university/KIT-BOOKS
1
0
Fork 0
Code Issues Pull requests Actions Packages Projects Releases Wiki Activity

fix(numerik summary): lagrange definition and indices

Browse source
This commit is contained in:
Orangerot 2025-04-17 00:00:17 +02:00
parent bbe3d07a2a
commit b61e2c88ed
1 changed files with 22 additions and 16 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

28
numerik/summary.typ
View file

@ -20,10 +20,13 @@ norm(A^(-1)) &= sup_(x != 0) norm(A^(-1) x)/ norm(x) =^(A^(-1)x = 1) sup_(z !=
$
#table(columns: 4, align: center + horizon,
[Spaltensummennorm], $ norm(A)_1 = max_(m=1,...,N) sum_(n=1)^N abs(a_"nm") $,
$ norm(x)_1 = sum_(n=1)^N abs(x_n) $, [1-Norm],
[Zeilensummennorm], $ norm(A)_1 = max_(n=1,...,N) sum_(m=1)^N abs(a_"nm") $,
$ max_(n=1,...,N) abs(x_n) $, [Maximumsnorm],
[Spaltensummennorm],
$ norm(A)_1 = max_(m=1,...,N) sum_(n=1)^N abs(a_"nm") $,
$ norm(x)_1 = sum_(n=1)^N abs(x_n) $,
[1-Norm],
[Zeilensummennorm],
$ norm(A)_oo = max_(n=1,...,N) sum_(m=1)^N abs(a_"nm") $,
$ norm(x)_oo = max_(n=1,...,N) abs(x_n) $, [Maximumsnorm],
[Spektralnorm], $ norm(A)_2 = sqrt("größter EW von " A^T A) $, $ norm(x)_2 = sqrt(sum_(n=1)^N x_n^2) $, [euklidische Norm]
)
@ -304,7 +307,7 @@ als:
$
p(x) = sum_(n=0)^N f_n L_n (x) #h(1cm)
, L_n (x) = sum_(m=0,m != n)^N (x-x_m)/(x_n-x_m) = sigma_(n m) = cases(1 ", falls "
, L_n (x) = product_(m=0,m != n)^N (x-x_m)/(x_n-x_m) = sigma_(n m) = cases(1 ", falls "
x_m = x_n, 0 ", falls " x_m != x_n)
$
@ -504,7 +507,7 @@ $
cases(reverse: #true,
s''_n (x_(n-1)) &= gamma_(n-1),
s''_n (x_n) &= gamma_n
) " damit " s''_n (x_n) = s''_(n-1) (x_n) = gamma_n
) " damit " s''_n (x_n) = s''_(n+1) (x_n) = gamma_n
$
Dies sind 2N Gleichungen für die N+1 Unbekannten $gamma_0, ..., gamma_N$. Durch
@ -587,9 +590,12 @@ Norm: $norm(f)_1 = I(abs(f))$
=== Quadraturformeln
$
integral_a^b f(x) d x approx integral_a^b p(x) d x = integral_a^b sum_(n=0)^N f(x_n)
L_n (x) = sum_(n=0)^N integral_a^b L_n (x) d x f(x) = (b-a) sum_(k=1)^s b_k f( a +
c_k (b-a))
integral_a^b f(x) d x
approx integral_a^b p(x) d x
= integral_a^b sum_(n=0)^N f(x_n) L_n (x)
= sum_(n=0)^N overbrace(integral_a^b L_n (x) d x, =(b-a)b_(n+1))
f underbrace((x_n), #box()#place(horizon+center, $script(=a + c_(n+1)(b-a))$))
= (b-a) sum_(k=1)^s b_k f( a + c_k (b-a))
$
#table(columns: 3,
@ -619,13 +625,13 @@ $
$
Für eine QF mit vorgegebenen Knoten $c_1 < ... c_s$ können die Gewichte genau
dann mi $b_k = integral_0^1 underbrace(L_k (x), #place(center, $L_k (x) = sum_(j=1,j != k)^s
dann mit $b_k = integral_0^1 underbrace(L_k (x), #place(center, $L_k (x) = product_(j=1,j != k)^s
(x-c_j)/(c_k - c_j)$)) d x$ eindeutig bestimmt werden wenn $p >= s$.
*Symmetrische QF*:
$
b_k = b_(s+1-k) && c_k = 1 - c_(s+1-k) "also symmetrisch um "1/2" verteilt"
b_k = b_(s+1-k) #h(1.5cm) ,c_k = 1 - c_(s+1-k) "also symmetrisch um "1/2" verteilt"
$
Die Ordnung p einer symmetrischen QF ist gerade. #TODO[Beweis]