Then in .NET

Maker QR-Code in .NET Then
Denso QR Bar Code barcode library on .net
using barcode encoding for .net vs 2010 control to generate, create qr code image in .net vs 2010 applications.
(5.1.23) (5.1.24)
Qr Bidimensional Barcode scanner on .net
Using Barcode decoder for .net framework Control to read, scan read, scan image in .net framework applications.
. ~H61) ( kJx~+z~ )
.net Vs 2010 Crystal bar code creator in .net
using visual .net crystal todeploy barcode on web,windows application
For the first three terms,
Attach barcode for .net
use visual .net barcode drawer toinclude bar code for .net
NT 2)1 ~al(Xd) ( ~~
Control qr codes size with visual c#
to deploy qrcode and qr barcode data, size, image with visual barcode sdk
QR Code JIS X 0510 barcode library in .net
generate, create qr-codes none for .net projects
Control qr code iso/iec18004 data with vb
to connect quick response code and denso qr bar code data, size, image with vb barcode sdk
= -4 H1 (kXd)T
Render barcode on .net
using .net vs 2010 toprint barcode in web,windows application
Visual Studio .NET pdf 417 drawer on .net
generate, create pdf417 none with .net projects
UPC - 13 barcode library for .net
using barcode integrated for .net vs 2010 crystal control to generate, create gs1 - 13 image in .net vs 2010 crystal applications.
If we retain three terms in the Taylor expansion (NT = 3),
Add ansi/aim code 39 on .net
use visual studio .net crystal ansi/aim code 39 drawer toprint 39 barcode on .net
= '""'" (Z(w) - Z(w)(O)) ~ mn mn
Insert cbc for .net
using .net framework crystal touse royal mail barcode for web,windows application
Control data matrix barcodes data in office excel
to access datamatrix and data matrix 2d barcode data, size, image with office excel barcode sdk
1.2 Formulation and Computational Procedure
Barcode Pdf417 drawer for .net
generate, create barcode pdf417 none in .net projects
(5.1.30) We make use of FFT in the calculation of Ym in (5.1.30). For example, for the second term in (5.1.30), -2(j(x m )) L:~=1 al~~d) f(xn)u n , we calculate in d the following manner: (1) pre-multiply Un by f(x n ) to get f(xn)u n (2) calculate L....J
Attach bar code on word
generate, create barcode none with office word projects
= Wn
Assign bar code for visual
using .net vs 2010 touse barcode with web,windows application
~ - - 2-w by FFT al(xd) n
Code128b barcode library in
use .net windows forms crystal code128b integrated torender code-128c in
1d Barcode generation for excel
use excel linear barcode integrated togenerate linear in excel
xd (3) post-multiply by -2f(xm )
Control barcode pdf417 image with excel spreadsheets
generate, create pdf417 none for excel spreadsheets projects
In the BMIA/CAG method, the Z is decomposed into a sum represented by the Taylor series expansion. Thus
Excel Spreadsheets 2d matrix barcode creation in excel spreadsheets
generate, create matrix barcode none on excel projects
= L....J Zm
'""'" =(w)
The m = 0 term corresponds to that of a flat surface. The form of Z m is such that it consists of terms that are products of a diagonal matrix TTl a translationally invariant matrix Zd, and a diagonal matrix T s : (5.1.32) where T s is a function of the coordinates x' of the scattering source, while T r is a function of the coordinates x of the field.
(B) Iteration Based on Updating the Right-Hand Side
Let X(O) and X(n) represent the zeroth-order solution and the nth-order solution, respectively. They obey the equations
=c =(S)-(n+l) -(n) Z X =c
(5.1.33) (5.1.34)
d n ) represents the updated right-hand side with
= C - L....J Zm X
=C- Z
'""'" =(w)-(n)
Note that for this case of iteration, only Z(s) is kept in the left-hand side of (5.1.34). A residual R(n) can be defined as follows such that its norm provides the stopping criterion for the iterative procedure:
-z X(n) + C = _(Z(s) + Z(Wlyx(n) + C
x 100%. From
where the normalized L-2 norm is defined as (5.1.33)-(5.1.36), it follows that
-(0) R =
=(s)-(O) =(w)-(O) X - Z X
+ C = -z
=(w)-(O) " =(w)-(O) X = - L..t Zm X
= -Z X
+ -C = -(n+l) C
Thus the residual vector can be computed readily from the updated righthand sides. In the numerical results illustrated in this section, the stopping criterion of the iterative solution is set at 0.1%.
Computational Complexity
For the TE case the matrix is symmetric. The bandwidth b is usually much smaller than the order of the matrix N. To take full advantage of the banded matrix Z(s), a direct banded matrix solver is used to solve (5.1.33) and (5.1.34). The LU decomposition requires O(b 2 N/2) operations, while the backsubstitution only requires O(2bN) operations. The
=(w)_ Z(w) X
product is
computed by the FFT. Therefore, we can evaluate Z X in r N (log N) + sN operations (where r accounts for the number of FFTs and s accounts for the number of pre- and post-multiplications before the FFT). The computational complexity up to the nth-order solutions are O(nb 2 N)+O(nrN log N +nsN).
(C) Solution Based on Complete Impedance Matrix and Conjugate Gradient Method (CGM)
Another iteration approach is to keep the entire impedance matrix on the left-hand side. Then we apply a conjugate gradient method (CGM) to the matrix equation with the matrix decomposition.
( =(S) Z
+ L..t Zm
~ =(W))
1.3 Weak Matrix and Unknown Column Vector
For the CGM version, an initial guess of XeD)
0 is chosen. Let N c be
=(s) =(w)
the number of CGM iterations. By decomposing into Z and Zm and the use of FFT in conjunction with CGM, the approach requires O(Nc(bN + rNlog N + sN)).
Memory Requirements
The memory requirement of the strong matrix Z(s) is O(bN). The coefficients am(xd) in the Taylor expansions are translationally invariant. The storage requirement for Zm , m = 0, 1,2, ... , M, is O((M +1)N). The total memory requirement for the algorithm is O(bN + (M + 1)N). In the simulations, the bandwidth b is an adjustable parameter. In the updated right-hand-side approach there is a minimum bandwidth bmin for which the iteration process works. It requires many more iteration steps to converge at b = bmin than at a larger bandwidth. Therefore, in the simulations, b is chosen to be greater than the bmin so as to reduce the number of iteration steps. For the CGM iterative approach discussed above, the bandwidth can be smaller than the one used in the updated right-hand-side approach. This is because the bandwidth in this case depends on the accuracy of the Taylor series expansion. As a result, this approach requires less computer memory, and therefore it is ideal for very large surface lengths. However, it usually takes more iteration steps to converge. In Section 1.5, only the 2500 wavelength surface examples are performed by applying the CGM iterative approach.