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| Stewart G.W. — Afternotes on Numerical Analysis |
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| Предметный указатель |
Absolute error 7
Absolute error as convergence criterion 7
Absolute value as a norm 113
Astrologer 150
Back substitution 100
Backward error analysis 55—57 125 128
Backward error analysis, backward stability 55
Backward error analysis, Gaussian elimination 122—124
Backward error analysis, stable algorithm 55
Backward error analysis, sum of n numbers 53—55
Backward error analysis, sum of two numbers 50—51
Backward stablity see "Backward error analysis"
Balanced matrix 120 136
Base see "Floating-point arithmetic"
Basic linear algebra subprogram see "BLAS"
Binary arithmetic see "Floating-point arithmetic"
BLAS 86—88
BLAS in Gaussian elimination 108—109
BLAS, axpy 87 98n 109
BLAS, copy 88
BLAS, dot 87
BLAS, ger 109
BLAS, imax 108
BLAS, level-three 109n
BLAS, level-two 109
BLAS, seal 88 109
BLAS, swap 108
Bracket for a root see "Nonlinear equations"
C 69 80n 86
C, storage of arrays 83 109
Cache memory 86
Cancellation 61—63 65 176
Cancellation in a sum of three numbers 61
Cancellation in ill-conditioned problems 63
Cancellation in the quadratic formula 62
Cancellation, death before cancellation 61—62 104
Cancellation, recovery from 62—63 65
Cancellation, revealing loss of information 61 62
Characteristic see "Floating-point arithmetic"
Chebyshev points 152
Cholesky algorithm 90—95 98
Cholesky algorithm and rounding error 98
Cholesky algorithm and solution of linear systems 95
Cholesky algorithm, Cholesky decomposition 90
Cholesky algorithm, Cholesky factor 90
Cholesky algorithm, Cholesky factorization 90
Cholesky algorithm, column orientation 94—95
Cholesky algorithm, economization of operations 92
Cholesky algorithm, economization of storage 92
Cholesky algorithm, equivalence of inner and outer product forms 98
Cholesky algorithm, implementation 92—95
Cholesky algorithm, inner product form 97—98
Cholesky algorithm, operation count 95 98
Cholesky algorithm, relation to Gaussian elimination 91—92
Column index see "Matrix"
Column orientation 84—85 87 94 98n
Column orientation and level-two BLAS 109
Column orientation, general observations 86
Column vector see "Vector"
Component of a vector see "Vector"
Computer arithmetic see "Floating-point arithmetic"
Condition number see "Condition"
Condition, artificial ill-conditioning 120—121
Condition, condition number 41 42
Condition, condition number with respect to inversion 120
Condition, generating matrices of known condition 129
Condition, ill-conditioning 41 51 57 63 65
Condition, ill-conditioning of Vandermonde 136
Condition, ill-conditioning revealed by cancellation 63
Condition, linear system 119—120 169
Condition, roots of nonlinear equations 41—42
Condition, roots of quadratic equations 63
Condition, sum of n numbers 57
Condition, well-conditioning 41
Conformity see "Partitioning" "Product" "Sum"
Constant slope method 17 22
Constant slope method as successive-substitution method 23
Constant slope method, convergence analysis 18—19
Constant slope method, failure 19
Constant slope method, linear convergence 19
Convergence see "Convergence of order p" "Cubic etc.
Convergence of order p 20 22
Convergence of order p and significant figures 20
Convergence of order p, limitations for large p 21
Convergence of order p, multipoint methods 33
Convergence of order p, noninteger orders 21
Convergence of order p, two-point methods 32
Cramer's rule 127 130
Cramer's rule, compared with Gaussian elimination 130—131
Cubic convergence 20 21
Decimal arithmetic see "Floating-point arithmetic"
Determinant 77
Determinant, unsuitability as measure of singularity 77—78
Diagonal matrix 71
Diagonal matrix, two-norm of 127
Difference equation 32
Difference equation, effects of rounding error 63—65
Difference quotient 27
Differentiation, compared with integration 157 181
Dimension of a vector see "Vector"
Divided difference 144—146
Divided difference and error in interpolant 148
Divided difference as coefficient of Newton interpolant 144
Divided difference, computation 145—146
Divided difference, operation count 146
Divided difference, relation to derivative 149
Economist see "Astrologer"
Element of a matrix see "Matrix"
Elementary permutation 105
Error bounds for floating-point operations see "Rounding error"
Error function 157
Euclidean length of a vector see "Norm" "Two-norm"
EXPONENT see "Floating-point arithmetic"
Fixed point 21
Fixed-point number 45
Floating-point arithmetic, advantage over fixed-point arithmetic 45
Floating-point arithmetic, avoidance of overflow 47—48
Floating-point arithmetic, base 45—46
Floating-point arithmetic, binary 46 49
Floating-point arithmetic, characteristic 45n
Floating-point arithmetic, decimal 46
Floating-point arithmetic, double precision 46
Floating-point arithmetic, error bounds for floating-point operations see "Rounding error"
Floating-point arithmetic, exponent exception 47
Floating-point arithmetic, floating-point number 45—46
Floating-point arithmetic, floating-point operations 49—50
Floating-point arithmetic, fraction 45
Floating-point arithmetic, guard digit 50
Floating-point arithmetic, hexadecimal 46
Floating-point arithmetic, high relative error in subtraction 50
Floating-point arithmetic, IEEE standard 46 47 50
Floating-point arithmetic, mantissa 45n
Floating-point arithmetic, normalization 45—46
Floating-point arithmetic, overflow 47—48 141 144
Floating-point arithmetic, quadruple precision 46
Floating-point arithmetic, rounding error qv
Floating-point arithmetic, single precision 46
Floating-point arithmetic, underflow 47—48 141 144
Floating-point arithmetic, unnormalized number 45
FORTRAN 94
FORTRAN, storage of arrays 84 109
Forward-substitution algorithm 80
FRACTION see "Floating-point arithmetic"
GAMS 187
Gauss, C.F. 31 169 176n
Gaussian elimination 92 98—100 108 127 128 135
Gaussian elimination and LU decomposition 100—102
Gaussian elimination and the Cholesky algorithm 91—92
Gaussian elimination for Hessenberg matrices 110—111
Gaussian elimination for tridiagonal matrices 111
| Gaussian elimination, back substitution 100
Gaussian elimination, backward error analysis 122—124
Gaussian elimination, BLAS in implementation 108—109
Gaussian elimination, compared with Cramer's rule 130—131
Gaussian elimination, compared with invert-and-multiply 130
Gaussian elimination, complete pivoting 105n
Gaussian elimination, exponential growth 125—126
Gaussian elimination, general assessment 126
Gaussian elimination, growth factor 125
Gaussian elimination, implementation 102
Gaussian elimination, multiplier 99 101—102 122 123
Gaussian elimination, operation count 102 111
Gaussian elimination, partial pivoting 104—108 125
Gaussian elimination, pivot 103
Gaussian elimination, pivoting 103—108 111
Gaussian elimination, stability 125
Gaussian quadrature 167
Gaussian quadrature, convergence 176
Gaussian quadrature, derivation 175
Gaussian quadrature, error formula 176
Gaussian quadrature, Gauss — Hermite quadrature 177
Gaussian quadrature, Gauss — Laguerre quadrature 176
Gaussian quadrature, Gauss — Legendre quadrature 176
Gaussian quadrature, introduction 169
Gaussian quadrature, positivity of coefficients 175—176
Guard digit see "Floating-point arithmetic"
Hessenberg matrix 110
Hessenberg matrix and Gaussian elimination 110—111
Hexadecimal arithmetic see "Floating-point arithmetic"
Identity matrix see "Matrix"
IEEE standard see "Floating-point arithmetic"
Ill-conditioning 123
Inner product 98n 171
Inner product, Cholesky algorithm 97—98
Inner product, computed by BLAS 87
Integral, definite 157
Integral, indefinite 157
Integration, compared with differentiation 157 181
Intermediate Value Theorem 4
Interpolation 35
Interpolatory method 34
Interval bisection 4—6
Interval bisection, combined with secant method 37—40
Interval bisection, convergence 6
Interval bisection, implementation 5—6
Inverse matrix 78 105 120n 130
Inverse matrix and linear systems 78
Inverse matrix of a product 78
Inverse matrix of a transpose 78
Inverse matrix, calculation of 95
Inverse matrix, lower triangular matrix 101
Invert-and-multiply algorithm see "Linear system"
Lagrange interpolation 144
Lagrange polynomials 137—138 161
Least squares 89 176
Legendre, A.M. 176n
linear combination 75
Linear convergence 19 20 22
Linear convergence, rate of convergence 20
Linear equations see "Linear system"
Linear fractional method 34—36
Linear independence 77
Linear system 77 104
Linear system and partial pivoting 105—108
Linear system and relative residual 128—129
Linear system and rounding error 120
Linear system, artificial ill-conditioning 120—121
Linear system, condition 119—120
Linear system, Cramer's rule qv
Linear system, existence of solutions 77
Linear system, invert-and-multiply algorithm 78 95 127 130
Linear system, lower triangular system 79—81 84—85 90 97
Linear system, matrix representation 72
Linear system, nonsymrnetric system 98
Linear system, operation count qv
Linear system, perturbation analysis 116—117
Linear system, positive-definite system 89 95
Linear system, solution by LU decomposition 78—79 89
Linear system, triangular system 95
Linear system, uniqueness of solutions 77
locality of reference 83 85 86
Lower triangular matrix 79
Lower triangular matrix, inverse 101
Lower triangular matrix, product 101
Lower triangular matrix, unit lower triangular matrix 102
Lower triangular system nonsingularity 143
LU decomposition 79 98 102 120n 123
LU decomposition and Gaussian elimination 100—102
LU decomposition with partial pivoting 106—107
LU decomposition, existence 103
LU decomposition, solution of linear systems 78—79 89
Mantissa see "Floating-point arithmetic"
MATLAB 127
Matrix 69
Matrix perturbation theory 116
Matrix, column index 69
Matrix, diagonal matrix qv
Matrix, element 69
Matrix, identity matrix 71
Matrix, inverse matrix qv
Matrix, lower triangular matrix qv
Matrix, nonsingular matrix qv
Matrix, normally distributed matrix 129
Matrix, operations see "Multiplication by a scalar" "Sum" "Product" "Transpose"
Matrix, order 69
Matrix, orthogonal matrix qv
Matrix, partitioning qv
Matrix, positive definite matrix qv
Matrix, rank-one matrix qv
Matrix, represented by upper-case letters 70
Matrix, row index 69
Matrix, symmetric matrix 89
Matrix, unit lower triangular matrix 102
Matrix, upper triangular matrix 79
Matrix, Vandermonde matrix qv
Matrix, zero matrix 70
Monic polynomial 172
Muller's method 33—34 135
Multiple zero 24
Multiple zero, behavior of Newton's method 24
Multiplication by a scalar and norms 113
Multiplication by a scalar and transposition 72
Multiplication by a scalar, matrix 70
Multiplication by a scalar, vector 71
Multipoint method 33 34 135
Multipoint method, rate of convergence 33
Natural basis interpolation 135—137 141
Natural basis interpolation, evaluation by synthetic division 141—142
Natural basis interpolation, ill-conditioning of Vandermonde 136
Natural basis interpolation, Vandermonde matrix qv
NETLIB 187
Newton Interpolation 142 144
Newton interpolation, addition of new points 144
Newton interpolation, coefficients as divided differences 144
Newton interpolation, computation of coefficients 145—146
Newton interpolation, evaluation by synthetic division 142
Newton interpolation, existence and uniqueness 143—144
Newton's method 9—15 22 34
Newton's method as confluent case of secant method 29
Newton's method as successive-substitution method 23
Newton's method, analytic derivation 10
Newton's method, calculating reciprocals 11
Newton's method, calculating square roots 11
Newton's method, convergence analysis 12—14 19
Newton's method, convergence to multiple zero 24
Newton's method, derivative evaluation 4 17
Newton's method, divergence 11
Newton's method, failure 37
Newton's method, geometric derivation 9
Newton's method, quadratic convergence 14 19 23
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