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Stewart G.W. — Afternotes on Numerical Analysis
Stewart G.W. — Afternotes on Numerical Analysis

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Название: Afternotes on Numerical Analysis

Автор: Stewart G.W.


There are many textbooks to choose from when teaching an introductory numerical analysis course, but there is only one Afternotes on Numerical Analysis. This book presents the central ideas of modern numerical analysis in a vivid and straightforward fashion with a minimum of fuss and formality. Stewart designed this volume while teaching an upper-division course in introductory numerical analysis. To clarify what he was teaching, he wrote down each lecture immediately after it was given. The result reflects the wit, insight, and verbal craftmanship which are hallmarks of the author. Simple examples are used to introduce each topic, then the author quickly moves on to the discussion of important methods and techniques. With its rich mixture of graphs and code segments, the book provides insights and advice that help the reader avoid the many pitfalls in numerical computation that can easily trap an unwary beginner.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1987

Количество страниц: 210

Добавлена в каталог: 28.10.2010

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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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