Numerical Methods Contents - SAM

Task:
efficient evaluation of matrix×vector product Tx, x ∈ K n
Note: this extended matrix is circulant (→ Def. 7.1.3)
⎛
⎞
u 0 u 1 · · · · · · u n−1 0 u 1−n · · · · · · u −1
u −1 u 0 u 1 . u n−1 0 . .. .
. . .. ... . .. . . ... . ..
. ... . .. ... . . ..
( )
. . .. ... u 1 . ... . .. u 1−n
T S
u
C = =
1−n · · · · · · u −1 u 0 u 1 u n−1 0
S T
0 u 1−n · · · · · · u −1 u 0 u 1 · · · · · · u n−1
u n−1 0 ... . u −1 u 0 u 1 .
. . .. ... . ... . .. ... .
⎜
... . . .. ... . .. .
⎟
⎝ . . .. ... u 1−n . ... . .. u ⎠ 1
u 1 u n−1 0 u 1−n · · · · · · u −1 u 0
Block-partitioned LSE, cf. Rem. 2.1.4, Rem. 2.2.8
⎛ ⎞⎛
⎞ ⎛ ⎞
u k
b 1
T k+1 x k+1 = ⎜ T k .
⎟⎜
˜x k+1
( )
⎟
⎝ u 1 ⎠⎝
⎠ = ⎜ .
⎟ ˜bk+1
⎝ b k
⎠ = b k+1
u k · · · u 1 1 x k+1 b
k+1 k+1
Reversing permutation: P k : {1,...,k} ↦→ {1,...,k}, P k (i) := k − i + 1
˜x k+1 = T −1
k (˜b k+1 − x k+1
k+1 P ku k ) = x k − x k+1
k+1 T−1 k P ku k ,
x k+1
k+1 = b k+1 − P k u k · ˜x k+1 = b k+1 − P k · x k + x k+1
k+1 P k · T −1
k P ku k .
Efficient algorithm by using auxiliary vectors:
( )
x k+1 ˜x k+1
=
x k+1 with
k+1
y k := T −1
k P ku k
(7.5.2)
(7.5.3)
x k+1
k+1 = (b k+1 − P k u k )/σ k
˜x k+1 = x k − x k+1
k+1 yk , σ k := 1 − P k u k · y k . (7.5.4)
This example demonstrates the case m = n
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In general:
T = toeplitz(u(0:-1:1-m),u(0:n-1));
S = toeplitz([0,u(n-1:-1:n-m+1)],[0,u(1-m:1:-1)]);
( ( )
x Tx
C =
0)
Sx
Computational effort O(n log n) for computing Tx (FFT based, Sect. 7.3)
7.5.2 The Levinson algorithm
Levinson algorithm
(recursive, u n+1 not used!)
Linear recursion:
Computational cost ∼ (n−k) on level k, k =
0, ...,n − 1
➣ Asymptotic complexity O(n 2 )
✄
MATLAB-CODE Levinson algorithm
function [x,y] = levinson(u,b)
k = length(u)-1;
if (k == 0)
x=b(1); y = u(1); return;
end
[xk,yk] = levinson(u(1:k),b(1:k));
sigma = 1-dot(u(1:k),yk);
t= (b(k+1)-dot(u(k:-1:1),xk))/sigma;
x= [ xk-t*yk(k:-1:1);t];
s= (u(k+1)-dot(u(k:-1:1),yk))/sigma;
y= [yk-s*yk(k:-1:1); s];
Given: S.p.d. Toeplitz matrixT = (u j−i ) n i,j=1 , generating vector u = (u −n+1, ...,u n−1 ) ∈ C 2n−1
(Symmetry ↔ u −k = u k , w.l.o.g u 0 = 1)
Task: efficient solution algorithm for LSE Tx = b, b ∈ C n
(Yule-Walker problem)
Remark 7.5.3 (Fast Toeplitz solvers).
FFT-based algorithms for solving Tx = b with asymptotic complexity O(n log 3 n) [37] !
Supplementary and further reading:
△
Recursive (inductive) solution strategy:
Define: T k := (u j−i ) k i,j=1 ∈ Kk,k (left upper block of T) ➣ T k is s.p.d. Toeplitz matrix ,
x k ∈ K k : T k x k = (b 1 , ...,b k ) T ⇔ x k = T −1
k bk ,
u k := (u 1 , ...,u k ) T
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[8, Sect. 8.5]: Very detailed and elementary presentation, but the discrete Fourier transform through
trigonometric interpolation, which is not covered in this chapter. Hardly addresses discrete convolution.
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