Hans Rudolf Schwarz | Norbert Köckler Numerische Mathematik
Hans Rudolf Schwarz | Norbert Köckler
Numerische Mathematik 7., überarbeitete Auflage STUDIUM
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Prof. Dr. Hans Rudolf Schwarz Universität Zürich Mathematisch-Naturwissenschaftliche Fakultät (MNF) Winterthurerstrasse 190 8057 Zürich Prof. Dr. Norbert Köckler Universität Paderborn Fakultät EIM – Institut für Mathematik 33098 Paderborn
[email protected] 1. Auflage 1986 6., überarbeitete Auflage 2006 7., überarbeitete Auflage 2009 Alle Rechte vorbehalten © Vieweg +Teubner |GWV Fachverlage GmbH, Wiesbaden 2009 Lektorat: Ulrike Schmickler-Hirzebruch |Susanne Jahnel Vieweg +Teubner ist Teil der Fachverlagsgruppe Springer Science+Business Media. www.viewegteubner.de Das Werk einschließlich aller seiner Teile ist urheberrechtlich geschützt. Jede Verwertung außerhalb der engen Grenzen des Urheberrechtsgesetzes ist ohne Zustimmung des Verlags unzulässig und strafbar. Das gilt insbesondere für Vervielfältigungen, Übersetzungen, Mikroverfilmungen und die Einspeicherung und Verarbeitung in elektronischen Systemen. Die Wiedergabe von Gebrauchsnamen, Handelsnamen, Warenbezeichnungen usw. in diesem Werk berechtigt auch ohne besondere Kennzeichnung nicht zu der Annahme, dass solche Namen im Sinne der Warenzeichen- und Markenschutz-Gesetzgebung als frei zu betrachten wären und daher von jedermann benutzt werden dürften. Umschlaggestaltung: KünkelLopka Medienentwicklung, Heidelberg Druck und buchbinderische Verarbeitung: MercedesDruck, Berlin Gedruckt auf säurefreiem und chlorfrei gebleichtem Papier. Printed in Germany ISBN 978-3-8348-0683-3
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7
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x2 1.3182 1.2654
1 6.8541 3.5267
−→
x1 1.2547 0
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l ) x1 2.1 −1.3 0.9
x2 2512 8.8 −6.2
x3 −2516 −7.6 4.6
1 6.5 −5.3 2.9
x1 2.1 −0.61905 0.42857
−→
x2 2512 1563.9 −1082.8
x3 −2516 −1565.1 1082.9
1 6.5 −1.2762 0.11430
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# .! 2+ . * 3 4 1 #! 4 & ( 5 678 . ai1 a1k " ) $ * -
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x2 2512 1563.9 −0.69237
a ˜ik n |˜ aik | ≈ 1, i = 1, 2, . . . , n. !" #$ k=1
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1 0.0012922 −0.29944 0.24786
! a31 2+ ( & +
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1 0.24786 −0.062780 −0.000053000
! x1 0.076923 0.0054274 −0.95480
x2 −0.52991 0.50227 −0.017481
x3 0.39316 −0.50232 −0.062771
1 0.24786 −0.000053000 −0.062781
. . . "# ! $ x3 = 1.0002, x2 = 1.0002 x1 = 5.0003 %& ' $# ( )! ! * ! &
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j=k
8 p = k % * p, 5 k , 5 ! " ! 9 & lik (i > k) '
* + , ! $#$ " ( ) " ! $ ! n X (k−1) *& si = |aij | j=k
qi = |a(k−1) |/si ik a31 ! " # $ x1 2.1 −1.3 0.9
x2 2512 8.8 −6.2
x3 −2516 −7.6 4.6
1 6.5 −5.3 2.9
si 5030.1 17.7 11.7
qi 0.00041749 0.073446 0.076923
x1 0.9 −1.4444 2.3333
x2 −6.2 −0.15530 2526.5
x3 4.6 −0.95580 −2526.7
x1 0.9 2.3333 −1.4444
x2 −6.2 2526.5 −0.000061468
1 2.9 −1.1112 −0.26660
x3 4.6 −2526.7 −1.1111
si − 1.1111 5053.2
qi − 0.13977 0.49998
1 2.9 −0.26660 −1.1112
. . . x3 = 1.0001 x2 = 1.0001 x1 = 5.0001 . 2 A !"# |A| = (−1) × 0.9 × 2526.5 × (−1.1111) = −2526.5 $ |A| = −2526.504
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k '
% ( pk = k 3 2 6 7 ) " ) $ + 2 (
% 8 % c b 9" bi - ci & " (% b & , $ #$% :- % x c ; b max : max = q; pk = i max = 0 : pk = k : det = − det j = 1, 2, . . . , n : h = akj ; akj = apk ,j ; apk ,j = h det = det ×akk i = k + 1, k + 2, . . . n : aik = aik /akk j = k + 1, k + 2, . . . , n : aij = aij − aik × akj det = det ×ann
pk = k : h = bk ; bk = bpk ; bpk = h
i = 1, 2, . . . , n : ci = b i
j = 1, 2, . . . , i − 1 : ci = ci − aij × cj
i = n, n − 1, . . . , 1 : s = ci
k = i + 1, i + 2, . . . , n : s = s − aik × xk xi = s/aii
!
! !
! m " # $ $ b1 % b2 , . . .% bm ! & % ! $' ! () *+ ,-
B = (b1 , b2 , . . . , bm ) ∈ Rn,m
!$$
() X ∈ R AX = B
n,m
.& (! *+ /-
0 X .& xμ ! 0 bμ
1 Z = (n3 − n) + mn2 . 3 ! "!## $ !% ! ! & # '!( A )! * X = A−1 +## ! '! #! ,
AX = I, −1
I ! )! - * A ! .! ! #!!! n /#! .! # '!( A +%+ ) & . #!"#!%! 0"! # ! 4 1 ZÁÒÚ = n3 − n. 1 3 3 )! ! ##! ! +% !!2 3!# ". ! ! I & # ! #. ## ) 4 5 ! ! # !. 4! !" !. 6!#! ! #. !
2 .! ! %! & "! )! '#!%! ! ! " .. % . )! - A−1 ! $ . 7"!"# + A 6 ! + I 2 7## % ! ! * ! ) 7"! ##! 2n2 4#
. /#! 8 . Ax = b .! .! . / 9&$# !. 2 # . / .!#! # ## (% % ˜ )! ! " ! /#! ! !. $##.! x ! x ## ## % ! r := A˜ x − b.
:
˜ ## ! (% x .! ;!# $ . %
% x
x=x ˜+z
<
.!# ) = % % z ! !.. 2 ! /#! +## ! 2 !# Ax − b = A(˜ x + z) − b = A˜ x + Az − b = 0.
- . ! ! /#! : 2 %
. 2 = % % z /#! 8 . Az = −r
.! # '!( A2 7! −r +## . )! = % z ! ! .! ! 4 5 & +% ! ! . ! & % r2 .! ""# / !%! . 2 ! -! "!#
! "# 0.29412x1 0.42857x1 0.36842x1 0.38462x1
+ 0.41176x2 + 0.57143x2 + 0.52632x2 + 0.53846x2
+ 0.52941x3 + 0.71429x3 + 0.42105x3 + 0.46154x3
+ 0.58824x4 + 0.64286x4 + 0.36842x4 + 0.38462x4
= 0.17642 = 0.21431 = 0.15792 = 0.15380
$ % ! & A %# ' & ( ) ( % % si " & * qi x1
x2
x3
x4
0.29412 0.42857 0.36842 0.38462
0.41176 0.57143 0.52632 0.53846
0.52941 0.71429 0.42105 0.46154
0.58824 0.64286 0.36842 0.38462
si 1.8235 2.3572 1.6842 1.7692
x1
x2
x3
x4
0.36842
0.52632
0.42105
0.36842
1.1633 0.79833 1.0440
−0.040840 −0.0084200 −0.011020
0.22448 0.19327 0.021960
x1 0.36842 1.0440 0.79833 1.1633 x1 0.36842 1.0440 1.1633 0.79833
x2 0.52632 −0.011020 0.76407 3.7060 x2 0.52632 −0.011020 3.7060 0.76407
x3 0.42105 0.021960 0.17649 0.14310 x3 0.42105 0.021960 0.14310 1.2333
0.21428 0.29412 −0.00001 x4
0.36842 −0.00001 0.29413 0.21432
si − 0.47960 0.49581 0.032980 si − − 0.47062 0.35742
qi 0.16129 0.18181 0.21875 0.21740 qi − 0.085154 0.016982 0.33414 qi − − 0.37502 0.40037
x4 0.36842 −0.00001 0.21432 0.029810
+ ,-.
. " / ( |A| = (−1)3 × 0.36842 × . (−0.011020) × 0.14310 × 0.029810 = 1.7319 × 10−5 $ % b / ' # ! ( 0( 1 1 P b = (0.15792 . 0.15380 0.21431 0.17642)T ! 2 1 c = (0.15792 −0.011070 . T ˜ = (−7.9333 0.071625 −0.029527) )( x T 4.9593 1.9841 −0.99051) . 2!# ( ) ) r˜ = (2, 3, −3, 7)T × 10−5 ! ( /3 ) . r = (2.3951, 7.1948, −4.5999, 5.0390)T × 10−5
r˜
r˜ ! r = Ax ˜ − b " # $ ! %& '() * + # $ + P r = (−4.5999, 5.0390, 7.1948, 2.3951)T × 10−5
. cr = (4.5999, −9.8413, 23.926, −28.056)T × 10−5 ,
, ! . z = (0.066142, 0.040360, 0.015768, −0.0094116)T . ˜ ˜ x z # x ˜+z =x ˜ # " + ." x $$ / 0 1 $ . ˜= ˜ x (−7.9994, 4.9997, 1.9999, −0.99992)T
x = (−8, 5, 2, −1)T # . ! r = (4.7062 6.5713 5.0525 5.3850)T × 10−5 ." 2 $ 3$ 4 ! r # 5 , $ # ˜ ˜ # x ˜ " # " " x 4 5 ! "5 , #
x ˜ Ax = b ! " # $ ! % x ˜ −x &
' & ( ' & ')#
* % " ( x ∈ Rn "
'& A ∈ Rn,n #
$ ( x ( x ∈ R
" % +* , &n
a)
x ≥ 0 x, x = 0 x = 0;
b) c)
cx = |c| · x c ∈ R x; x + y ≤ x + y x, y .3 1.
./#001 ./#021 ./#01
,
x∞ := max |xk |, x2 :=
k
n
12 x2k
k=1
x1 :=
n
(L1 ).
|xk |,
!
k=1
"#$% & '% ( (" ) * +, & $- (" x ∈ Rn $ . % % % √ 1 √ x2 ≤ x∞ ≤ x2 ≤ nx∞ , n 1 x1 ≤ x∞ ≤ x1 ≤ nx∞ , n √ 1 √ x1 ≤ x2 ≤ x1 ≤ nx2 . n
. A A ∈ Rn,n / ' & - '% ( (-0 a)
A ≥ 0 (" A, A = 0 (" A = 0;
b) c)
cA = |c| · A (" c ∈ R A; A + B ≤ A + B (" A, B ) % %;
d)
A · B ≤ A · B.
1 2
) %( '% ( + ( (" 3 - % - % 4 %#+ 5
|aik |,
6
|aik |,
*
7
AG := n · max |aik |, i,k
Az := max i
⎡ AF := ⎣
k=1
As := max k
n
n i=1
n
⎤ 12
a2ik ⎦ ,
/ #
i,k=1
) %%# '% ( 1& 2 (" & 8 ) '% ( - - (" 5
n n A · BG = n · max aij bjk ≤ n · max |aij | · |bjk | i,k i,k j=1 j=1 ≤ =
n · max i,k
n
{max |alm |} · {max |brs |}
j=1
r,s
l,m
n2 · {max |alm |} · {max |brs |} = AG · BG . r,s
l,m
! " A ∈ Rn,n # 1 AG ≤ Az,s ≤ AG ≤ nAz,s , n 1 AG ≤ AF ≤ AG ≤ nAF . n $ " % &
" !" ' & % !
( A ) & x% # Ax ≤ A x x ∈ Rn A ∈ Rn,n .
*+ ,-
. / AG Az ! x∞ ; AG As ! x1 ;
*+ 01*+ 02*+ 0+-
AG AF ! x2 .
/! " " 3 n n Ax∞ = max aik xk ≤ max |aik | · |xk | i i k=1 k=1 n [max |ars |] · [max |xl |] = AG · x∞ ≤ max i
k=1
r,s
l
4 ! &/ # 5 3" # & ⎡ ⎤1 n n 12 n n 2 2 n 2 2 ⎣ ⎦ Ax2 = aik xk ≤ aik xk i=1
=
i=1
k=1
n n i=1 k=1
12 a2ik
n k=1
k=1
12 x2k
= AF · x2 .
k=1
!" # $ " % & ' ()
() * " x = 0 + ("
& " "
,"
% A := max x=0
* " " & -
Ax = max Ax x x=1
./
" # $ " " " 00 "
%
x
! " # $ ! % & $ ! % ' " # !
( !
1 ! &% 2 ( # $ "
# x = 0 Ax ≥ 0 () A ∈ Rn,n
x > 0 3" max Ax/x ≥ x=0 0 1 % % ' A = 0 A = 0 (" 1 + '
A = 0
$
apq = 0 3) x 4 q 5
2 ! " eq = 0' () Aeq = 0 3) * " Aeq /eq > 0 # $ ./ ' " 1 0 !" ( +
% 2 ( * " " cA := max cAx = max {|c| · Ax} = |c| · A. x=1
x=1
6 % () * " " (" A + B
:= ≤
max (A + B)x ≤ max {Ax + Bx}
x=1
x=1
max Ax + max Bx = A + B.
x=1
x=1
0 ! 4 ," % ' % A = 0
B = 0 !"
( 7 ! ()
A · B := ≤ ≤
max x=0
ABx = x
max
x = 0 Bx = 0
A(Bx) Bx Bx x
A(Bx) Bx · max Bx=0 x=0 x Bx Ay Bx · max = A · B. max y=0 y x=0 x max
!" # $ % & # '() ()" x = 0 " Ax = A · x
*+ , x∞ # A∞ () n A∞ := max Ax∞ = max max aik xk i x=1 x=1 k=1 n n = max max aik xk = max |aik | = Az . i i x=1 k=1
k=1
- & . /0 / i 1+ " /
xk = (aik ) $/ * % & # )
2 " % () 3 # () x2 #)1 0 () A2 )# " /
4) $ / () 1
1
A2 := max Ax2 = max {(Ax)T (Ax)} 2 = max {xT AT Ax} 2 x2 =1
x2 =1
x2 =1
# $( / AT A ' 5 () % " . /0 #)1 6 () Q(x) := xT (AT A)x ≥ 0 /0
x = 0 () 7. μi % AT A
() %" n 7% x1 , x2 , . . . , xn %
" ) - Rn μi ∈ R,
AT Axi = μi xi ,
μi ≥ 0;
8!
xTi xj = δij
x ∈ Rn
9 7% xi x=
n
!
ci x i
i=1
()
-0(() % 8! ⎛ ⎞ n T n xT AT Ax = ci xi AT A ⎝ cj xj ⎠ i=1
=
n i=1
T
ci xi
⎛ ⎝
j=1 n j=1
⎞
cj μj xj ⎠ =
n
c2i μi .
i=1
7. μi *1+ () " μ1 ≥ μ2 ≥ . . . ≥ μn ≥ 0
x2 = 1
n
c2i = 1
i=1
A2 = max
x2 =1
n
12
≤ max
c2i μi
x2 =1
i=1
μ1
n
12 c2i
=
√ μ1 .
i=1
√ μ1 x = x1 c1 = 1, c2 = . . . = cn = 0 ! √ A2 := max Ax2 = μ1 , " #$% x2 =1
μ1 & ! ' AT A ( )( A2 * + ,, ( ( )( '- +.( '- +. A λ1 , λ2 , . . . , λn ! ' A AT A = AA = A2 ( ! μi = λ2i ≥ 0 " #$% A2 = |λ1 |,
" ##%
|λ1 | = max |λi |. i
+.( / A & ! λ1 ) +.( √ 0 ' A−1 - A * " #$% A−1 2 = ψ1 , ψ1 T & ! ' A−1 A−1 = (AAT )−1 ' C −1 ( .( ! ' C ψ1 .( ( ". ' % ! . ' 1 AAT - AT A A−1 (AAT )A = AT A AAT AT A ! √ A−1 2 = 1/ μn , " #2% μn ( ! . ' 1 AT A 3 / - A−1 2 = 1/|λn |,
AT = A,
|λn | = min |λi |. i
" #4%
3 5 ( *- x ˜ 6 x 7 - +/ Ax = b 8 9 : ;( 5 & ; '( r = A˜ x − b 3 z := x − x ˜ ( A x '- )( " 0 i = 1, 2, . . . , n; a2ik < aii akk i = k; i, k = 1, 2, . . . n;
k
max |aij | = akk . i,j
x = 0 ! " x = ei i# $ Q(x) = aii > 0 % x = ξei +ek & ξ ∈ R & i = k & ' ( " Q(x) aii ξ 2 + 2aik ξ + akk > 0 ) ξ ∈ R ' aii ξ 2 + 2aik ξ + akk = 0 $ *+ ξ & $ " 4a2ik − 4aii akk < 0& , "" " - & +- . / "
& 0" "
1 ) 2 3"" # . / " " . 4 $ ' ( " !"" 5 ) a11 > 0 " & % A ! 67 2 ( 8 " & x1 & " % 5 % 9 n n n Q(x) = a11 x21 + 2 ai1 x1 xi + aik xi xk i=2
=
=
i=1
l11
=
(1)
=
aik
i=2 k=2
2 $ n # n n √ ai1 ai1 a1k aik − xi xk a11 x1 + xi + √ a11 a11 i=2 i=2 k=2 2 2 n n n n (1) li1 xi + aik xi xk = li1 xi + Q(1) (x(1) ) i=2 k=2
6
i=1
√ ai1 ai1 a11 ; li1 = √ = , i = 2, 3, . . . , n; a11 l11 ai1 a1k aik − = aik − li1 lk1 , i, k = 2, 3, . . . , n. a11
:
Q(1) (x(1) ) ' ( " (n − 1) ; x2 , x3 , . . . , xn " " a11 ! + . / 3 "
!
(1)
A
A = (aik ) a11 > 0 (1) (1) = (aik ) ∈ R(n−1),(n−1) aik "#
$% &'
()* A 2 ? @ " ; $ x(1) = (x2 , x3 , . . . , xn )T = 0 $
x1
a11 > 0 l11 = 0
n
li1 xi = 0
i=1
x = (x1 , x2 , . . . , xn )T = 0 0 < Q(1) (x(1) ) !
A(1) " # $
% & A(1) " # $ ' x = 0 Q(x) ≥ 0 & Q(x) = 0 '
! # ! ( Q(1) (x(1) ) = 0 ) x2 = x3 = . . . = xn = 0 '
# ! l11 = 0 x1 = 0 * + , A " # $ ( ' (
" ! - # . /
! " # $ '. #
A = (aik ) ∈ Rn,n
!" #$ %
& % 0 A " # $ a11 > 0 1 # * * + , A(1) ! ' (1) " # $ a22 > 0 2
1 # * (n−1) + A(k) k = 2, 3, . . . , n − 1 ! ann 1 # " # (1)
(n−1)
' 1 # a11 > 0, a22 > 0, . . . , ann > 0 + , (n−1) A(n−1) = (ann ) " # $
# 23 ( # ' ! + A(n−2) , . . . , A(1) , A " # $ 4 ! ' 5 6
.
! " # $ + A * ! 7 3/( 8 # ! ! * ! 9 + 7 !
. .
! :!" , : :! 8 ;2
A = (aik ) ∈ Rn,n ' ( ) Q(x) * n + n 2 n n n Q(x) = aik xi xk = lik xi ! ! &2 9 73 2 k / :
!
lkk (k)
aij
% (k−1) = akk ; (k−1)
= aij
(k−1)
lik =
− lik ljk ,
aik , lkk
i = k + 1, k + 2, . . . , n,
i, j = k + 1, k + 2, . . . , n.
A
! "#$% lik & # '# i ≥ k '# & # # &# (#)#* ⎞ ⎛ l11 0 0 ... 0 ⎜ l21 l22 0 ... 0 ⎟ ⎟ ⎜ ⎜ l31 l32 l33 . . . 0 ⎟ L=⎜ + ⎟ ⎟ ⎜
⎠ ⎝
ln1
ln2
ln3
. . . lnn
! " " !"# " $ % A
,
A = LLT .
&' - . / 0# 1 #) Q(x) & # # ) # (#)#* L + Q(x) = xT Ax = (LT x)T (LT x) = xT LLT x.
2 # 3 # # ,
! , ( )*+ " " # 4))# !#* A
" 5 ( ) 67 .8 #'
! 9 # : 4; m.
( )*+,
# m %
L A = LL
!" # m $ lik = 0 %& i k i − k > m. ( )*-, T
' .& ( /, ( /0, 1 L m (1)
% A(1) = (aik ) # m # 2 ( /, li1 = 0 i i − 1 > m 3 ai1 = 0 (1) # 1! aik 1 (i, k) i ≥ k ≥ 2 i − k > m 3 aik = 0 4 li1 = 0 i − 1 > i − k > m ( /0, 5 (1) aik = 0 i, k ≥ 2 |i − k| > m 1 * 6 7 &!8' ! " #8 % 6 m % L 9 5 % A & ' & : .& # & # ; m ' ' 3 ) .& # % m = 4 . .& 7 "$ 6 ? 3 AA B8 CDE 4 % " @' !
. 7 % F B% A/ > C/ 8 CD/ 8 CC/ = C/ = CA/ CG/ ( CA / @ HH/ 0 C/ # HH/ ". CAE - = $ )! % 7 4F * $ 0' % ( B C E
! " # $ % & '# ( ) * + ,) " %% - ../$ % 0* $ $ $
' 1 2# ) # !) ) * + 2#
! % ! 3
# ) * 45 3 ) ) 1# % $ ) LR 6 -7 . / % * ) ! P × Q " '# # 8 * 9 , ' % ,& % $ ) ) * % % % ) $ 6, % ) 5 + ! * ' mb × nb 1 6 * 1 % '# ) % " * + ) 5
2 × 3" 12 × 12 1 %%* 0*0 % % * + ! % % 1 : #
'# * + 3 %% ;
" $ # 1 * + %% ,? @
* ! * " " " ! ! ! 1 1 " 2 1 ! +" 1
* " !"" ! ! ; ( ' 'E F '
/ / A / B / B C ! " * ! ! LR') 345557 > A + RQ') *
! ! R Q ! - ! & " ! R Q
n = 8 ⎛ 1 ⎜ r2 1 ⎜ ⎜ r3 1 ⎜ ⎜ r4 1 r5 ⎜ R := ⎜ ⎜ 1 r6 ⎜ ⎜ 1 r7 ⎜ ⎝ 1 r8 1
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎠
⎛
q1 b 1 ⎜ q2 b2 ⎜ ⎜ q3 b3 ⎜ ⎜ q4 ⎜ Q := ⎜ ⎜ c5 q5 ⎜ ⎜ c6 q6 ⎜ ⎝ c7 q7 c8 q8
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎠
A = RQ ! "
#
$ % R " & % Q' q1 = a1 ; r2 q1 = c2 , r3 q2 = c3 , r4 q3 = c4 ,
r2 b1 + q2 = a2 ; r3 b2 + q3 = a3 ; r4 b3 + q4 + r5 c5 = a4 ;
r5 q5 q5 + r6 c6 = a5 , r6 q6 q6 + r7 c7 = a6 , r7 q7 q7 + r8 c8 = a7 , r8 q8 q8 = a8 .
= b4 ; = b5 ; = b6 ; = b7 ;
(
( ) % * # % + , q1 - r2 - q2 - r3 - q3 - r4 %. - !. / q8 - r8 - q7 - r7 - q6 - r6 q5 - r5 0 &
#
1 q4 ! & RQ23 #
%
- n = 2m, m ∈ N∗ - ! ! ' q1 = a1
" i = 2, 3, . . . , m : ri = ci /qi−1 qi = ai − ri × bi−1
qn = an
" i = n − 1, n − 2, . . . , m + 1 : ri+1 = bi /qi+1 qi = ai − ri+1 × ci+1 rm+1 = bm /qm+1 qm = qm − cm+1 × rm+1
4
5
!
" # " " $ " % "" &" " qm ' rm+1 ( n = 2m + 1, m ∈ N∗ ) RQ*$ " " + , + - m (m + 1) " R Q . /0 12 34 RQ*$ " " + " 5 RQ*$ " 4 6++ 7 8 5 M *5 /9 9 2 RQ*$ " : (n − 1) 5" 7") # ++ 7 + +: LR*$ " # ; RQ*$ " " 5 4 A ;" *
6 + Ax = d RQx = d 9 - 9?4 "> ), 6; "' $%, - 1 ? @ "# $%,
0- A 2 3 - p % 7 p
(p) Pn (x) # & $ x 2 % 8 1 f (x) 4 , 1 6 % 0 9:; ? $ $#
1 P2 (x) $ # (x0 , y0 ), (x2 , y2 ) $ # $ @ A
$ #
(x1 , y1 )
! "
#
$% &
"" ' " " $ (
) " & * + & ," "
1 [−11y0 + 18y1 − 9y2 + 2y3 ] 6h 1 [−2y0 − 3y1 + 6y2 − y3 ] f (x1 ) ≈ 6h 1 [y0 − 27y1 + 27y2 − y3 ] f (xM ) ≈ 24h 1 f (xM ) ≈ 2 [y0 − y1 − y2 + y3 ] 2h 1 xM = (x0 + x3 ) 2 f (x0 ) ≈
f (x) = sinh(x) x = 0.6 !
" # $%&' ( ) * h $ + * )
" ! $ , - . h, x0 , x2 , y0 , y2 /- ( f (0.6) = sinh(0.6) = 0.636653582 $ h
x0
x2
y0
y2
f (x1 ) ≈
0.1 0.01 0.001 0.0001 0.00001
0.50 0.59 0.599 0.5999 0.59999
0.7 0.61 0.601 0.6001 0.60001
0.521095305 0.624830565 0.635468435 0.636535039 0.636641728
0.758583702 0.648540265 0.637839366 0.636772132 0.636665437
0.637184300 0.636660000 0.637000000 0.700000000 10.00000000
0 h 1 ( h = 0.00001 /- $
"
" + . + % $ & '
& / " &
0 , + 1" 2 0 $"" $ 3 4 & & 1" . %
" ) , " 0 & , $
f (x) p4 "
+
" 5
$"" ( 0 $ 0 6 ) 4* & $
y1 = f (x1 ) = f (x0 + h) !" #$ h2 h3 h4 f (x0 ) + f (3) (x0 ) + f (4) (x0 ) + . . . 2! 3! 4! % ! y0 = f (x0 ) y1 = f (x0 ) + hf (x0 ) +
y1 − y0 h h2 h3 = f (x0 ) + f (x0 ) + f (3) (x0 ) + f (4) (x0 ) + . . . h 2! 3! 4! % f (x0 ) ! & % !% ' h !%% (%% & ! ) ! !" # $ y2
y0
=
f (x2 ) = f (x1 + h)
=
f (x1 ) + hf (x1 ) +
=
f (x0 ) = f (x1 − h)
h2 h3 h4 f (x1 ) + f (3) (x1 ) + f (4) (x1 ) + + . . . 2! 3! 4!
h2 h3 h4 f (x1 ) − f (3) (x1 ) + f (4) (x1 ) − + . . . 2! 3! 4! 2 4 6 h h h 1 [y2 − y0 ] = f (x1 ) + f (3) (x1 ) + f (5) (x1 ) + f (7) (x1 ) + . . . 2h 3! 5! 7! ! ) ! * % +, f (x1 ) & % ' ! h - (%%
! & % . ( &
% . % +// 0 ! %
% ! % % - =
f (x1 ) − hf (x1 ) +
% ! !% $% ! 2h2 (4) 2h4 (6) 2h6 (8) y2 − 2y1 + y0 = f (x1 ) + f (x1 ) + f (x1 ) + f (x1 ) + . . . 2 h 4! 6! 8! 1 ,%% % %! !%% B(t) . '! ! t !, ( A +, % !// 0 & % !% ' t ! % % !%% % 23 a1 , a2 , . . . B(t) = A + a1 t + a2 t2 + a3 t3 + . . . + an tn + . . .
4
+% % 5 !, !% 5 % + ! % % % 6 , ,! 5 67 B(t) % - '! ! t ,% !%% B(t) +// 0 ! A ! % !% ,%
! '! ! t0 > t1 > t2 > . . . > tn > 0 * B(tk ) , ! 6 8 /! %/" Pk (t) %-%%. ! !7 !,
9 % 9 t = 0 !% : k % * Pk (0) ,%% ;( % % * B(0) = A ! 10 !/!
!,, %,! 0 !/ * Pk (0) % * A
. , 5!- ! % % 10 !/! %/ %%% -! ; #8 /!
f (x) = sinh(x) x = 0.6
! " # t = h2 B(tk ) := [y2 − 2y1 + y0 ]/tk $ % & ' & & (
) yi = sinh(xi ) & ) % *$ $ + , Pk (0) - % , . hk
tk = h2k
0.30 0.20 0.15 0.10
0.09 0.04 0.0225 0.01
*$ &/ tk B(tk ) 0.6414428889 0.6387786000 0.6378482222 0.6371843000
0.05328577800 0.05316444571 0.05311377600
0.001797515344 0.001688990476
0.001356560849
0 12 cj ' j = 0, 1, 2, 3' # 3 Pk (t)' k = 0, 1, 2, 3 f (0.6) = sinh(0.6) = 0.6366535821482413 k
1
2
3
Pk (0) |Pk (0) − sinh(0)|
0.6366471689 6.4133 · 10−6
0.6366536399 5.7787 · 10−8
0.6366535300 5.2094 · 10−8
Pk (0) *0 B(tk ) P3 (0) , y $
! " ! ! # $ "! ! %" ! & & ' ( ! ! ! ) "! ! " ! * " + ,-. / & 0 $ ! ! 1 ! $ ( ) $ ! ! ! 2 3 . " 4 ! !5 $ 6 " ! %" ! ! $ 2 & 7 ! *! 8 • 2 ! " ) +9 ! k • 2 ! p !5 • ! $ ! ! ! *! ! : $ ! ! ,
/ ; n l .)) 8 (
F (x, y) !
" #$ % &'% ( ) %* + % , &'% ( ) % , !
% %* % -. / $ * 0 123 , %
F (x, y)
S(x, y)
y ! "# F (x, y) := −2
5 8 1 a12 )2 a + 65 a2 + 12 a12 (a1 + a2 + 168 5 1 + 2 5 5 25 5 5 25 (1 + 8 a1 + 6 a2 + 2352 a12 )2 1 + 8 a1 + 6 a2 + 2352 a12
$$"
128 54 182 8 6 14 y − x), a2 := exp( y − x), a12 := exp( y− x). 125 5 125 5 125 5 % &
'( (& [−8, 7] × [−8, 7] & ( ) &
* + &
* ) ,, - & '( (& % & *. * x y ' ( ( ( )( ' & / , 0 ,* ,* , " % , 20 × 20 = 400 & % 1.( * * ) 31 × 31 2 & , ( ( " % 3
* 45 & 6(& ) % * ( & a1 := exp(
n + 1 x(i) ∈ Rm (i)
(i)
x(i) = (x1 , x2 , . . . , x(i) m ), i = 0, 1, . . . , n.
! "# "! t ∈ R $ %& & $ "
" m ""! '
x1 (t), x2 (t), . . . , xm (t), t ∈ [0, 1],
(
% ) !* " ti *$ '+ , (i)
xk (ti ) = xk , i = 0, 1, . . . , n, k = 1, 2, . . . , m.
n + 1 x(i) ti t0
:=
ti
:=
0
+ ,m , (i) (i−1) 2 ti−1 + - (xk − xk ) , i = 1, . . . , n.
k=1
ti ! " [0, 1] ti :=
ti , i = 1, . . . , n. tn
# m $% % & $ & (i) ti ' $ & & k ( $% xk (t) ) xk m * & +" R (x1 (t), x2 (t), . . . , xm (t)),
t ∈ [0, 1].
,
! "# # $ r(t) = c t (1 +
bct sin(a t) ). d
% &'(
) (r, t) * + , " t ! ϕ -) * . / $ d = 500 $ c=2$ b = 0.2 $ a = 5.03 $
0# "/ 1) 2 1) 23 #
* ) x(t) y(t)
= =
r cos(t) , r sin(t) .
% &4(
5 6 # ) '7 8774 * " ) )) 84 2 ) , "
'7 * # + ) 8774 * ) , " * 9# # +
0) -# + ) 6 + . 2 : * . 2 /$ ;+ · * ! 7 >' $ (f, f ) = f 2 ∀f ∈ H ! i=1
0 . ' (π 1 1 sin{(j + k)x} + x =π 2 j+k −π
.
! " # $ )π
1 sin(jx) sin(kx) dx = − 2
−π
)π [cos{(j + k)x} − cos{(j − k)x}] dx, −π
% & ' " $ ( ) ) $ ' * + % ( , ' ) + ' "- $ . 2π - + + ' $ [a, b] "- & " t = 2π (x − a)/(b − a)
" ) / 1, cos(kt), sin(kt), . . . 0 * 1#2 .! 3 4 αi = (f, ϕi )/(ϕi , ϕi ) 5 + + % " #
' 1 ' 3 4 1 ak = π 1 bk = π
)π f (x) cos(kx) dx,
k = 0, 1, . . . , n,
−π )π
5 f (x) sin(kx) dx,
k = 1, 2, . . . , n.
−π
2 6 (2π),7 ' ) f (x) + ) gn (x) ,3 4 ! ,( / )! "- + + 6 n 8 9 (2π),7 ' -') ) f (x) ' #-+ '
g(x) :=
∞
1 a0 + {ak cos(kx) + bk sin(kx)} 2
k=1
" + ) * :; %
f (x) (2π) ! " # g(x) $% &'&( !( ) f (x0 ) *! f (x) ! + x0 1
( {y0− + y0+ } , $% &-.( *! f (x) ! + x0 + 2 ? 2 @ ' 6 ' -') 1 ) # 7 ) & " + 7 f (x) = |x| "- x = 0
y
x −π
π
0
2π
g3 (x)
(2π) f (x) [−π, π] f (x) = |x|,
−π ≤ x ≤ π.
! "#$
% & &' ( ) !' $ % ' * '
* +,) bk = 0 - . ' ak ( & Zπ Zπ 1 2 a0 = |x| dx = x dx = π π π −π 0 3 2 ˛π Zπ Zπ ˛ 2 2 41 1 ˛ x cos(kx) dx = sin(kx) dx5 x sin(kx)˛ − ak = ˛ π π k k 0 0 0 ˛π ˛ 2 2 ˛ = cos(kx)˛ = [(−1)k − 1], k > 0 ˛ πk2 πk2 0
) '/ ' 0 ff j 1 cos(3x) cos(5x) 4 cos(x) g(x) = π − + + + . . . . 2 π 12 32 52
! "# $
g3 (x) ') 1 f (x) ' f 2 ) ' % ) # (3 * ' % ) # g(0) = 0 ' 1 1 1 1 π2 = 2 + 2 + 2 + 2 + ... 8 1 3 5 7
(2π) f (x) (0, 2π) 2
f (x) = x ,
0 < x < 2π.
! "#4$
% ) (3 xk = 2πk, (k ∈ Z) % ' * (3 2 ) ' % ) # ) '/ ' +,) 5 6 ( ' 1 a0 = π
Z2π 0
x2 dx =
8π 2 3
y 4π 2
f (x)
2π 2
g4 (x)
x −π
π
0
2π
1 π
ak =
1 bk = π
Z2π
x2 cos(kx) dx =
0
Z2π
4 , k2
x2 sin(kx) dx = −
0
4π , k
k = 1, 2, . . .
k = 1, 2, . . .
ff ∞ j 4π 2 X 4 4π g(x) = cos(kx) − + sin(kx) . 3 k2 k k=1
!"#$
% & % ' f (x) ( g4 (x) ) ' * + ' & , -
ak bk ! " # $ # $% $
& %' & ( ( ( ) * ++ , ( - + " . ( / ( 0 1 ! )
( [0, 2π] N 2 ( & 3 $ h xj h=
2π , N
xj = hj =
2π j, N
j = 0, 1, 2, . . . , N
& $
2 + 1 ! ak
=
1 π
)2π f (x) cos(kx) dx 0
!
≈
⎫ ⎧ N −1 ⎬ 1 2π ⎨ f (xj ) cos(kxj ) + f (xN ) cos(kxN ) . f (x0 ) cos(kx0 ) + 2 ⎭ π 2N ⎩ j=1
(2π) f (x) cos(kx) ak bk a∗k
N 2 := f (xj ) cos(kxj ), N j=1
b∗k
N 2 := f (xj ) sin(kxj ), N j=1
k = 0, 1, 2, . . . ,
! "#$ k = 1, 2, 3, . . . .
% &' ' () a∗k b∗k *
+ ! , - . ! "$ ! /$ 0 ' % !
xj
N
!
cos(kxj ) =
j=1 N
sin(kxj ) = 0
0 N
k/N ∈/ Z
k/N ∈ Z
! "/$
k ∈ Z.
! "1$
j=1
2 '3 4 5''
S :=
N N N {cos(kxj ) + i sin(kxj )} = eikxj = eijkh , j=1
j=1
! #$
j=1
' 6 ' ' '3 4$ 7 q := eikh = e2πik/N
! * %
. ' 2
''! 8
k/N ∈ / Z
q = 1 5''' S = eikh
eikhN − 1 e2πki − 1 = eikh ikh = 0, ikh e −1 e −1
k/N ∈ / Z.
8
k/N ∈ Z q = 1
S = N
! , *
' 6 ' 8' S ! #$ 3 ! "/$ ! "1$!
xj ! ⎧ k+l k−l ⎪ ⎪ 0, ∈ / Z ∈ /Z
⎪ ⎪ N N ⎪ N
⎪ ⎨
k+l k−l ∈ Z ∈ Z N N j=1 k N+ l ∈ Z k N− l ∈ Z k N+ l ∈/ Z k N− l ∈/ Z N k N+ l ∈ Z k N− l ∈ Z sin(kxj ) sin(lxj ) = ⎪ ⎪ N k+l k−l j=1 ⎪ ⎪ ∈ Z ∈ /Z − ,
⎪ ⎪ 2 N N ⎪ ⎪ ⎪ ⎩ N, k N+ l ∈/ Z k N− l ∈ Z 2 N cos(kxj ) sin(lxj ) = 0, k, l ∈ N cos(kxj ) cos(lxj ) =
N , ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ N, ⎧ ⎪ ⎪ 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
"
j=1
#"
! "
cos(kxj ) cos(lxj ) =
1 [cos{(k + l)xj } + cos{(k − l)xj }] 2
sin(kxj ) sin(lxj ) =
1 [cos{(k − l)xj } − cos{(k + l)xj }] 2
cos(kxj ) sin(lxj ) =
1 [sin{(k + l)xj } − sin{(k − l)xj }] 2
# $ %& & ' ( & ) & * + ! ", -& ak bk . & * & ! * / 0 a∗k b∗k 1 ' / * %* " (f, ϕj ) 2 ϕ2k (x) = cos(kx) )&' ϕ2k+1 (x) = sin(kx)( k = 0, 1, 2, . . .(
' ' % & 3 & 4 * #&" %* ) ( . & * 5) $ * & " # 6 ) ( ) * 4 * 5) # 6 7 * ) ' ! ", -& a∗k b∗k ) * % & * #
y 4π 2
f (x)
g3∗ (x)
2π 2
x −π
π
0
2π
N ∗ gm (x) :=
= 2n, n ∈ N∗
1 ∗ a + 2 0
m
{a∗k cos(kx) + b∗k sin(kx)}
k=1
m < n ! f (x) ! " #$ % N &' xj ( & ) *+,#$ - F :=
N
∗ {gm (xj ) − f (xj )}2
j=1
(2π) f (x) = x2 , 0 < x < 2π! "
# f (0) = f (2π) = 2π 2 ! N = 12 $% . . . . 4.092652, a∗2 = 1.096623, a∗3 = 0.548311, a∗0 = 26.410330, a∗1 = . . . b∗1 = −12.277955, b∗2 = −5.698219, b∗3 = −3.289868. & ' ( ) g3∗ (x) f (x)
m ! " # $ % & ' $ ()% *+
N gn∗ (x) :=
= 2n, n ∈ N∗
1 ∗ a + 2 0
n−1 k=1
1 {a∗k cos(kx) + b∗k sin(kx)} + a∗n cos(nx) 2
,
-( &' xj &' f (xj )( j = 1, 2, . . . , N
y π
x x1
0
x2
x3
π = x4
x5
x6
x7
2π
g4 (x)
f (xj ) ! xj " # yj− yj+ $ $% N ! &'( )* a∗k b∗k +, -.
! f (x) " #$%& ! N = 8' h = π/4' xj = πj/4' j = 1, 2, . . . 8' ( . . a∗0 = π, a∗1 = −1.34076, a∗2 = 0, a∗3 = −0.230038, a∗4 = 0, b∗1 = b∗2 = b∗3 = 0. g4∗ (x) . g4∗ (x) = 1.57080 − 1.34076 cos(x) − 0.230038 cos(3x) ) * % + , !
/ & !!0 |a∗k − ak | |b∗k − bk | 1 2 $ $ 3 4
f (x) 2π N = 2n n ∈ N∗ f (xj ) xj ! " #$ %&' ( ) * + a∗k b∗k , a∗k b∗k
= ak + aN −k + aN +k + a2N −k + a2N +k + . . . , (0 ≤ k ≤ n), = bk − bN −k + bN +k − b2N −k + b2N +k + . . . , (1 ≤ k ≤ n − 1),
|a∗k − ak | ≤
∞
{|aμN −k | + |aμN +k |},
μ=1
(0 ≤ k ≤ n),
5 667 5 687 5 6-7
|b∗k − bk | ≤
∞
{|bμN −k | + |bμN +k |},
(1 ≤ k ≤ n − 1).
μ=1
N ! " #! $ ak bk % ! f (x) !& ' ( N ) &! ε > 0 |a∗k − ak | ≤ ε |b∗k − bk | ≤ ε % k = 0, 1, 2, . . . , m < n
f (x) !" # $ % & ( 4 − 2 , $ k &
' a0 = π; ak = ( πk 0, $ k > 0 &
) N = 8 *& # $ & + $ $ $ a∗0 = a0 + 2(a8 + a16 + a24 + a32 + . . .) = a0 , a∗1 = a1 + a7 + a9 + a15 + a17 + . . . = a1 −
4 π
j
ff 1 1 1 1 + + + + ... . 49 81 225 289
,$ $ - * N +&$." ' $ $$ $ * *$ /" ak $ a∗k ' +$ * ε = 10−6
k &
k N & j ff 1 4 1 1 1 a∗k − ak = − + + + + . . . π (N − k)2 (N + k)2 (2N − k)2 (2N + k)2 j ff 2 2 2 2 8N + 2k 4 2N + 2k + + ... = − π (N 2 − k2 )2 (4N 2 − k2 )2 j ff 1 1 1 8 + + + . . . ≈ − π N2 (2N )2 (3N )2 j ff 4π 1 1 1 8 π2 8 =− + + + . . . =− · . = − 2 2 2 2 2 πN 1 2 3 πN 6 3N 2 0 1
& " 2 & + $ $ 3 4 x = 0 $ # & & |a∗k − ak | ≤ ε = 10−6 & N > 2046
( ! *+ , ! "#! $ ak bk (2π)"- ! ! f (x) & . *+ / &! !0 , - 12 3 ! , & . " 4, 5 1 *6 N ! !0 ! . ) 7 " 7 , . ! % , N = 2n #! $ a∗k k = 0, 1, . . . , n b∗k k = 1, 2, . . . , n − 1 N 2 8 - ! N 2 !! ! % )0 ( &! N N 1000 7 -! & !0 ( 9!
! "# $ " "% & '# # (' # )" " # # " *#% + % , - , . ## + - + + /0 12 / , !
3 4 % ak := bk :=
N −1 j=0 N −1
N , 2
f (xj ) cos(kxj ),
k = 0, 1, 2, . . . ,
f (xj ) sin(kxj ),
N − 1, k = 1, 2, . . . , 2
j=0
5- /6
2π j
N xj = N
! " # $ $ %&"'() j * ! N − 1" + !
n = N/2 , yj := f (x2j ) + if (x2j+1 ),
j = 0, 1, . . . , n − 1,
n=
N . 2
%&"'(&)
, - yj .
/ n ck :=
n−1
yj e−ijk
j=0
wn := e
−i 2π n
2π n
n−1
yj wnjk , k = 0, 1, . . . , n − 1 j=0 # $ # $ 2π 2π − i sin . = cos n n =
%&"'(0)
- 123 wn
n 4 " - ,
ck
123 ak bk
ak bk ck 1 1 ak − ibk = (ck + c¯n−k ) + (ck − c¯n−k )e−ikπ/n %&"'(5) 2 2i 1 1 ck + cn−k ) + (¯ ck − cn−k )eikπ/n an−k − ibn−k = (¯ %&"'(6) 2 2i k = 0, 1, . . . , n, b0 = bn = 0 cn = c0
n−1 n−1 1 1 1 j(n−k) {yj wnjk + y¯j wn }= (yj + y¯j )wnjk , (ck + c¯n−k ) = 2 2 j=0 2 j=0
n−1 n−1 1 1 1 j(n−k) (ck − c¯n−k ) = {yj wnjk − y¯j wn }= (yj − y¯j )wnjk . 2i 2i j=0 2i j=0
!" # $%
=
1 1 (ck + c¯n−k ) + (ck − c¯n−k )e−ikπ/n 2 2i n−1 3 2 f (x2j )e−ijk2π/n + f (x2j+1 )e−ik(2j+1)π/n j=0
=
n−1
& f (x2j )[cos(kx2j ) − i sin(kx2j )]
j=0
+ f (x2j+1 )[cos(kx2j+1 ) − i sin(kx2j+1 )] = ak − ibk .
! " $% # k n−k
% & ' $ ( ak bk ) * + k ,
% - $ ' $
!. ( / ak an−k # % bk bn−k $ ( 0 - / / 1 ,$ ! $ 2 23 4 ! wn n2 5 # $ 6 wnjk 5 /+ 5% % 5 / $78$ n25 ! 5+/
% n 2 9 % $ % n = 4 ! 23 4 : + W4 ∈ C4,4 3 ⎛ ⎞ ⎛ ⎞⎛ ⎞ c0 1 1 1 1 y0 ⎜ c1 ⎟ ⎜ 1 w 1 w 2 w 3 ⎟ ⎜ y 1 ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ , w = w4 ; c = W4 y ; ⎝ c2 ⎠ = ⎝ 1 w 2 1 w 2 ⎠ ⎝ y2 ⎠ c3
1
w3
w2
w1
y3
; / c & W4 # 7
/ : + .
⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ 1 1 1 1 c0 c˜0 y0 ⎜ c˜1 ⎟ ⎜ c2 ⎟ ⎜ 1 w2 1 ⎜ ⎟ w2 ⎟ ⎜ ⎟=⎜ ⎟=⎜ ⎟ ⎜ y1 ⎟ = 2 3 ⎠⎝ y ⎠ ⎝ c˜2 ⎠ ⎝ c1 ⎠ ⎝ 1 w w w 2 c˜3 1 ⎜ 1 ⎜ ⎝ 0 0 ⎛
1 w2 0 0
0 0 1 1
c3 0 0 1 w2
1
⎞⎛
1 ⎟⎜ 0 ⎟⎜ ⎠⎝ 1 0
w3 0 1 0 w1
w2 1 0 w2 0
w1 ⎞⎛
y3 ⎞ 0 y0 ⎜ ⎟ 1 ⎟ ⎟ ⎜ y1 ⎟ . ⎠ ⎝ y2 ⎠ 0 3 y3 w
! " "# $ % " # & ' " " # $ " ! " ( % " )# * !
) + " ),
-) -
. / 0 y ! *" 0 z 1 / z 0 = y0 + y2 ,
2
z1 = y1 + y3 ,
z2 = (y0 − y2 )w0 ,
23
z3 = (y1 − y3 )w1 ,
, "# w = −1 w = −w " 4 5 ) 0' " " 23 / w0 = 1 )"), 4 -
5) z ! / . 2
3
1
c˜0 = c0 = z0 + z1 ,
c˜1 = c2 = z0 + w2 z1 ,
2
c˜2 = c1 = z2 + z3 ,
c˜3 = c3 = z2 + w2 z3 .
26
7 2 26 " 4 5 ) 0" ' " / w2 = −1 "), . w42 = w21 , ' "# # 2 26 ( /! '-)
% " '-) % "
2 23 ) '-) % " , "), 8 /! '-) " % " ) '-) ( % " 9"
' " " / -) ! "" :
" 5) " $ ) " * n = 2m, m ∈ N∗ " ), /! -) ck ; " 4 k = 2l# l = 0, 1, . . . , m − 1# c2l =
2m−1
yj wn2lj =
j=0
4 7
m−1
m−1
j=0
j=0
(yj + ym+j )wn2lj =
2l(m+j) wn
zj := yj + ym+j ,
(yj + ym+j )(wn2 )lj .
= wn2lj wn2lm = wn2lj m 5)
j = 0, 1, . . . , m − 1,
2
wn2 = wm m c2l =
m−1
jl zj wm ,
l = 0, 1, . . . , m − 1,
j=0
m ! " zj # ck $ k = 2l + 1% l = 0, 1, . . . , m − 1% " c2l+1
=
2m−1
yj wn(2l+1)j
j=0
=
=
m−1
{yj wn(2l+1)j + ym+j wn(2l+1)(m+j) }
j=0
m−1
{yj −
ym+j }wn(2l+1)j
j=0
=
m−1
{(yj − ym+j )wnj }wn2lj .
j=0
& m ! " zm+j := (yj − ym+j )wnj ,
j = 0, 1, . . . , m − 1,
m c2l+1 =
m−1
jl zm+j wm ,
'
l = 0, 1, . . . , m − 1,
j=0
m ! " zm+j ( )#*+ #, + -". n = 2m + -". m " " *, /*, m + -". &" -" + $ n = 2γ , γ ∈ N∗ %
+0 1 m "1 2 , "1 #*+ #, %
3 *, /+ 0 " *, $ "" n = 32 = 25 1 4 , " 5 *, 1 16
2·8
4·4
8·2
16·1
F T32 → 2(F T16 ) → 4(F T8 ) → 8(F T4 ) → 16(F T2 ) → 32(F T1 ),
6
"*, F Tk k 1% ) ," " *, + -". &" -" + # 17 /+
*, 1 ( 8 9 #1 % "" *, " /+
*, , " ) ," *, ck 8 + -". n = 2γ % γ ∈ N∗ 4 "" : 6 γ /+
*, n 8 #*+ #,1 ( 2 ;*, n/2 + -". &" -" + % 1 " /*, ZF T n =
1 1 nγ = n log2 n 2 2
n ! "!!#$ % &' () % * &+ (,) -. ! # / 0 $ 1 2 * n 3 n2 1 4 % 0 56$ 1 3 ZF T n 0 5$ 2 γ= n= n2 = ZF T n =
5 32 1024 80
6 64 4096 192
8 256 65536 1024
9 512 2.62 · 105 2304
10 1024 1.05 · 106 5120
11 2048 4.19 · 106 11264
12 4096 1.68 · 107 24576
Faktor
12.8
21.3
64
114
205
372
683
0 $
! # 2 n2 !
/ * 0 $ * - 7 ! # 1 * 4 1 * # y 8 c 8
# 00 2 ! n = 16 = 24 !2 % w := e−i2π/16 = e−iπ/8 = cos(π/8) − i sin(π/8) 9 7: zj 0 0$ zm+j 0 $ yj ym+j 1 8 * 1 +7: 8 * ; %
* w 3 + * y 8 ! # ck * w / ! # ck > @ + 0 k = j = 0
⎫ ⎪ ⎬ ⎪ ⎭
,
k, j ∈ N
x = cos(ϕ), Tk (x) = cos(kϕ), Tj (x) = cos(jϕ) dx = − sin(ϕ)dϕ )1 −1
dx Tk (x)Tj (x) √ =− 1 − x2 )π
=
)0
1 cos(kϕ) cos(jϕ)dϕ = 2
0
cos(kϕ) cos(jϕ)
sin(ϕ)dϕ sin(ϕ)
π
)π cos(kϕ) cos(jϕ)dϕ. −π
! " # $ # " % & [−1, 1] " '( f (x) % )
* +% gn (x) n, - . % /,* +% 1 c0 T0 (x) + ck Tk (x) 2 n
gn (x) =
0
k=1
112 % . ( - 3,#112 % % % ( , 1( )1 (f, g) :=
)1 f (x)g(x) √
f (x)g(x)w(x) dx := −1
−1
dx . 1 − x2
. 45 ck 2 ck = π
)1 f (x)Tk (x) √ −1
dx , 1 − x2
k = 0, 1, 2, . . . , n.
! "# ! $
cos(ϕ)
x=
!% % " & ' ('! '
2 ck = π
)π f (cos ϕ) cos(kϕ)dϕ,
k = 0, 1, . . . , n.
0
F (ϕ) := f (cos ϕ) (2π) ϕ ck =
1 π
)π f (cos ϕ) cos(kϕ)dϕ,
k = 0, 1, 2, . . . , n.
−π
!" ck #$ %& ' gn (x) !" ak % ( (2π) F (ϕ) := f (cos ϕ) ) ) * + !" ", ), $( - $ . $ !" ck "/ 01 gn (x) / ), 2 n 3 $( " 4 5+ [−1, 1] f (x) 6", "7 8#$ ∞
g(x) =
1 c0 T0 (x) + ck Tk (x) 2
9
k=1
$ +" : 9 ; " + "$ *" f (x) 5+ [−1, 1] , $ ( ), F (ϕ) " ? >"+" + " f (x) / 8#$ 9 ), x ∈ [−1, 1] @ f (x) ' ) f (x) − gn (x) ";" A" ;" $
ck ∞ . f (x) |ck | k=1 !" #$%& ' x ∈ [−1, 1] () f (x) |f (x) − gn (x)| ≤
∞
|ck |
' x ∈ [−1, 1].
B
k=n+1
* A" + 8: 9 $ f (x) ), |x| ≤ 1 , *$ ;2 +" " $ 8(
ε > 0 N ∞
|ck | < ε
n > N.
k=n
n > N |x| ≤ 1
|Tk (x)| ≤ 1 ∞ ∞ |f (x) − gn (x)| = ck Tk (x) ≤ |ck | < ε. k=n+1
k=n+1
!
f (x) = e
[−1, 1] gn (x) ! " # " $% & ' Zπ 2 ck = ecos ϕ cos(kϕ)dϕ = 2Ik (1), k = 0, 1, 2, . . . , & π x
0
" Ik (x) (% k) * ) +,' -&. ,/ 0 % /1 Ik (x) " . . . c2 = 0.2714953395, c0 = 2.5321317555, c1 = 1.1303182080, . . . c3 = 0.0443368498, c4 = 0.0054742404, c5 = 0.0005429263, & . . . c8 = 0.0000001992, c6 = 0.0000449773, c7 = 0.0000031984, . . c9 = 0.0000000110, c10 = 0.0000000006. 2 $% ck '
3 / 4 % 5 6 7) 8 % ! f (x) ' ' / 9 % ' 4 : % ; % < /1 1 g6 (x) " & 0
!"#$ (n + 1) ( k = j = 0 !"#$ ) *
!""'$ !""'$ + , -
. !""/$
0 1 0 2 1 γ0 , γ1 , . . . , γn
- ,0 34 !""/$ 0 l5 - !""/$ Tj (xl ) j ( , 0 ≤ j ≤ n (n + 1) - 6 1 !""'$ ( , 7 8 γk
γk
= =
n+1 2 f (xl )Tk (xl ) n+1 l=1 $$ # $ # n+1 # 2l − 1 π 2l − 1 π 2 cos k , f cos n+1 n+1 2 n+1 2
!" $
l=1
k = 0, 1, . . . , n.
) 1 1 8 γk 0 9 . Pn∗ (x) : 5 . ;5*
Tn+1 (x)
7 8 c∗k !"" $ * 7 8 1 9 . n5 - 1 ∗ 1 c0 T0 (x) + c∗k Tk (x) + c∗n Tn (x) 2 2 n−1
gn∗ (x) :=
!""$
k=1
N # 0 ) 5 2 # $ jπ (e)
(n + 1) ), xj = cos Tn (x) ( n ( 01 ±1 ( 0 0 . gn∗ (x) gn∗ (x) * !"" 2m* - 0 "
* Im,n = 0 - . 1 # $% &%2' $% &%' "
* - ! m = n ! 3 d2n [(x2 − 1)n ] = (2n)! dx2n
) $% &%' n # "
)1 In,n
(x − 1)n (x + 1)n dx
n
= (−1) (2n)! −1
1 1 n+1 = (−1) (2n)! (x − 1) (x + 1) n+1 −1 ⎤ 1 ) n − (x − 1)n−1 (x + 1)n+1 dx⎦ = · · · n+1 n
n
−1
n(n − 1)(n − 2) . . . 1 = (−1) (2n!) (n + 1)(n + 2)(n + 3) . . . (2n)
)1
2n
(x + 1)2n dx −1
22n+1 . = (n!)2 · 2n + 1
!"# $% Pn (x) &$ x '() !* n ($ ($ +## &$ x , &$ #$)% ) n , -() Pn (−x) = (−1)n Pn (x),
.
n ∈ N. Pn (x) n ≥ 1
(−1, 1) n
) /()% (x2 − 1)n 0 x = ±1 1 n () 2 ,% () n # 3 4 , dk ,()% [(x2 − 1)n ] 0 x = ±1 k = 1, 2, . . . , n − 1 1 (n − k) () dxk 2 -5 # n ' () 2 # 6 [−1, 1] !"# n 7 n 2 80($() ) 9()) ,% 8)'
2 Pn (x) $: 0 # n () # & / !"# () () &# , 6) # ,' ;, % 3() % 3 < ,
! "# P0 (x)
=
1,
P1 (x) = x,
Pn+1 (x)
=
2n + 1 n x Pn (x) − Pn−1 (x), n+1 n+1
= n = 1, 2, . . . .
- $ # % 1# () L2 3$'$ 3"# ) !"# , 3"# 0 /# 4$ 3"# >, ,##% ) ; ? ,
> e− . n
+ − In /n, < e « j„ ff & '(9) In−1 = > 1 > : − In /n, e+ . n
+ e
.1 3 N IN = 0 * 1 " " + 1 N = 24 " 7 : % 9 4
I0 % I6 " ;!" c0 % c6 : % 9 !" k
Ik
ck
< @ '
' 00===' ( ===(9''9<
