
Contents
Preface xiii
About the Authors xvii
List of Figures xix
List of Tables xxi
List of Algorithms xxiii
1 Basic Linear Algebra Subprograms - BLAS 1
1.1 An Introductory Example..................................... 1
1.2 Matrix Notations ........................................... 3
1.3 IEEE Floating Point Systems and Computer Arithmetic . . 4
1.4 Vector֊Vector Operations: Level-1 BLAS ..................... 5
1.5 Matrix-Vector Operations: Level-2 BLAS ..................... 8
1.6 Matrix-Matrix Operations: Level-3 BLAS..................... 11
1.6.1 Matrix Multiplication Using GAXPYs.................. 12
1.6.2 Matrix Multiplication Using Scalar Products......... 13
1.6.3 Matrix Multiplication Using External Products .... 13
1.6.4 Block Multiplications .............................. 13
1.6.5 An Efficient Data Management........................ 14
1.7 Sparse Matrices: Storage and Associated Operations......... 15
1.8 Exercises ................................................. 19
1.9 Computer Project: Strassen Algorithm ...................... 25
2 Basic Concepts for Matrix Computations 27
2.1 Vector Norms .............................................. 27
2.2 Complements on Square Matrices ............................ 28
2.2.1 Definition of Important Square Matrices............. 29
2.2.2 Use of Orthonormal Bases ........................... 29
2.2.3 Gram-Schmidt Process................................ 30
2.2.4 Determinants........................................ 33
2.2.5 Eigenvalue-Eigenvector and Characteristic Polynomial 34
2.2.6 Schurs Decomposition................................ 36
2.2.7 Orthogonal Decomposition of Symmetric Real and
Complex Hermitian Matrices.......................... 39
vii
viii Contents
2.2.7.1 A Real and Symmetric: A — AT.............. 39
2.2.7.2 A Complex Hermitian: A = A*............... 40
2.2.8 Symmetric Positive Definite and Positive Semi-Definite
Matrices............................................. 41
2.3 Rectangular Matrices: Ranks and Singular Values ............ 42
2.3.1 Singular Values of a Matrix.......................... 43
2.3.2 Singular Value Decomposition................... 44
2.4 Matrix Norms ............................................... 47
2.5 Exercises .................................................. 51
2.6 Computer Exercises ......................................... 53
3 Gauss Elimination and LU Decompositions of Matrices 57
3.1 Special Matrices for LU Decomposition ...................... 57
3.1.1 Triangular Matrices ................................. 57
3.1.2 Permutation Matrices................................. 58
3.2 Gauss Transforms ......................................... 60
3.2.1 Preliminaries for Gauss Transforms................... 60
3.2.2 Definition of Gauss Transforms....................... 61
3.3 Naive LU Decomposition for a Square Matrix with Principal
Minor Property (pmp) ..................................... 62
3.3.1 Algorithm and Operations Count....................... 65
3.3.2 LDL Decomposition of a Matrix Having the Principal
Minor Property (pmp)........................... 66
3.3.3 The Case of Symmetric and Positive Definite Matrices:
Cholesky Decomposition............................... 66
3.3.4 Diagonally Dominant Matrices......................... 68
3.4 PLU Decompositions with Partial Pivoting Strategy .... 69
3.4.1 Unsealed Partial Pivoting............................ 69
3.4.2 Scaled Partial Pivoting ............................. 71
3.4.3 Solving a System Ax — b Using the LU Decomposition 72
3.5 MATLAB Commands Related to the LU Decomposition .... 73
3.6 Condition Number of a Square Matrix................... 73
3.7 Exercises .................................................. 75
3.8 Computer Exercises ......................................... 77
4 Orthogonal Factorizations and Linear Least Squares
Problems 79
4.1 Formulation of Least Squares Problems: Regression Analysis 79
4.1.1 Least Squares and Regression Analysis................ 79
4.1.2 Matrix Formulation of Regression Problems............ 80
4.2 Existence of Solutions Using Quadratic Forms ............... 80
4.2.1 Full Rank Cases: Application to Regression Analysis . 82
4.3 Existence of Solutions through Matrix Pseudo-Inverse .... 83
4.3.1 Obtaining Matrix Pseudo-Inverse through Singular
Value Decomposition.................................. 85
Contents ix
4.4 The OR Factorization Theorem ............................ 87
4.4.1 Householder Transforms................................ 87
4.4.2 Steps of the QR Decomposition of a Matrix............. 91
4.4.3 Particularizing When m n ........................... 92
4.4.4 Givens Rotations...................................... 93
4.5 Gram-Schmidt Orthogonalization .............................. 94
4.6 Least Squares Problem and QR Decomposition................... 98
4.7 Householder QR with Column Pivoting ......................... 99
4.8 MATLAB Implementations ...................................... 99
4.8.1 Use of the Backslash Operator ........................ 99
4.8.2 QR Decompositions..................................... 99
4.9 Exercises .................................................. 101
4.10 Computer Exercises ........................................ 102
5 Algorithms for the Eigenvalue Problem 105
5.1 Basic Principles ........................................... 105
5.1.1 Why Compute the Eigenvalues of a Square Matrix? . 105
5.1.2 Spectral Decomposition of a Matrix................... 107
5.1.3 The Power Method and its By-Products................. 112
5.2 QR Method for a Non-Symmetrie Matrix....................... 115
5.2.1 Reduction to an Upper Hessenberg Matrix ............. 115
5.2.2 QR Algorithm for an Upper Hessenberg Matrix .... 117
5.2.3 Convergence of the QR Method......................... 119
5.3 Algorithms for Symmetric Matrices .......................... 121
5.3.1 Reduction to a Tridiagonal Matrix.................... 121
5.3.2 Algorithms for Tridiagonal Symmetric Matrices .... 122
5.4 Methods for Large Size Matrices............................. 124
5.4.1 Rayleigh-Ritz Projection............................. 125
5.4.2 Arnoldi Procedure.................................... 126
5.4.3 The Arnoldi Method for Computing Eigenvalues of a
Large Matrix......................................... 128
5.4.4 Arnoldi Method for Computing an Eigenpair............ 130
5.4.5 Symmetric Case: Lanczos Algorithm.................... 131
5.5 Singular Value Decomposition ............................... 134
5.5.1 Full SVD............................................. 134
5.5.2 Singular Triplets for Large Matrices................. 136
5.6 Exercises ...................................................138
5.7 Computer Exercises ...........................................Ml
6 iterative Methods for Systems of Linear Equations 149
6.1 Stationary Methods ......................................... 150
6.1.1 Splitting............................................ 150
6.1.2 Classical Stationary Methods......................... 150
6.2 Krylov Methods ............................................. 152
6.2.1 Krylov Properties ................................... 152
x Contents
6.2.2 Subspace Condition .................................. 153
6.2.3 Minimization Property for spd Matrices............... 154
6.2.4 Minimization Property for General Matrices........... 155
6.3 Method of Steepest Descent for spd Matrices................. 156
6.3.1 Convergence Properties of the Steepest Descent Method 157
6.3.2 Preconditioned Steepest Descent Algorithm............ 157
6.4 Conjugate Gradient Method (CG) for spd Matrices ............ 158
6.4.1 Krylov Basis Properties.............................. 159
6.4.2 CG Algorithm ........................................ 161
6.4.3 Convergence of CG.................................... 162
6.4.4 Preconditioned Conjugate Gradient ................... 163
6.4.5 Memory and CPU Requirements in PCG .................. 164
6.4.6 Relation with the Lanczos Method..................... 164
6.4.7 Case of Symmetric Indefinite Systems: SYMMLQ
Method............................................... 165
6.5 The Generalized Minimal Residual Method..................... 165
6.5.1 Krylov Basis Computation ............................ 166
6.5.2 GMRES Algorithm...................................... 166
6.5.3 Convergence of GMRES................................. 167
6.5.4 Preconditioned GMRES................................. 168
6.5.5 Restarted GMRES...................................... 169
6.5.6 MINRES Algorithm..................................... 169
6.6 The Bi-Conjugate Gradient Method ........................... 169
6.6.1 Orthogonality Properties in BiCG..................... 170
6.6.2 BiCG Algorithm....................................... 172
6.6.3 Convergence of BiCG ................................. 172
6.6.4 Breakdowns and Near-Breakdowns in BiCG............... 173
6.6.5 Complexity of BiCG and Variants of BiCG.............. 173
6.6.6 Preconditioned BiCG ................................. 173
6.7 Preconditioning Issues...................................... 174
6.8 Exercises .................................................. 175
7 Sparse Systems to Solve Poisson Differential Equations 177
7.1 Poisson Differential Equations ............................. 177
7.2 The Path to Poisson Solvers ................................ 179
7.3 Finite Differences for Poisson-Dirichlet Problems.......... 179
7.3.1 One-Dimensional Dirichlet-Poisson.................... 180
7.3.2 Two-Dimensional Poisson-Dirichlet on a Rectangle . . 187
7.3.3 Complexity for Direct Methods: Zero-Fill Phenomenon 192
7.4 Variational Formulations ................................... 195
7.4.1 Integration by Parts and Green s Formula............. 195
7.4.2 Variational Formulation to One-Dimensional Poisson
Problems............................................. 197
7.4.3 Variational Formulations to Two-Dimensional Poisson
Problems............................................. 198
Contents xi
7.4.4 Petrov-Galerkin Approximations..................... 200
7.5 One-Dimensional Finite-Element Discretizations ........... 201
7.5.1 The Pi Finite-Element Spaces ...................... 202
7.5.2 Finite-Element Approximation Using Si (II)......... 203
7.5.3 Implementation of the Method....................... 205
7.5.4 One-Dimensional Բշ Finite-Elements................. 211
7.6 Exercises ................................................ 214
7.7 Computer Exercises ....................................... 215
Bibliography 227
Index 233
|
| any_adam_object | 1 |
| author | Nassif, Nabil R. Erhel, Jocelyne Philippe, Bernard |
| author_facet | Nassif, Nabil R. Erhel, Jocelyne Philippe, Bernard |
| author_role | aut aut aut |
| author_sort | Nassif, Nabil R. |
| author_variant | n r n nr nrn j e je b p bp |
| building | Verbundindex |
| bvnumber | BV042738088 |
| classification_rvk | SK 220 |
| ctrlnum | (OCoLC)920396761 (DE-599)BVBBV042738088 |
| discipline | Mathematik |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>02007nam a2200469 c 4500</leader><controlfield tag="001">BV042738088</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20150821 </controlfield><controlfield tag="007">t|</controlfield><controlfield tag="008">150805s2015 xxu |||| 00||| eng d</controlfield><datafield tag="010" ind1=" " ind2=" "><subfield code="a">015011621</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9781482258691</subfield><subfield code="9">978-1-4822-5869-1</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)920396761</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV042738088</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">aacr</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="044" ind1=" " ind2=" "><subfield code="a">xxu</subfield><subfield code="c">US</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-739</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 220</subfield><subfield code="0">(DE-625)143224:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Nassif, Nabil R.</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Introduction to computational linear algebra</subfield><subfield code="c">Nabil Nassif ; Jocelyne Erhel ;XXIV, Bernard Philippe</subfield></datafield><datafield tag="246" ind1="1" ind2="3"><subfield code="a">Computational linear algebra</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2015</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XXVI, 235 S.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">"A CRC title."</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Algebras, Linear</subfield><subfield code="v">Textbooks</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Lineare Algebra</subfield><subfield code="0">(DE-588)4035811-2</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Numerische Mathematik</subfield><subfield code="0">(DE-588)4042805-9</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Computeralgebra</subfield><subfield code="0">(DE-588)4010449-7</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Computeralgebra</subfield><subfield code="0">(DE-588)4010449-7</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="689" ind1="1" ind2="0"><subfield code="a">Numerische Mathematik</subfield><subfield code="0">(DE-588)4042805-9</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="1" ind2="1"><subfield code="a">Lineare Algebra</subfield><subfield code="0">(DE-588)4035811-2</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="1" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Erhel, Jocelyne</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Philippe, Bernard</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Passau - ADAM Catalogue Enrichment</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028168996&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Klappentext</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Passau - ADAM Catalogue Enrichment</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028168996&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="943" ind1="1" ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-028168996</subfield></datafield></record></collection> |
| id | DE-604.BV042738088 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T17:18:35Z |
| institution | BVB |
| isbn | 9781482258691 |
| language | English |
| lccn | 015011621 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028168996 |
| oclc_num | 920396761 |
| open_access_boolean | |
| owner | DE-739 |
| owner_facet | DE-739 |
| physical | XXVI, 235 S. |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| record_format | marc |
| spellingShingle | Nassif, Nabil R. Erhel, Jocelyne Philippe, Bernard Introduction to computational linear algebra Algebras, Linear Textbooks Lineare Algebra (DE-588)4035811-2 gnd Numerische Mathematik (DE-588)4042805-9 gnd Computeralgebra (DE-588)4010449-7 gnd |
| subject_GND | (DE-588)4035811-2 (DE-588)4042805-9 (DE-588)4010449-7 |
| title | Introduction to computational linear algebra |
| title_alt | Computational linear algebra |
| title_auth | Introduction to computational linear algebra |
| title_exact_search | Introduction to computational linear algebra |
| title_full | Introduction to computational linear algebra Nabil Nassif ; Jocelyne Erhel ;XXIV, Bernard Philippe |
| title_fullStr | Introduction to computational linear algebra Nabil Nassif ; Jocelyne Erhel ;XXIV, Bernard Philippe |
| title_full_unstemmed | Introduction to computational linear algebra Nabil Nassif ; Jocelyne Erhel ;XXIV, Bernard Philippe |
| title_short | Introduction to computational linear algebra |
| title_sort | introduction to computational linear algebra |
| topic | Algebras, Linear Textbooks Lineare Algebra (DE-588)4035811-2 gnd Numerische Mathematik (DE-588)4042805-9 gnd Computeralgebra (DE-588)4010449-7 gnd |
| topic_facet | Algebras, Linear Textbooks Lineare Algebra Numerische Mathematik Computeralgebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028168996&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028168996&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT nassifnabilr introductiontocomputationallinearalgebra AT erheljocelyne introductiontocomputationallinearalgebra AT philippebernard introductiontocomputationallinearalgebra AT nassifnabilr computationallinearalgebra AT erheljocelyne computationallinearalgebra AT philippebernard computationallinearalgebra |