Parallel Algorithms for Linear Models: Numerical Methods and Estimation ProblemsSpringer Science & Business Media, 31 janv. 2000 - 182 pages Parallel Algorithms for Linear Models provides a complete and detailed account of the design, analysis and implementation of parallel algorithms for solving large-scale linear models. It investigates and presents efficient, numerically stable algorithms for computing the least-squares estimators and other quantities of interest on massively parallel systems. The monograph is in two parts. The first part consists of four chapters and deals with the computational aspects for solving linear models that have applicability in diverse areas. The remaining two chapters form the second part, which concentrates on numerical and computational methods for solving various problems associated with seemingly unrelated regression equations (SURE) and simultaneous equations models. The practical issues of the parallel algorithms and the theoretical aspects of the numerical methods will be of interest to a broad range of researchers working in the areas of numerical and computational methods in statistics and econometrics, parallel numerical algorithms, parallel computing and numerical linear algebra. The aim of this monograph is to promote research in the interface of econometrics, computational statistics, numerical linear algebra and parallelism. |
Table des matières
LINEAR MODELS AND QR DECOMPOSITION | 1 |
21 THE ORDINARY LINEAR MODEL | 2 |
22 THE GENERAL LINEAR MODEL | 7 |
3 FORMING THE QR DECOMPOSITION | 10 |
31 THE HOUSEHOLDER METHOD | 11 |
32 THE GIVENS ROTATION METHOD | 13 |
33 THE GRAMSCHMIDT ORTHOGONALIZATION METHOD | 16 |
4 DATA PARALLEL ALGORITHMS FOR COMPUTING THE QR DECOMPOSITION | 17 |
24 QRD OF STRUCTURED BANDED MATRICES | 82 |
25 RECURSIVE LEASTSQUARES WITH LINEAR EQUALITY CONSTRAINTS | 87 |
3 ADDING EXOGENOUS VARIABLES | 90 |
4 DELETING OBSERVATIONS | 92 |
41 PARALLEL STRATEGIES | 94 |
5 DELETING EXOGENOUS VARIABLES | 99 |
THE GENERAL LINEAR MODEL | 105 |
2 PARALLEL ALGORITHMS | 108 |
42 THE HOUSEHOLDER METHOD | 19 |
43 THE GRAMSCHMIDT METHOD | 21 |
44 THE GIVENS ROTATION METHOD | 22 |
45 COMPUTATIONAL RESULTS | 23 |
51 THE CPP GAMMA SIMD SYSTEM | 24 |
52 THE HOUSEHOLDER QRD ALGORITHM | 25 |
53 QRD OF SKINNY MATRICES | 27 |
QRD OF A SET OF MATRICES | 29 |
62 MATRICES WITH DIFFERENT NUMBER OF COLUMNS | 34 |
OLM NOT OF FULL RANK | 39 |
2 THE QLD OF THE COEFFICIENT MATRIX | 40 |
21 SIMD IMPLEMENTATION | 41 |
TRIANGULARIZING THE LOWER TRAPEZOID | 43 |
32 THE GIVENS METHOD | 46 |
4 COMPUTING THE ORTHOGONAL MATRICES | 49 |
5 DISCUSSION | 54 |
UPDATING AND DOWNDATING THE OLM | 57 |
2 ADDING OBSERVATIONS | 58 |
21 THE HYBRID HOUSEHOLDER ALGORITHM | 60 |
22 THE BITONIC AND GREEDY GIVENS SEQUENCES | 67 |
23 UPDATING WITH A MATRIX HAVING A BLOCK LOWERTRIANGULAR STRUCTURE | 75 |
3 IMPLEMENTATION AND PERFORMANCE ANALYSIS | 111 |
SEEMINGLY UNRELATED REGRESSION EQUATIONS MODELS | 117 |
2 THE GENERALIZED LINEAR LEAST SQUARES METHOD | 121 |
3 TRIANGULAR SURE MODELS | 123 |
31 IMPLEMENTATION ASPECTS | 127 |
4 COVARIANCE RESTRICTIONS | 129 |
41 THE QLD OF THE BLOCK BIDIAGONAL MATRIX | 133 |
42 PARALLEL STRATEGIES | 138 |
43 COMMON EXOGENOUS VARIABLES | 140 |
SIMULTANEOUS EQUATIONS MODELS | 147 |
1 GENERALIZED LINEAR LEAST SQUARES | 149 |
11 ESTIMATING THE DISTURBANCE COVARIANCE MATRIX | 151 |
12 REDUNDANCIES | 152 |
13 INCONSISTENCIES | 153 |
2 MODIFYING THE SEM | 154 |
3 LINEAR EQUALITY CONSTRAINTS | 157 |
31 BASIS OF THE NULL SPACE AND DIRECT ELIMINATION METHODS | 158 |
4 COMPUTATIONAL STRATEGIES | 160 |
References | 163 |
| 177 | |
| 179 | |
Expressions et termes fréquents
3SLS estimator Algorithm 2.1 annihilation scheme argmin banded matrix bitonic algorithm block column-based column-layout computing the factorization computing the orthogonal computing the QRD constraints data parallel deleted denotes diagonal dimension downdating E. J. Kontoghiorghes Econometrics efficient equivalent execution exogenous factorization 2.4 Figure full column rank Givens rotations Givens sequence GLLSP Greedy sequence H₁ Householder algorithm Householder method Householder reflections Householder transformations implementation LALIB linear least squares linear model lower-triangular MasPar method for computing n₁ non-singular number of CDGRS orthogonal factorization orthogonal matrix parallel algorithms Parallel Computing Parallel strategies partitioned performance permutation matrix QR decomposition QR_FACTOR required to compute rows sequence for computing SIAM Journal SIMD SIMD systems simultaneous equations models SK sequence solution solving stage structure submatrix SURE model SURE-CC model tion total number triangular system UGSS updating upper triangular variables variance-covariance matrix vector
Fréquemment cités
Page 174 - VK Srivastava and DEA Giles. Seemingly Unrelated Regression Equations Models: Estimation and Inference (Statistics: Textbooks and Monographs), volume 80. Marcel Dekker, Inc., 1987.