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Handscomb D.C. — Methods of numerical approximation

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Название: Methods of numerical approximation

Автор: Handscomb D.C.

Аннотация:

Methods of numerical approximation lectures delivered at a Summer School held at Oxford University

Язык:

Рубрика: Математика/Численные методы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1966

Количество страниц: 218

Добавлена в каталог: 01.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

approximation      70
$\delta^2$ process      137
$\tau$ method      55
$\varepsilon$ algorithm      129
$\varepsilon$ algorithm, convergence of      138
$\varepsilon$ algorithm, relation to $\delta^2$ process      137
$\varepsilon$ algorithm, relation to Pade table      129
$\varepsilon$ algorithm, used to accelerate converhence      136
$\varepsilon$ -capacity      84
$\varepsilon$ -entropy      184
$\varepsilon$ -entropy, bounds on      186
Acceleration of convergence by a algorithm      136
Achieser, N.I.      7
Aitken, A.C.      137
Algebraic conjugate      170
Approximation of experimental data      4 27 166
Approximation of linear functionals      169
Approximation, best      4 11
Approximation, least-squares      11
Approximation, linear      11 15
Approximation, rational      99 ff.
Approximation, relative      196
Bernstein polynomials      17
Bernstein polynomials, practical value of      20
Best approximations      4 11
Best approximations existence      61
Best approximations uniqueness      64; see also Least-squares Minimax
Best approximations, convexity of set of      63
Blanch, G.      100
Bounded linear functional      170
Bounded linear functional, representation of      171
Buck, R.C.      7
Chebyshev norm      10
Chebyshev polynomials      21 37 42 47 117
Chebyshev polynomials, extrema of, used as reference      144
Chebyshev polynomials, interpolation at roots of      23
Chebyshev polynomials, minimax property of      44 53
Chebyshev polynomials, second kind      71
Chebyshev series      42 49
Chebyshev series for rational functions      54
Chebyshev series, compared with Fourier series      45
Chebyshev series, cornpardi with power series      53
Chebyshev series, determination of      51 ff.
Chebyshev series, evaluation of      49
Chebyshev series, integration of      50
Chebyshev set      67 174
Chebyshev set, approximation by      71
Chebyshev set, minimax approximation by      68 83
Chebyshev set, non-existence in many variables      192
Chebyshev, P.L.      3
Clenshaw, C.W.      56 58
complete      31
Computers, automatic      4 196
Continued fractions      99
Continued fractions, convergence of      101
Continued fractions, economization      117
Continued fractions, equivalent      100
Continued fractions, evaluation of      100
Continued fractions, even and odd parts of      100
Continued fractions, interpolation by      108
Continued fractions, S-fraction      100
Continued fractions, Thiete's expansion in      114
Continuity condition for convergence of minimax approximation      16
Continuity, modulus of      11 91 151 191
Convex hull      8
Convex set of functions      8
Curtis, A.R.      157 158
Curve-fitting      27 166
Davis, P.J.      7
Differences, divided      107
Differences, inverted      109
Differences, reciprocal      111
Discontinuity, effects of      16 41
Discrete problem, least-squares      32 42 57
Discrete problem, minimax      73
Economization of continued fraction      117
Economization of power series      53
Error bounds of optimal formulae      172 179
Error bounds, a priori and a posteriori      198
Error, absolute and relative      196
Essential supremum      10
Exchange algorithms      73 83 158
Exchange algorithms and linear prograxmuing      78
Exchange algorithms, convergence of      76 86 159
Experimental data, fitting of      4 166
Exponential interpolation      135
Extrema of error curve, search for      85 144
Extrema of error curve, sign-alternation of      64 84
Favard, J.      93
Fejer's theorem      16
floating-point numbers      196
Fourier coefficient      29 39
Fourier series      16 39
Fourier series, compared with Chebyshev series      45
Fourier series, compared with minimax approximation      95
Fourier series, pointwise convergence of      40
Function-space      7
Functional      9 169
Functional analysis      7
Functional, bilinear      177
Functional, linear      169
Functions, equal almost everywhere      10
Functions, orthogonal      28
Gauss quadrature      34
Golomb, M.      177
Gram — Schmidt process      31
Green's function      149
Guerra, S.      95
Haar's condition for Chebyshev set      67
Hammer, P.C.      7 194
Hankel determinant      126
Haselgrove, C.B.      194
Hermite interpolation      35
Hilbert space      177
Hoelder inequality      12 170
Hoelder norms      10
Hypercircle      178
Ill-conditioning of multivariable interpolation      193
Ill-conditioning of normal equations      58
Inequality, Hoelder's      12
Inequality, Minkowski's      10
Inequality, Schwarz's      12
Infinite-range approximations      156 197
Inner product      12 29 177
Interpolation at Chebyshev points      23 71
Interpolation by continued fractions      108
Interpolation by exponentials      135
Interpolation by polynomials, convergence of      23
Interpolation by rational functions      105
Interpolation by spline functions      165
Interpolation, Hermite      35
Interpolation, Lagrange      21 33
Jackson's theorem      92 151
Jackson's theorem in many variables      191
Kolmogorov, A.N.      190
Lagrange interpolation      21 33
Lagrange quadrature      34
Laguerre polynomials      198
Lanczos, C.      56
Least-squares approximation      11 27ff. 136 166
Legendre polynomials      37 44
Levelled reference error      84
Linear equations, over-determined      73

Linear functional      169
Linear programming and exchange algorithm      78
Linear vector space      8
LR transformation, acceleration of      135
LR transformation, relation to QR algorithm      134
Maehly, H. J.      84 117 141
Many variables, functions of      191 ff.
Markov set      67
Metric      9
Minimax approximation      11
Minimax approximation in many variables      191 ff.
Minimax approximation, general non-linear      155
Minimax approximation, general non-linear, characterization      157
Minimax approximation, general non-linear, convergence of exchange algorithm      159
Minimax approximation, general non-linear, existence      156
Minimax approximation, linear, by Chebyshev set      66
Minimax approximation, linear, by non-Chebyshev set      69
Minimax approximation, linear, by polynomials      64
Minimax approximation, linear, characterization      66
Minimax approximation, linear, compared with Fourier series      95
Minimax approximation, linear, convergence of      91
Minimax approximation, linear, determination of      73 83
Minimax approximation, linear, uniqueness of      65
Minimax approximation, rational      139
Minimax approximation, rational, convergence, to analytic function      150
Minimax approximation, rational, convergence, to continuous function      151
Minimax approximation, rational, existence, uniqueness, characterization of      141
Minimax approximation, rational, Maehly's algorithm for      141
Minimax solution of linear equations      76
Minkowski's inequality      10
Modulus of continuity      11 91 151
Modulus of continuity, partial      191
Montessus de Ballore, R. de      138 151
Natanson, I.P.      7
Newman, D.J.      151
Newton's process      133
Norm      9 29
Norm of linear functional      170
Norm, , $L_{\infty}$       10
Norm, Chebyshev      10
Norm, convergence in      11
Norm, dual      11
Norm, HoeIder      10
Norm, strict      9 63
Norm, weighted      10 27
Normal equations      27
Normal equations, ill-conditioning of      58
Normal Pade table      126
Normed conjugate      170
Optimal      169
Optimal, approximation      173 177
Orthogonal functions      28
Orthogonal polynomials      31
Orthogonal polynomials in many variables      194
Orthogonal polynomials over continuous and discrete ranges      32
Orthogonal polynomials, recurrence relation between      31
Over-determined linear equations      73
Pace, J.R.      7 71
Pade table      101 125
Pade table and $\varepsilon$ algorithm      129
Pade table, abnormal      130
Pade table, convergence in      138
Pade table, extended      129
Pade table, normal      126
Parseval identity      31
Periodic spline      167
Piesz, F.      171
Polynomials, Bernstein      17
Polynomials, Chebyshev      see under Chebyshev
Polynomials, Lagrangian      21
Polynomials, Laguerre      198
Polynomials, Legendm      37
Polynomials, trigonometric      16
Polynomials, ultraspherical      36
Power series, conversion to Chebyshev series      53
Prony, R. de      135
Quadrature formulae      33 169
Quadrature formulae, multiple      194
Quotient-difference (QD) algorithm      127
Quotient-difference (QD) algorithm, convergence      131
Quotient-difference (QD) algorithm, instability      133
Quotient-difference (QD) algorithm, relation to LR transformation      134
Quotient-difference (QD) algorithm, used in Prony's method      136
Quotient-difference (QD) algorithm, used to locate poles and zeros      131
Reference      76 83 141
Reference, error, levelled      84
Relative error      139 196
Representation of linear functionals      171
Representation of linear functionals on Hilbert space      177
Rhombus rules      127 129
Runge phenomenon      24
Rutishauser, H.      127 134
S-fractions      100 126
Sard, A.      171
Schoenberg, I.J.      64 166 177
Seminorm      9 163 171 180
Shapiro, H.S.      151
Simplex method      78
Simpson's rule      172
Smoothest approximation      163 166
Smoothest approximation in many variables      180 194
Space vector      8
Space,       10
Space, $L_{\infty}(X)$       10
Space, B(X)      8
Space, C(X)      8
Space, conjugate      170
Space, function      7
Space, inner-product      29
Spanned      8
Spline Function      163
Spline function, convergence of      166
Spline function, determination of      165
Spline function, equivalence of definitions      164
Spline function, multidimensional      180 194
Spline function, periodic      167
Spline function, uniqueness of      164
Spline function, used to smooth data      166
Sprecher, D.A.      190
Spurious singularities in rational approximations      107 143 150
Staircase in Pade table      126
Steklov's transformation      95
Stiefel, E.L.      78
Stone, M.H.      191
Strict norm      9 63
Thiele's expansion      101 114
Timan, A.F.      7
Trigonometric approximations, uniform convergence of      16 92;
Tschebyseheff (etc.)      see Chebyshev
Ultraspherical polynomials      36
Uniform norm      10
Uniqueness of best approximations in strict norm      64
Uniqueness of minimax approximation      65
Uniqueness of minimax rational approximation      141
Van Vleck, E.B.      101
Vector space      8
Vitushikin, A.G.      183
Walsh, J.L.      149
Weierstrass's theorem      16 197
Weierstrass's theorem in many variables      191
Weierstrass's theorem, proof of      19
Weights      10 27 139
Wynn, P.      129
z-transform      135

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