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Figure 2.4.6 g(n) and g(n - 1).
Define
X2 (n)
g'(n)
x2(n) =
{ g'(n - 2N)
for 1 ~ n ~ N for n = N + 1 for N + 2 ~ n
(2.4.29)
for n = 1,2, ... , 2N. Let x2(n) be the periodic version of x2(n) with period 2N. Then
X2 (n) = g' (n ) for - N
Note that
+2 ~ n ~ N
1 ~n ~ N n=N +1 N +2 ~ n
(2.4.30)
g(n - 1)
x2(n) =
Hence
{ g(n - 2N - 1)
(2.4.31)
y(n) =
x2(n - m + l)x(m)
(2.4.32)
4.2 FFT for Product of Toeplitz Matrix and Column Vector
Define
y(n) =
x2(n - m
+ l)x(n)
(2.4.33)
for all n. The result of y( n) is periodic with period 2N. Also
y(n)=y(n)
l~n~N
(2.4.34)
Now (2.4.33) satisfies the properties of periodic convolution. We apply periodic convolution.
X(k)
L x(n)W~~-l)(n-l)
n=l 2N
(2.4.35)
X2(k)
L x2(n)W~~-1)(n-l)
(2.4.36) (2.4.37)
Then
Y(k)
= X 2(k)X(k)
-( ) _ _ '"' Y(k)uT-(k-l)(n-l) 1 y n - 2N ~ vV2N
We can extend to the case of three-dimensional convolution with three indices as needed for the discrete dipole approximation. Let N x , Ny and N z points, respectively, in X, fJ and z directions with N x , Ny and N z all equal to powers of 2. Let
y(n, m, l)
L L L g(n - n ' , m - m ' , l-l')x(n' , m ' , l')
n'=l m'=l/'=l
(2.4.38)
is to be computed. For the sake of simplicity, we illustrate the scalar case. The vector case follows by a simple extension. Then we have 3-D periodic sequences x and X2 with period M x = 2Nx , My = 2Ny and M z = 2Nz , respectively in x, fJ and z directions. For one period of x(n, m, l) it is x(n, m, l) 1 ~ n ~ N x and 1 ~ m ~ Ny and 1 ~ l ~ N z
x(n,m,l)
when N x + 1 ~ n ~ 2Nx or Ny + 1 ~ m ~ 2Ny or N z
+1 ~ l ~
(2.4.39) The X2 for nonzero values can be computed as in Table 2.4.1. Then the 3-D DFT and inverse DFT can be performed accordingly.
2 INTEGRAL EQUATION FORMULATIONS AND NUMERICAL METHODS
X2(n,m, l) g(n-l,m-l,I-I) g(n - 2Nx g(n - 2Nx g(n - 2Nx g(n - 2Nx
[1,Nx J [Nx [Nx
[1, Ny] [1, Ny]
[1,Nz J [1,Nz ] [1,Nz ] [1,Nz ]
+ 2,2Nx ]
[1,Nx ] [Ny
1, m - 1, 1- 1)
+ 2,2Nx ]
[1,Nx ]
+ 2,2Ny] [Ny + 2,2Ny]
[1, Ny]
g(n - 1, m - 2Ny - 1, 1- 1)
1, m - 2Ny - 1,1- 1) 1, m - 1, 1- 2Nz 1) 1)
[Nx [Nx
+ 2,2Nx
[1,Nx ]
[1,Ny] [Ny [Ny
+ 2,2Nx ]
+ 2,2Ny] + 2,2Ny]
+ 2,2Nz ] [Nz + 2,2Nz ] [Nz + 2,2Nz ] [Nz + 2,2Nz ]
g(n-l,m-l,l- 2Nz -1)
g(n - 1, m - 2Ny - 1,1- 2Nz
1, m - 2Ny - 1, 1- 2Nz
Table 2.4.1 Computations for the nonzero values of x2(n,m,I).
Conjugate Gradient Method
Consider a matrix equation of the form (2.5.1) where A is a N x N nonsingular matrix, x is the unknown column vector and b is the right hand side. Both x and bare N x 1 column vectors. In the following we briefly describe the conjugate gradient method. Details can be found in textbooks on matrix computation [Hestenes and Stiefel, 1952; Golub and Van Loan, 1996].
5.1 Steepest Descent Method
Let A be a real symmetric matrix and positive definite and (x) be the functional
A-(-) - -x t A- - -t-b x - l_ x x
(2.5.2)
where t denotes transpose so that index notation
xt is a row vector of dimension 1 x N.
(2.5.3) Taking the derivatives
- ax =
-"2 LAijxj -"2 LXjA ji + bi
5.1 Steepest Descent Method
= bi - LAijxj
(2.5.4)
The second equality is due to the fact that
A ij
= A ji
(2.5.5) (2.5.6) (2.5.7)
Thus the gradient is
-V' =b-Ax
Optimizing with respect to
x gives
O=-V' =b-Ax Ax=b.
The residual is the "left over" or the "remainder". Let (i - 1)th iterative solution. The residual is
Ti-l
This means that optimizing is equivalent to solving the matrix equation
Xi-l
be the (2.5.8) (2.5.9)
= b- AXi-l
The direction vector di gives the next solution
= Xi-l + D:idi
In the method of steepest descent, the direction vector is chosen to be the same as the residual vector.
(2.5.10) Then the ith solution is (2.5.11) Substituting in (2.5.2), we have
(Xi)
(Xi-l
+ D:idi)
+ D:idi ) t = A
-t =
Xi-l
1 =-
Xi-l
+ D:idi )
Xi-l
+ O'idi ) t -b
(Xi-l)
+ D:idi AXi-l + tdi Adi +
2 D: -t =D: 2
(Xidi b
D:iTi_l
A-(-) = <p Xi-l A-(-) = <p Xi-l
= -t AD:iTi-l Xi-l
+ 2~ -t - 1 A- Ti-l Ti
(25. 12) .
-t D:iTi_lTi-l
D:Lt + 2 Ti - 1 ATi-l D:i
Optimizing (Xi) by taking its derivative with respect to to zero gives
and setting it
(2.5.13)
2 INTEGRAL EQUATION FORMULATIONS AND NUMERICAL METHODS
In the steepest descent method, the direction vector is in the residual direction. On the other hand, in the conjugate gradient , the direction vector is more general. 5.2 Real Symmetric Positive Definite Matrix
We first summarize the results for the conjugate gradient method for real symmetric positive definite matrix. At the zeroth step, let the solution be
xa =
fi-1
(2.5.14)
At the (i - 1) th step, let Xi-1 be the approximate solution. Then
= b - AXi-1
(2.5.15) (2.5.16) (2.5.17)
is the residual at the (i - 1) th step. In particular,
To get a new solution Xi, i = 1,2, ... , let
= Xi-1 + O;idi
where di is the unknown direction vector and O;i is the unknown scalar to denote the movement in the di direction. To determine O;i and di , we note that
(Xi-1
+ O;idd = (Xi-1) + O;idi A Xi-1 + -;]:di A di -