add ch16

diff --git a/BookGPU/BookGPU.tex b/BookGPU/BookGPU.tex
index edd313e11bf3e095a15b65546402d4f76e35b955..fab2a67befe268b85bac975761b1bb7872004c6a 100755 (executable)
--- a/BookGPU/BookGPU.tex
+++ b/BookGPU/BookGPU.tex
@@ -3,6 +3,7 @@
 \usepackage{amssymb}
 \usepackage{amsfonts,amssymb}
 \usepackage{amsmath}
 \usepackage{amssymb}
 \usepackage{amsfonts,amssymb}
 \usepackage{amsmath}

+\usepackage{amscd}

 \usepackage{graphicx}
 \usepackage{subfigure}
 \usepackage{makeidx}
 \usepackage{graphicx}
 \usepackage{subfigure}
 \usepackage{makeidx}


@@ -27,6 +28,12 @@
 \usepackage{pgfplots}
 \usepackage{pgf}
 \usepackage{tikz}
 \usepackage{pgfplots}
 \usepackage{pgf}
 \usepackage{tikz}

+\usepackage{url}
+\usepackage{dsfont}
+\usepackage{commath}
+\usepackage{stmaryrd}
+\usepackage{moreverb}
+\usepackage{commath}

 %% \usepackage{listings}
 %% \usepackage{subfigure}
 
 %% \usepackage{listings}
 %% \usepackage{subfigure}
 


@@ -98,6 +105,11 @@
 \captionsetup[lstlisting]{format=listing,labelfont=white,textfont=white, singlelinecheck=false, margin=0pt, font={bf,footnotesize}}
 
 
 \captionsetup[lstlisting]{format=listing,labelfont=white,textfont=white, singlelinecheck=false, margin=0pt, font={bf,footnotesize}}
 
 

+\DeclareGraphicsRule{.pdftex}{pdf}{*}{}
+
+\DeclareGraphicsRule{.pstex}{eps}{*}{}
+
+

 
 \makeindex
 
 
 \makeindex
 


@@ -130,17 +142,19 @@
 
 \setcounter{page}{1}
 \part{This is a Part}
 
 \setcounter{page}{1}
 \part{This is a Part}

-\include{Chapters/chapter1/ch1}
-\include{Chapters/chapter2/ch2}
-\include{Chapters/chapter3/ch3}
-\include{Chapters/chapter5/ch5}
-\include{Chapters/chapter6/ch6}
-\include{Chapters/chapter7/ch7}
-\include{Chapters/chapter8/ch8}
-\include{Chapters/chapter9/ch9}
-\include{Chapters/chapter11/ch11}
-\include{Chapters/chapter14/ch14}
-\include{Chapters/chapter15/ch15}
+%% \include{Chapters/chapter1/ch1}
+%% \include{Chapters/chapter2/ch2}
+%% \include{Chapters/chapter3/ch3}
+%% \include{Chapters/chapter5/ch5}
+%% \include{Chapters/chapter6/ch6}
+%% \include{Chapters/chapter7/ch7}
+%% \include{Chapters/chapter8/ch8}
+%% \include{Chapters/chapter9/ch9}
+%% \include{Chapters/chapter11/ch11}
+%% \include{Chapters/chapter14/ch14}
+%% \include{Chapters/chapter15/ch15}
+ \include{Chapters/chapter16/ch16}
+%% \include{Chapters/chapter18/ch18}

 
 \bibliographystyle{hep}
 %%%\bibliography{biblio}
 
 \bibliographystyle{hep}
 %%%\bibliography{biblio}

diff --git a/BookGPU/Chapters/chapter16/bdf.tex b/BookGPU/Chapters/chapter16/bdf.tex
new file mode 100644 (file)
index 0000000..dcf6313
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/bdf.tex
@@ -0,0 +1,179 @@
+\subsection{Gear-2 based sensitivity calculation}
+\label{sec:gear}
+
+The Gear-2 integration method is a backward differentiation
+formula (BDF), which approximates the derivative of a function
+using information from past few steps.  Gear-2 method has been
+shown to be more suitable for many practical problems such as
+stiff problems where circuit behavior is affected by different time constants
+(fast ones with large poles and slow ones with small poles)~\cite{Vlach:Book'94}.
+Switching power converters and RF systems are typically stiff systems
+as waveforms changing in two different time scales are involved.
+
+% Although on one time step, Gear-2
+% involves more computation than backward-Euler and trapezoidal method,
+% it has better convergence and stability, and usually allows larger
+% time step.  As a result, fewer time points are needed for a given
+% transient sweep range, and the simulation is speeded up with the same
+% accuracy
+
+Given the DAE~(\ref{eq:dae}), at the $i$-th time step,
+Gear-2 integration\index{Gear-2 integration}
+approximates the derivative $\ud q(x(t))/\ud t$
+by a two-step backward finite difference,
+\[
+   \frac{\ud q(x(t_i))}{\ud t}
+   %\ud q(x(t_i)) / \ud t
+   = \frac{1}{h_i}
+     \left[ q(x(t_{i})) - \alpha_1^i q(x(t_{i-1})) - \alpha_2^i q(x(t_{i-2}))\right],
+\]
+where the coefficients $\alpha$'s are chosen to satisfy Gear's
+backward differentiation formula~\cite{Gear:book'71}.
+For uniformly discretised time steps,
+the index $i$ in $h_i$, $\alpha_1^i$ and $\alpha_2^i$ can be omitted.
+
+For the first step $t_1$,
+only the initial condition at $t_0$ is available.
+Therefore backward-Euler is used, i.e.,
+\begin{equation}\label{eq:BE}
+   \tfrac{1}{h_1}[q(x_1)-q(x_0)] + f(x_1) = b_1.
+\end{equation}
+Since $x_0$ directly affects the solution of $x_1$,
+the sensitivity matrix\index{sensitivity matrix} up to now is
+\begin{equation}\label{eq:sens1}
+   \frac{\ud x_1}{\ud x_0}
+   = \left[\frac{1}{h_1}C_1+G_1\right]^{-1} \frac{C_0}{h_1}
+   = J_1^{-1} \frac{C_0}{h_1}.
+\end{equation}
+%For the sake of simplicity,
+Let $J_i$ denote $(1/h_i)C_i + G_i$
+in the remaining part of this chapter.
+In addition, the LU\index{LU} factorizations of $J_i$ are already calculated
+since they are used to solve the circuit state at
+each time step before we calculate the sensitivity.
+
+Next, for the solution at $t_2$,
+with the previous two steps information available,
+Gear-2 integration can be applied,
+\begin{equation}\label{eq:Gear_t2}
+   \tfrac{1}{h_2}[q(x_2) - \alpha_1 q(x_1) - \alpha_2 q(x_0)] + f(x_2) = b_2.
+\end{equation}
+In view of the sensitivity of $x_2$ with respect to the changes of $x_0$,
+\eqref{eq:Gear_t2} indicates that $x_2$'s perturbation can be
+traced back to $x_0$ along two paths:
+$x_2$ is directly affected by $x_0$,
+and $x_2$ is also affected indirectly by $x_0$ via $x_1$.
+That is,
+\[ %begin{equation}
+  \frac{\ud x_2}{\ud x_0} = \frac{\partial x_2}{\partial x_1}\frac{\ud x_1}{\ud x_0}
+                          + \frac{\partial x_2}{\partial x_0},
+\] %end{equation}
+where the two partial derivatives are
+\[
+   \frac{\partial x_2}{\partial x_1} = J_2^{-1} \frac{\alpha_1}{h_2} C_1, \qquad
+   \frac{\partial x_2}{\partial x_0} = J_2^{-1} \frac{\alpha_2}{h_2} C_0,
+\]
+and $\ud x_1/\ud x_0$ %, the sensitivity of $x_1$ with respect to $x_0$,
+is calculated previously in \eqref{eq:sens1}.
+Thus, the sensitivity matrix deduced from \eqref{eq:Gear_t2} is
+\begin{equation}\label{eq:sens2}
+   \frac{\ud x_2}{\ud x_0}
+   = J_2^{-1} \frac{\alpha_1}{h_2} C_1 \cdot J_1^{-1} \frac{C_0}{h_1}
+   + J_2^{-1} \frac{\alpha_2}{h_2} C_0.
+\end{equation}
+
+Likewise, for the third time step $t_3$,
+apply the Gear-2 integration formula to DAE~(\ref{eq:dae}),
+\begin{equation}\label{eq:Gear_t3}
+   \tfrac{1}{h_3} [q(x_3) - \alpha_1 q(x_2) - \alpha_2 q(x_1)] + f(x_3) = b_3,
+\end{equation}
+and the chain rule for sensitivity is
+\[ %begin{equation}\label{eq:sens3}
+% \begin{split}
+%    \frac{\ud x_3}{\ud x_0} &= \frac{\partial x_3}{\partial x_2}\frac{\ud x_2}{\ud x_0}
+%                             + \frac{\partial x_3}{\partial x_1}\frac{\ud x_1}{\ud x_0} \\
+%    &=  J_3^{-1} \frac{\alpha_1}{h_3} C_2 \frac{\ud x_2}{\ud x_0}
+%      + J_3^{-1} \frac{\alpha_2}{h_3} C_1 \frac{\ud x_1}{\ud x_0}.
+% \end{split}
+   \frac{\ud x_3}{\ud x_0} \!=\! \frac{\partial x_3}{\partial x_2}\frac{\ud x_2}{\ud x_0}
+                            + \frac{\partial x_3}{\partial x_1}\frac{\ud x_1}{\ud x_0}
+   \!=\!  J_3^{-1} \!\left[ \frac{\alpha_1}{h_3} C_2 \frac{\ud x_2}{\ud x_0}
+                   \!+\!\frac{\alpha_2}{h_3} C_1 \frac{\ud x_1}{\ud x_0}\right],
+\] %end{equation}
+where both $\ud x_2/\ud x_0$ and $\ud x_1/\ud x_0$
+are ready from the previous two time steps, i.e., Eqs.~(\ref{eq:sens2}) and~(\ref{eq:sens1}).
+Follow this chain rule in the aforementioned recursive style,
+the sensitivity matrix for Gear-2 integration is computed
+along the remaining time steps up to the $M$-th step,
+\begin{equation}\label{eq:sensM}
+% \begin{split}
+%    J = \frac{\ud x_M}{\ud x_0}
+%      & =  J_M^{-1} \frac{\alpha_1}{h_M} C_{M-1} \frac{\ud x_{M-1}}{\ud x_0}  \\
+%      &\quad + J_M^{-1} \frac{\alpha_2}{h_M} C_{M-2} \frac{\ud x_{M-2}}{\ud x_0}.
+% \end{split}
+J\! =\! \frac{\ud x_M}{\ud x_0}
+ \! =\!  J_M^{-1} \! \left[ \frac{\alpha_1}{h_M} C_{M-1} \frac{\ud x_{M-1}}{\ud x_0}
+     \! + \! \frac{\alpha_2}{h_M} C_{M-2} \frac{\ud x_{M-2}}{\ud x_0}\right].
+\end{equation}
+
+%% \begin{algorithm}
+%% \caption{The matrix-free\index{matrix-free} method for
+%%  Krylov subspace\index{Krylov subspace} construction.}
+%% \label{alg:mf_Gear}
+%% \begin{algorithmic}[1]
+%%   \REQUIRE current Krylov subspace basis vector $v$,
+%%            time step lengths $h_i$,
+%%            saved $C_i$ matrices and LU\index{LU} factors of $J_i$, $i=0,\ldots,M$
+%%   \ENSURE matrix-vector product $r$, such that $r = A v$
+%%   \STATE solve $J_1 p_2 = h_1^{-1} C_0 v$ for $p_2$
+%%   \STATE solve $J_2 p_1 = h_2^{-1} \left( \alpha_1 C_1 p_2 + \alpha_2 C_0 v \right)$ for $p_1$
+%%   \FOR{$i=3 \ldots M$}
+%%     \STATE solve
+%%        $J_i p_0 = \alpha_1 h_i^{-1} C_{i-1} p_1 + \alpha_2 h_i^{-1} C_{i-2} p_2$
+%%        for $p_0$ \label{line:mf_Gear_loop}
+%%     \STATE $p_2 = p_1$
+%%     \STATE $p_1 = p_0$
+%%   \ENDFOR
+%%   \STATE $r = (k-1) p_0 - k v$ \label{line:shift}
+%% \end{algorithmic}
+%% \end{algorithm}
+
+
+%It is easy to judge that, as $J$ is dense, explicitly
+%computing $J$ is expensive and impractical for large problems.
+%In addition, with a dense matrix,
+%both direct or iterative solvers will be inefficient.
+%Fortunately, one iterative solver, GMRES,
+%only requires $J x$ (actually $Ax$),
+%instead of an explicit $J$, to fulfill the solving process.
+%This virtue greatly saves computation cost.
+Note that as the matrix-free\index{matrix-free} GMRES method is applied,
+which only requires matrix-vector multiplication and no explicit $J$ is required,
+as explained in Algorithm~\ref{alg:mf_Gear}.
+% Next section will explain how to use this technique for
+% our proposed Gear-2 envelope-following.
+
+With matrix-free method, the matrix-vector multiplication
+in Line~\ref{line:mvp} of Algorithm~\ref{alg:GMRES}
+is replaced by the iteration shown in Algorithm~\ref{alg:mf_Gear},
+which calculates the multiplication product of the Gear-2 sensitivity we
+encounter in envelope-following and a basis vector in the Krylov
+subspace.
+Note that the scaling and shift of $J$ in $A=(k-1)J-kI$,
+as described in Eq.~\eqref{eq:A},
+is taken into consideration by Line~\ref{line:shift}
+of Algorithm~\ref{alg:mf_Gear}.
+
+For the matrix-free generation of new basis vectors, it is
+straightforward to apply CUBLAS and CUSPARSE routines, and some
+customized GPU kernel functions to implement
+Algorithm~\ref{alg:mf_Gear} with the stored LU matrices of $J_i$ and
+$C_i$ mentioned earlier. For example in Line~\ref{line:mf_Gear_loop},
+as the iteration index $i$ traverses all the $M$ time points'
+matrix information,
+sparse matrix-vector multiplication \verb|csrmv|
+is first called to calculate $C_{i-1} p_1$ and $C_{i-2} p_2$.
+And after the two resulted vectors are combined by \verb|axpy| of CUBLAS,
+$p_0$ is solved for by \verb|trsv|, since as we noted before,
+$J_i$ is already in LU factorization form when transient simulation
+at step $i$ finished.

diff --git a/BookGPU/Chapters/chapter16/biblio16.bib b/BookGPU/Chapters/chapter16/biblio16.bib
new file mode 100644 (file)
index 0000000..1252282
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/biblio16.bib
@@ -0,0 +1,171 @@
+
+@Book{Trzynadlowski:Book'10,
+  author = "A. M. Trzynadlowski",
+  title = "Introduction to Modern Power Electronics",
+  publisher = "Wiley",
+  edition = "Second",
+  year = "2010"
+}
+
+@InProceedings{Kundert:ICCAD'88,
+  author =      {K. Kundert and J. White and A. Sangiovanni-Vincentelli},
+  title =       {An envelope following method for the efficient transient simulation
+                  of switching power and filter circuits},
+  booktitle =   {Proc. ICCAD},
+  pages =       {446--449},
+  year =        1988,
+  month =       {Oct.}
+}  
+
+
+@Article{White:TPE'91,
+  author =      {J. White and S. Leeb},
+  title =       {An envelope-following approach to switching
+                  power converter simulation},
+  journal =     {IEEE Trans. Power Electron.},
+  year =        1991,
+  volume =      6,
+  number =      2,
+  pages =       {303--307},
+  month =       {Apr.}
+}
+
+@InProceedings{Feldmann:ICCAD'96,
+  author =      {P. Feldmann and J. Roychowdhury},
+  title =       {Computation of circuit waveform envelopes using an efficient,
+                  matrix-decomposed harmonic balance algorithm},
+  booktitle =   {Proc. ICCAD},
+  pages =       {295--300},
+  year =        1996,
+  month =       {Nov.}
+}
+
+@Book{Krein:book'97,
+  author =      {P. Krein},
+  title =       {Elements of Power Electronics},
+  publisher =   {Oxford University Press},
+  year =        1997
+}
+
+@InProceedings{Telichevesky:DAC'95,
+  author =       {R. Telichevesky and K. Kundert and J. White},
+  title =        {Efficient steady-state analysis based on matrix-free {K}rylov-subspace methods},
+  booktitle =    dac,
+  pages =       {480--484},
+  year =     1995,
+}
+
+@Misc{IntelMC'06,
+  author = {{Intel Corporation}},
+  title =    {Intel Multi-Core Processors, Making the move to quad-core and beyond ({W}hite {P}aper)},
+  year =     2006,
+  note =     {\url{http://www.intel.com/multi-core}}
+}
+
+@Misc{AMDMC'06,
+  author = {{AMD Inc.}},
+  title =    {Multi-Core Processors---The Next Evolution in Computing ({W}hite {P}aper)},
+  year =     2006,
+  note =     {\url{http://multicore.amd.com}}
+}
+
+@MISC{nvidia,
+    author = {{NVIDIA Corporation}},
+    key = {openmp},
+    note    = {{h}ttp://www.nvidia.com},
+    year = 2011
+}
+
+@Book{Kirk:Book'10,
+  Author="D. B. Kirk and W.-M. Hwu",
+  Title="Programming Massively Parallel Processors: A Hands-on Approach",
+  Year="2010",
+  Publisher="Morgan Kaufmann Publishers Inc.",
+  Address="San Francisco, CA",
+}
+
+@MISC{gpgpu,
+  author = {D. G\"{o}ddeke},
+  title =        {General-Purpose Computation Using Graphics Harware},
+  howpublished = {\url{http://www.gpgpu.org/}},
+  year = 2011
+}
+
+@misc{cuda,
+    author = {{{NVIDIA} Corporation}},
+    title   = {{CUDA} ({C}ompute {U}nified {D}evice {A}rchitecture)},
+    note    = {\url{http://www.nvidia.com/object/cuda_home.html}},
+    year = 2011
+}
+
+@MISC{streamSDK,
+  author = {{AMD Inc.}},
+  title =        {{AMD} {S}team {SDK}},
+  howpublished = {\url{http://developer.amd.com/gpu/ATIStreamSDK}},
+  year = 2011
+}
+
+@MISC{openCL,
+  author = {{Khronos Group}},
+  title =        {Open {C}omputing {L}anguage ({O}pen{CL})},
+  howpublished = {\url{http://www.khronos.org/opencl}},
+  year = 2011
+}
+
+@Misc{CUDA_Zone,
+  title =       {{CUDA} Community Showcase},
+  howpublished = {\url{http://www.nvidia.com/}}
+}
+
+@BOOK{Golub:Book'96,
+  AUTHOR =       "G. H. Golub and C. Van Loan",
+  TITLE =        "Matrix Computations",
+  PUBLISHER =    "The Johns Hopkins University Press",
+  YEAR =         "1996",
+  edition =      "3rd",
+}
+
+@Misc{CUDASDK,
+  author =      {{NVIDIA}},
+  year =        2011,
+  note =        {\url{http://developer.nvidia.com/cuda-toolkit-40}}
+}
+
+@Book{Gear:book'71,
+  author =      {C. W. Gear},
+  title =       {Numerical Initial Value Problems in Ordinary Differential Equations},
+  publisher =   {Prentice-Hall},
+  year =        1971,
+  address =     {Englewood Cliffs, NJ}
+}
+
+@Misc{ngspice,
+  title =       {{NGSPICE}},
+  howpublished = {\url{http://ngspice.sourceforge.net/}}
+}
+
+@INPROCEEDINGS{Kato:COMPEL'06,
+  author={Kato, T. and others},
+  booktitle={IEEE COMPEL'06},
+  title={Envelope Following Analysis of an Autonomous Power Electronic System},
+  year={2006},
+  month={July},
+  pages={29--33}
+}
+
+@Book{Vlach:Book'94,
+  author =   {J. Vlach and K. Singhal},
+  title =        {Computer Methods for Circuit Analysis and Design},
+  publisher =    {Van Nostrand Reinhold},
+  year =         {1995},
+  address =      {New York, NY},
+}
+
+@conference{LiuTan1:DATE'12,
+author={X.-X. Liu and S. X.-D. Tan and H. Wang and H. Yu},
+title={A {GPU}-accelerated envelope-following method for switching power converter simulation”},
+booktitle= date,
+year={2012},
+pages = {1349-1354},
+month ={March},
+}

diff --git a/BookGPU/Chapters/chapter16/ch16.aux b/BookGPU/Chapters/chapter16/ch16.aux
new file mode 100644 (file)
index 0000000..7d1293a
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/ch16.aux
@@ -0,0 +1,101 @@
+\relax 
+\@writefile{toc}{\author{X.-X. Liu}{}}
+\@writefile{toc}{\author{S. X.-D. Tan}{}}
+\@writefile{toc}{\author{H. Wang}{}}
+\@writefile{toc}{\author{H. Yu}{}}
+\@writefile{loa}{\addvspace {10\p@ }}
+\@writefile{toc}{\contentsline {chapter}{\numberline {1}GPU-Accelerated Envelope-Following Method}{3}}
+\@writefile{lof}{\addvspace {10\p@ }}
+\@writefile{lot}{\addvspace {10\p@ }}
+\@writefile{toc}{\contentsline {section}{\numberline {1.1}Introduction}{3}}
+\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
+\newlabel{fig:ef1}{{1.1(a)}{5}}
+\newlabel{sub@fig:ef1}{{(a)}{5}}
+\newlabel{fig:ef2}{{1.1(b)}{5}}
+\newlabel{sub@fig:ef2}{{(b)}{5}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.1}{\ignorespaces Transient envelope-following analysis. (Both two figures reflect backward-Euler style envelope-following.)\relax }}{5}}
+\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {Illustration of one envelope skip.}}}{5}}
+\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {The envelope changes in a slow time scale.}}}{5}}
+\newlabel{fig:ef_intro}{{1.1}{5}}
+\@writefile{toc}{\contentsline {section}{\numberline {1.2}The envelope-following method in a nutshell}{6}}
+\newlabel{sec:ef}{{1.2}{6}}
+\newlabel{eq:dae}{{1.1}{6}}
+\newlabel{eq:Newton}{{1.2}{7}}
+\newlabel{eq:A}{{1.3}{7}}
+\@writefile{toc}{\contentsline {section}{\numberline {1.3}New parallel envelope-following method}{8}}
+\newlabel{sec:gmres}{{1.3}{8}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.1}GMRES solver for Newton update equation}{8}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.2}{\ignorespaces The flow of envelope-following method.\relax }}{9}}
+\newlabel{fig:ef_flow}{{1.2}{9}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.2}Parallelization on GPU platforms}{10}}
+\newlabel{sec:gpu}{{1.3.2}{10}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.3}{\ignorespaces GPU parallel solver for envelope-following update.\relax }}{11}}
+\newlabel{fig:gmres}{{1.3}{11}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.3.3}Gear-2 based sensitivity calculation}{12}}
+\newlabel{sec:gear}{{1.3.3}{12}}
+\newlabel{eq:BE}{{1.4}{12}}
+\newlabel{eq:sens1}{{1.5}{12}}
+\newlabel{eq:Gear_t2}{{1.6}{13}}
+\newlabel{eq:sens2}{{1.7}{13}}
+\newlabel{eq:Gear_t3}{{1.8}{13}}
+\newlabel{eq:sensM}{{1.9}{13}}
+\@writefile{toc}{\contentsline {section}{\numberline {1.4}Numerical examples}{14}}
+\newlabel{sec:exp}{{1.4}{14}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.4}{\ignorespaces Diagram of a zero-voltage quasi-resonant flyback converter.\relax }}{15}}
+\newlabel{fig:flyback}{{1.4}{15}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.5}{\ignorespaces Illustration of power/ground network model.\relax }}{15}}
+\newlabel{fig:pg}{{1.5}{15}}
+\@writefile{lot}{\contentsline {table}{\numberline {1.1}{\ignorespaces CPU and GPU time comparisons (in seconds) for solving Newton update equation with the proposed Gear-2 sensitivity. \relax }}{15}}
+\newlabel{table:circuit}{{1.1}{15}}
+\newlabel{fig:flybackWhole}{{1.6(a)}{16}}
+\newlabel{sub@fig:flybackWhole}{{(a)}{16}}
+\newlabel{fig:flybackZoom}{{1.6(b)}{16}}
+\newlabel{sub@fig:flybackZoom}{{(b)}{16}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.6}{\ignorespaces Flyback converter solution calculated by envelope-following. The red curve is traditional SPICE simulation result, and the back curve is the envelope-following output with simulation points marked.\relax }}{16}}
+\@writefile{lof}{\contentsline {subfigure}{\numberline{(a)}{\ignorespaces {The whole plot}}}{16}}
+\@writefile{lof}{\contentsline {subfigure}{\numberline{(b)}{\ignorespaces {Detail of one EF simulation period}}}{16}}
+\newlabel{fig:flyback_wave}{{1.6}{16}}
+\@writefile{lof}{\contentsline {figure}{\numberline {1.7}{\ignorespaces Buck converter solution calculated by envelope-following.\relax }}{17}}
+\newlabel{fig:buck_wave}{{1.7}{17}}
+\@writefile{toc}{\contentsline {section}{\numberline {1.5}Summary}{17}}
+\newlabel{sec:summary}{{1.5}{17}}
+\@writefile{toc}{\contentsline {section}{\numberline {1.6}Glossary}{18}}
+\@writefile{toc}{\contentsline {section}{Bibliography}{18}}
+\@setckpt{Chapters/chapter16/ch16}{
+\setcounter{page}{20}
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+}

diff --git a/BookGPU/Chapters/chapter16/ch16.tex b/BookGPU/Chapters/chapter16/ch16.tex
new file mode 100644 (file)
index 0000000..bfa5d6c
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/ch16.tex
@@ -0,0 +1,85 @@
+\chapterauthor{X.-X. Liu}{Dept. Electrical Engineering,
+  University  of California, Riverside, CA 92521}
+\chapterauthor{S. X.-D. Tan}{Dept. Electrical Engineering,
+  University  of California, Riverside, CA 92521}
+\chapterauthor{H. Wang}{Univ. of Electronics Science and Technology of China,
+  Chengdu, Sichuan, China}
+\chapterauthor{H. Yu}{School of Electrical \& Electronic Engineering,
+  Nanyang Technological University, Singapore}
+
+%  \thanks{
+%    This research was supported in part by NSF grants under
+%    No.~CCF-1017090 
+%    No.~OISE-1051797, and
+%    No.~OISE-0929699.
+%  }
+
+\newcommand{\ud}{\,\mathrm{d}}
+
+%\renewcommand{\algorithmicrequire}{\textbf{Input:}}
+%\renewcommand{\algorithmicensure}{\textbf{Output:}}
+%\renewcommand{\algorithmiccomment}[1]{\% \textit{#1}}
+
+\chapter[GPU-Accelerated Envelope-Following Method]{A GPU-Accelerated Envelope-Following Method for Switching Power Converter Simulation}
+
+% \section{Abstract}
+% % Power converters have seen a surge of new trends and novel
+% % applications due to their widespread use  in renewable energy
+% % systems and emerging  hybrid and purely-electric vehicles. More
+% % efficient simulation  techniques for power converters are urgently
+% % needed to meet more design constraints.
+% In this chapter, we propose a new envelope-following parallel transient analysis method for 
+% the general switching power converters. The new method first exploits 
+% the parallelisim in the envelope-following method
+% and parallelize the Newton update solving part,
+% which is the most computational expensive, in GPU platforms 
+% to boost the simulation performance.
+% To further speed up the iterative GMRES
+% solving for Newton update equation in the envelope-following
+% method, we apply the matrix-free Krylov basis generation
+% technique, which was previously used for RF simulation. 
+% Last, the new method also applies more robust
+% Gear-2 integration to compute the sensitivity matrix instead of
+% traditional integration methods. 
+% %Furthermore, the resulted Gear-2 and matrix-free GMRES have been
+% Experimental results from several integrated on-chip power
+% converters show that the proposed GPU envelope-following algorithm leads to
+% about 10$\times$ speedup compared to its CPU counterpart,
+% and 100$\times$ faster than the traditional envelop-following methods
+% while still keeps the similar accuracy.
+
+\input{Chapters/chapter16/intro.tex}
+\input{Chapters/chapter16/ef.tex}
+%\input bdf.tex  % now inside gpu.tex now
+\input{Chapters/chapter16/gpu.tex}
+\input{Chapters/chapter16/exp.tex}
+
+\section{Summary}
+\label{sec:summary}
+In this chapter, we present a new envelope-following method for
+transient analysis of switching power converters.  First, the
+computationally expensive step, the solving of Newton update equation,
+has been parallelized on CUDA-enabled GPU platforms with iterative
+GMRES solver to boost performance of the analysis method.  To further
+speed up the GMRES solving for Newton update equation, we have
+employed the matrix-free Krylov basis generation technique.  The
+proposed method also applies the more robust Gear-2 integration to
+compute the sensitivity matrix.  Experimental results from several
+integrated on-chip power converters have shown that the proposed GPU
+envelope-following algorithm can lead to about 10$\times$ speedup
+compared to its CPU counterpart, and 100$\times$ faster than the
+traditional envelope-following methods while still keeps the similar
+accuracy.
+
+
+\section{Glossary}
+\begin{Glossary}
+\item[Envelope-Following] In transient simulation of switching power circuits,
+nodal voltage waveforms in neighboring high frequency clock cycles are similar,
+but not exactly the duplicates. Envelope-following technique approximates
+the slowly changing transient trend over a lot of clock cycles
+without calculating waveforms in all cycles.
+\end{Glossary}
+
+\putbib[Chapters/chapter16/biblio16]
+%\bibliography{./envelope,../../bib/interconnect,../../bib/architecture,../../bib/simulation}

diff --git a/BookGPU/Chapters/chapter16/date12envelope.tex b/BookGPU/Chapters/chapter16/date12envelope.tex
new file mode 100644 (file)
index 0000000..67f9016
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/date12envelope.tex
@@ -0,0 +1,139 @@
+\documentclass[conference]{IEEEtran}
+
+\usepackage{graphicx}
+\usepackage{algorithm,algorithmic}
+\usepackage{multirow}
+\usepackage{subfigure}
+\usepackage{booktabs}
+\usepackage{epstopdf}
+\usepackage{graphicx}
+\usepackage[dvips]{color}
+\usepackage{psfrag}
+\usepackage{pstricks}
+\usepackage{amssymb,amsmath}
+%\usepackage{authblk}
+\usepackage{cite}
+\usepackage{url}
+
+\DeclareGraphicsRule{.pdftex}{pdf}{*}{}
+\graphicspath{{matlab/}}
+
+
+\newtheorem{myformulation}{Formulation}
+\newtheorem{mytheorem}{Theorem}
+\newtheorem{mylemma}{Lemma}
+\newtheorem{mydefinition}{Definition}
+%\ninept
+
+% correct bad hyphenation here
+%\hyphenation{op-tical net-works semi-conduc-tor IEEEtran}
+%\renewcommand{\baselinestretch}{0.99}
+
+
+\renewcommand{\algorithmicrequire}{\textbf{Input:}}
+\renewcommand{\algorithmicensure}{\textbf{Output:}}
+\renewcommand{\algorithmiccomment}[1]{\% \textit{#1}}
+
+
+\newcommand{\setRn}{\mathbb{R}_{}^N}
+\newcommand{\setR}{\mathbb{R}}
+\newcommand{\ud}{\,\mathrm{d}}
+\newcommand{\res}[1]{\mathbf{r}^{(#1)}}
+\newcommand{\vtr}[1]{\mathbf{#1}}
+\newcommand{\mtx}[1]{\mathbf{#1}}
+\newcommand{\mtxPhi}{\boldsymbol{\phi}}
+\newcommand{\mT}{\mathrm{T}}
+
+\IEEEoverridecommandlockouts
+
+
+\renewcommand{\baselinestretch}{0.97}
+
+\begin{document}
+\title{
+A GPU-Accelerated Envelope-Following Method for Switching Power Converter Simulation
+}
+\author{
+  \IEEEauthorblockN{
+    Xue-Xin Liu\IEEEauthorrefmark{1},
+    Sheldon X.-D. Tan\IEEEauthorrefmark{1},
+    Hai Wang\IEEEauthorrefmark{1}, and
+    Hao Yu\IEEEauthorrefmark{2}
+  }
+  \IEEEauthorblockA{
+    \IEEEauthorrefmark{1}Dept. Electrical Engineering,
+    University  of California, Riverside, CA 92521
+  }
+  \IEEEauthorblockA{
+    \IEEEauthorrefmark{2}School of Electrical \& Electronic Engineering,
+    Nanyang Technological University
+  }
+  \thanks{
+    This research was supported in part by NSF grants under
+    No.~CCF-1017090 
+    No.~OISE-1051797, and
+    No.~OISE-0929699.
+  }
+}
+
+%\IEEEpubid{978-3-9810801-8-6/DATE12/@2012 EDAA}
+
+\maketitle
+
+\begin{abstract}
+% Power converters have seen a surge of new trends and novel
+% applications due to their widespread use  in renewable energy
+% systems and emerging  hybrid and purely-electric vehicles. More
+% efficient simulation  techniques for power converters are urgently
+% needed to meet more design constraints.
+In this paper, we propose a new envelope-following parallel transient analysis method for 
+the general switching power converters. The new method first exploits 
+the parallelisim in the envelope-following method
+and parallelize the Newton update solving part,
+which is the most computational expensive, in GPU platforms 
+to boost the simulation performance.
+To further speed up the iterative GMRES
+solving for Newton update equation in the envelope-following
+method, we apply the matrix-free Krylov basis generation
+technique, which was previously used for RF simulation. 
+Last, the new method also applies more robust
+Gear-2 integration to compute the sensitivity matrix instead of
+traditional integration methods. 
+%Furthermore, the resulted Gear-2 and matrix-free GMRES have been
+Experimental results from several integrated on-chip power
+converters show that the proposed GPU envelope-following algorithm leads to
+about 10$\times$ speedup compared to its CPU counterpart,
+and 100$\times$ faster than the traditional envelop-following methods
+while still keeps the similar accuracy.
+\end{abstract}
+
+\input intro.tex
+\input ef.tex
+%\input bdf.tex  % now inside gpu.tex now
+\input gpu.tex
+\input exp.tex
+
+\section{Conclusion} \label{sec:conclusion}
+A new envelope-following method for transient analysis of
+switching power converters has been introduced.
+First, the computationally expensive step,
+the solving of Newton update equation,
+has been parallelized on CUDA-enabled GPU platforms
+with iterative GMRES solver to boost performance of the analysis method.
+To further speed up the GMRES solving for Newton update equation,
+we have employed the matrix-free Krylov basis generation technique.
+The proposed method also applies the more robust Gear-2 integration
+to compute the sensitivity matrix.
+Experimental results from several integrated
+on-chip power converters have shown that the proposed GPU
+envelope-following algorithm can lead to about 10$\times$
+speedup compared to its CPU counterpart,
+and 100$\times$ faster than the traditional
+envelop-following methods while still keeps the similar accuracy.
+
+
+\bibliographystyle{unsrt} % IEEEtranS
+\bibliography{./envelope,../../bib/interconnect,../../bib/architecture,../../bib/simulation}
+% ./rfsim.bib
+
+\end{document}

diff --git a/BookGPU/Chapters/chapter16/ef.tex b/BookGPU/Chapters/chapter16/ef.tex
new file mode 100644 (file)
index 0000000..b0967a9
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/ef.tex
@@ -0,0 +1,146 @@
+\section{The envelope-following method in a nutshell}
+\label{sec:ef}
+
+In transient analysis\index{transient analysis},
+a nonlinear circuit's behavior can be represented
+by a coupled set of nonlinear first-order
+differential algebraic equations\index{differential algebraic equations}
+(DAE) of the form
+\begin{equation}\label{eq:dae}
+   \frac{\ud}{\ud t} q(x(t)) + f(x(t)) = b(t),
+\end{equation}
+where $x(t)\in \mathbb{R}^N$ is the vector of circuit variables,
+usually comprising node voltages and possibly branch currents,
+$f(\cdot)$ is a nonlinear function that maps $x(t)$ to a vector
+of $N$ entries most of
+which are sums of resistive currents at a node,
+$q(\cdot)$ %is also a nonlinear function that
+maps $x(t)$ to a vector of $N$ entries of
+capacitor charges or inductor fluxes,
+and $b(t)$ contains the input source values.
+For many power electronics circuits, the input
+switching signal is known %as a circuit parameter
+and is periodic with clock period $T$.
+
+Assume the time inside one period $T$ has been discretised
+into $M$ time steps, $0=t_0 < t_1 < t_2 < \cdots < t_M=T$, and the
+$i$-th step length is $h_i = t_i-t_{i-1}$.
+In practice, the discretisation is nonuniform to
+control truncation error\index{truncation error} and convergence.
+Then, given an initial condition $x(0)$ at $t=0$,
+numerical integration is applied to find the time domain solution of
+circuit state $x(t)$ at each time step till $t=T$.
+%, in a consequent time series.
+
+For those circuits whose carrier waveforms\index{carrier waveforms}
+vary slowly in a large number of periods,
+the envelope-following\index{envelope-following} method only integrates the
+DAE in several selected periods and then jumps over to a new time point.
+By repeating this ``simulate and skip'' action,
+envelope-following method attains its efficiency
+compared to conventional transient analysis, but still can accurately
+obtain the slow varying envelope.
+
+For example, if the clock period is $T$, and the suitable jump
+interval for the envelope is $k$ periods, then the envelope step is
+$kT$.  Suppose the state at time $t=mT$ is known as $x(mT)$, and the
+envelope-following wants to obtain the state at the next envelope
+step, $x((m+k)T)$.  If the envelope step is much larger than the clock
+period ($k$ is much bigger than one), envelope-following will lead to
+significant savings in simulation time.
+
+% One simple way to estimate $x((m+k)T)$ is to use the information at
+% $mT$ and $(m-1)T$, which leads to the forward-Euler method as
+To estimate $x((m+k)T)$,
+a forward-Euler\index{forward-Euler} style jumping relies only on $x(mT)$ and $x((m-1)T)$,
+i.e.,
+\[
+   x((m+k)T)
+   =
+   x(mT) + k\left[ x(mT)-x((m-1)T) \right].
+\]
+However, this approach is inefficient due to its restriction on
+envelope step $k$, since larger $k$ usually causes instability.
+Instead, backward-Euler\index{backward-Euler} jumping,
+%and the equation becomes
+\[
+  x((m+k)T)-x(mT) = k\left[x((m+k)T)-x((m+k-1)T)\right],
+\]
+allows larger envelope steps.
+Here $x((m+k-1)T)$ is the unknown variable to be solved
+by Newton iteration\index{Newton iteration},
+and $x((m+k)T)$ is dependent on $x((m+k-1)T)$
+in each iteration.
+% Forward-Euler may be used to generate the initial guess.
+The Newton update equation in each iteration is thus expressed as
+% x^{j+1}((m+k-1)T) = x^{j}((m+k-1)T) - A^{-1} \left[(k-1)x^j((m+k)T) - kx^j((m+k-1)T)+ x(mT)\right],
+%\begin{equation} \label{eq:Newton}
+%\begin{split}
+\begin{multline} \label{eq:Newton}
+  \Delta x((m+k-1)T)= A^{-1} \left[(k-1)x^j((m+k)T) \right. \\
+   \left. - kx^j((m+k-1)T)+ x(mT)\right],
+\end{multline}
+%\end{split}
+%\end{equation}
+where the Jacobian matrix\index{Jacobian matrix} $A$ is computed as
+a combination of circuit sensitivity matrix\index{sensitivity matrix}
+and identity matrix, as
+\begin{equation} \label{eq:A}
+  A = (k-1)\frac{\ud x((m+k)T)}{\ud x((m+k-1)T)} -kI
+    = (k-1)J-kI.
+\end{equation}
+In each Newton iteration,
+$x((m+k-1)T)$ is used as initial condition to calculate
+$x((m+k)T)$ by integrating the DAE in one clock period.
+Meanwhile, %circuit matrices, for example,
+the conductance matrix $G_i=\ud f(x(t_i))/\ud x(t_i)$
+and the capacitance matrix $C_i=\ud q(x(t_i))/\ud x(t_i)$
+at each time step %in this period,
+are used to derive the %circuit
+sensitivity matrix $J= \ud x((m+k)T) / \ud x((m+k-1)T)$.
+% i.e.,
+% \[
+%   x((m+k)T) = \Phi(x((m+k-1)T),(m+k-1)T,(m+k)T).
+% \]
+
+Different integration rules can be applied to
+the computation of sensitivity matrix.
+It can be easily derived that,
+%for one signal period with $M$ time steps,
+if the DAE is integrated with backward-Euler rule,
+the sensitivity is
+\[
+   J = \frac{\ud x_M}{\ud x_0}
+   = \prod_{i=1}^{M}
+     \left( \frac{1}{h_i} C_i + G_i \right)^{-1}
+     \frac{1}{h_i}C_{i-1}
+%  \frac{\ud x_n}{\ud x_0}
+%  = \left(\frac{1}{h_n} C_n + G_n\right)^{-1}
+%    \frac{1}{h_n}C_{n-1} \frac{\ud x_{n-1}}{\ud x_0}
+\]
+%is the sensitivity matrix
+% In the next section, we will formulate the sensitivity in
+% Gear-2 scheme.
+% and for trapezoidal integration,
+% \[
+%   \frac{\ud x_n}{\ud x_0}
+%  = \left(\frac{1}{h_n} C_n + \frac{1}{2} G_n\right)^{-1}
+%    \left(\frac{1}{h_n} C_{n-1} - \frac{1}{2} G_{n-1}\right)
+%    \frac{\ud x_{n-1}}{\ud x_0}.
+% \]
+
+In summary, the envelope-following method is fundamentally
+a boosted version of traditional transient analysis,
+with certain skips over several periods and a Newton iteration
+to update or correct the errors brought by the skips,
+as is exhibited by Fig.~\ref{fig:ef_flow}.
+\begin{figure}[!tb]
+\centering
+\resizebox{.7\textwidth}{!}{\input{./Chapters/chapter16/figures/ef_flow.pdf_t}}
+\caption{The flow of envelope-following method.}
+\label{fig:ef_flow}
+%\end{center}
+\end{figure}
+
+
+

diff --git a/BookGPU/Chapters/chapter16/exp.tex b/BookGPU/Chapters/chapter16/exp.tex
new file mode 100644 (file)
index 0000000..6fd3eb9
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/exp.tex
@@ -0,0 +1,121 @@
+\section{Numerical examples}
+\label{sec:exp}
+
+The presented algorithm has been prototyped and numerical experiments
+have been carried out on a server which has an Intel Xeon quad-core
+CPU with 2.0~GHz clock speed, and 24~GBytes memory. The GPU card
+mounted on this server is NVIDIA's Tesla~C2070 (Fermi), which contains
+448 cores (14 MPs $\times$ 32 cores per MP) running at a 1.30~GHz and
+has 4~GBytes on-chip memory. Some initial results have been published
+in~\cite{LiuTan1:DATE'12}.
+
+The envelope-following method with the proposed Gear-2
+sensitivity matrix computation is added to an open-source
+SPICE\index{SPICE}, implemented in C~\cite{ngspice}.
+Our envelope-following program is implemented by following
+the algorithm mentioned in~\cite{Kato:COMPEL'06}.
+To solve the Newton update equation,
+different methods are used to compare the computation time,
+such as direct LU, GMRES with explicitly formed matrix,
+and GMRES with implicit matrix-vector multiplication (matrix-free).
+Moreover, the matrix-free method is also incorporated to the same SPICE
+simulator using CUDA C programming interface, as described in
+Section~\ref{sec:gpu}.
+
+\begin{figure}
+  \centering
+  \resizebox{.8\textwidth}{!}{\input{./Chapters/chapter16/figures/resonant_flyback.pdf_t}}
+  \caption{Diagram of a zero-voltage quasi-resonant flyback converter.}
+  \label{fig:flyback}
+\end{figure}
+
+
+\begin{figure}%[hbfp]
+  \centering
+  \resizebox{.6\textwidth}{!}{\input{./Chapters/chapter16/figures/pgMesh.pdf_t}}
+  \caption{Illustration of power/ground network model.}
+  \label{fig:pg}
+\end{figure}
+
+\begin{figure}
+\centering
+\subfigure[The whole plot]{
+  \includegraphics[width=.6\textwidth]{./Chapters/chapter16/figures/flyback_wave_emb.eps}
+  \label{fig:flybackWhole}
+}
+\subfigure[Detail of one EF simulation period]{
+  \includegraphics[width=.6\textwidth]{./Chapters/chapter16/figures/flyback_zoomin_emb.eps}
+  \label{fig:flybackZoom}
+}
+\caption{Flyback converter solution calculated by envelope-following.
+The red curve is traditional SPICE simulation result, and
+the back curve is the envelope-following output with simulation points
+marked.}
+\label{fig:flyback_wave}
+\end{figure}
+
+
+\begin{figure}%[hbfp]
+  \centering
+  \includegraphics[width=.6\textwidth]{./Chapters/chapter16/figures/buck_wave_emb.eps}
+  \caption{Buck converter solution calculated by envelope-following.}
+  \label{fig:buck_wave}
+\end{figure}
+
+We use several integrated on-chip converters as simulation examples
+to measure running time and speedup. They include a Buck converter,
+a quasi-resonant flyback converter (shown in Fig.~\ref{fig:flyback}),
+and two boost converters.
+Each converter is directly integrated with on-chip power grid networks,
+since the performance of converters should be studied with their loads and
+we can easily observe the waveforms at different nodes in a power
+grid (see Fig.~\ref{fig:pg} for a simplified power grid structure).
+
+Fig.~\ref{fig:flyback_wave}
+and Fig.~\ref{fig:buck_wave}
+shows the waveform at output node of the resonant flyback converter
+and the Buck converter.
+Note that on the envelope curve, the darker
+dots in separated segments indicate the real simulation points were
+calculated in those cycles, and the segments without dots are the
+envelope jumps where no simulation were done.
+It can be verified that the proposed Gear-2 envelope-following method
+produces a envelope matching the original waveform well.
+
+\begin{table}
+\centering
+\caption{CPU and GPU time comparisons (in seconds) for solving Newton update equation
+  with the proposed Gear-2 sensitivity.
+}
+\vspace{0.1in}
+\label{table:circuit}
+{%\normalsize
+\begin{tabular}{@{}c|c|c|c|c|c|c@{}}
+\hline\hline
+Circuit & Nodes & Direct & Explicit & \multicolumn{3}{c}{Implicit GMRES}\\ \cline{5-7}
+         &       & LU   & GMRES     & CPU  & GPU & X \\
+\hline
+Buck       &  910   & 423.8  & 420.3 &  36.8  &  3.9 & 9.4$\times$  \\
+Flyback    &  941   & 462.4  & 459.6 &  64.5  &  7.4 & 8.7$\times$  \\
+Boost-1    &  976   & 695.1  & 687.7 &  73.2  &  6.2 & 11.8$\times$ \\
+Boost-2    & 1093   & 729.5  & 720.8 &  71.0  &  8.5 & 9.9$\times$ \\
+\hline\hline
+\end{tabular}
+}
+\end{table}
+
+For the comparison  of running time spent in solving
+Newton update equation, Table~\ref{table:circuit} lists the time
+costed by direct method, explicit GMRES, matrix-free GMRES,
+and GPU matrix-free GMRES. All methods carry out the Gear-2 based
+envelope-following method, but they handle the sensitivity and
+equation solving in different implementation steps.
+It is obvious that as long as the sensitivity matrix is explicitly formed,
+such as the cases in direct method and explicitly GMRES,
+the cost is much higher than the implicit methods.
+When matrix-free technique is applied to generate matrix-vector
+products implicitly, the computation cost is greatly reduced.
+Thus, for the same example, implicit GMRES would be one order
+of magnitude faster than explicit GMRES. Furthermore, our GPU parallel
+implementation of implicit GMRES makes this method even faster,
+with a further 10$\times$ speedup.

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+/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4
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+  0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef
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+  vradius add translate hradius vradius scale 0 0 1 180 270 arc 
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+  hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix
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+  0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub
+  translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix
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+  newpath tMatrix currentmatrix pop ulx rad add uly rad add translate
+  rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad
+  sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix
+  closepath} bdef
+/FlrRR {MlrRR stroke } bdef
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+/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def
+  newpath tMatrix currentmatrix pop ulx rad add uly rad add translate
+  rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad
+  sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix
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diff --git a/BookGPU/Chapters/chapter16/figures/buck_wave_emb-eps-converted-to.pdf b/BookGPU/Chapters/chapter16/figures/buck_wave_emb-eps-converted-to.pdf
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diff --git a/BookGPU/Chapters/chapter16/figures/buck_wave_emb.eps b/BookGPU/Chapters/chapter16/figures/buck_wave_emb.eps
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+/pdfEndPage { end } def
+% PDF color state
+/cs { /pdfFillXform exch def dup /pdfFillCS exch def
+      setcolorspace } def
+/CS { /pdfStrokeXform exch def dup /pdfStrokeCS exch def
+      setcolorspace } def
+/sc { pdfLastFill not { pdfFillCS setcolorspace } if
+      dup /pdfFill exch def aload pop pdfFillXform setcolor
+     /pdfLastFill true def /pdfLastStroke false def } def
+/SC { pdfLastStroke not { pdfStrokeCS setcolorspace } if
+      dup /pdfStroke exch def aload pop pdfStrokeXform setcolor
+     /pdfLastStroke true def /pdfLastFill false def } def
+/op { /pdfFillOP exch def
+      pdfLastFill { pdfFillOP setoverprint } if } def
+/OP { /pdfStrokeOP exch def
+      pdfLastStroke { pdfStrokeOP setoverprint } if } def
+/fCol {
+  pdfLastFill not {
+    pdfFillCS setcolorspace
+    pdfFill aload pop pdfFillXform setcolor
+    pdfFillOP setoverprint
+    /pdfLastFill true def /pdfLastStroke false def
+  } if
+} def
+/sCol {
+  pdfLastStroke not {
+    pdfStrokeCS setcolorspace
+    pdfStroke aload pop pdfStrokeXform setcolor
+    pdfStrokeOP setoverprint
+    /pdfLastStroke true def /pdfLastFill false def
+  } if
+} def
+% build a font
+/pdfMakeFont {
+  4 3 roll findfont
+  4 2 roll matrix scale makefont
+  dup length dict begin
+    { 1 index /FID ne { def } { pop pop } ifelse } forall
+    /Encoding exch def
+    currentdict
+  end
+  definefont pop
+} def
+/pdfMakeFont16 {
+  exch findfont
+  dup length dict begin
+    { 1 index /FID ne { def } { pop pop } ifelse } forall
+    /WMode exch def
+    currentdict
+  end
+  definefont pop
+} def
+/pdfMakeFont16L3 {
+  1 index /CIDFont resourcestatus {
+    pop pop 1 index /CIDFont findresource /CIDFontType known
+  } {
+    false
+  } ifelse
+  {
+    0 eq { /Identity-H } { /Identity-V } ifelse
+    exch 1 array astore composefont pop
+  } {
+    pdfMakeFont16
+  } ifelse
+} def
+% graphics state operators
+/q { gsave pdfDictSize dict begin } def
+/Q {
+  end grestore
+  /pdfLastFill where {
+    pop
+    pdfLastFill {
+      pdfFillOP setoverprint
+    } {
+      pdfStrokeOP setoverprint
+    } ifelse
+  } if
+} def
+/cm { concat } def
+/d { setdash } def
+/i { setflat } def
+/j { setlinejoin } def
+/J { setlinecap } def
+/M { setmiterlimit } def
+/w { setlinewidth } def
+% path segment operators
+/m { moveto } def
+/l { lineto } def
+/c { curveto } def
+/re { 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto
+      neg 0 rlineto closepath } def
+/h { closepath } def
+% path painting operators
+/S { sCol stroke } def
+/Sf { fCol stroke } def
+/f { fCol fill } def
+/f* { fCol eofill } def
+% clipping operators
+/W { clip newpath } def
+/W* { eoclip newpath } def
+% text state operators
+/Tc { /pdfCharSpacing exch def } def
+/Tf { dup /pdfFontSize exch def
+      dup pdfHorizScaling mul exch matrix scale
+      pdfTextMat matrix concatmatrix dup 4 0 put dup 5 0 put
+      exch findfont exch makefont setfont } def
+/Tr { /pdfTextRender exch def } def
+/Ts { /pdfTextRise exch def } def
+/Tw { /pdfWordSpacing exch def } def
+/Tz { /pdfHorizScaling exch def } def
+% text positioning operators
+/Td { pdfTextMat transform moveto } def
+/Tm { /pdfTextMat exch def } def
+% text string operators
+/cshow where {
+  pop
+  /cshow2 {
+    dup {
+      pop pop
+      1 string dup 0 3 index put 3 index exec
+    } exch cshow
+    pop pop
+  } def
+}{
+  /cshow2 {
+    currentfont /FontType get 0 eq {
+      0 2 2 index length 1 sub {
+        2 copy get exch 1 add 2 index exch get
+        2 copy exch 256 mul add
+        2 string dup 0 6 5 roll put dup 1 5 4 roll put
+        3 index exec
+      } for
+    } {
+      dup {
+        1 string dup 0 3 index put 3 index exec
+      } forall
+    } ifelse
+    pop pop
+  } def
+} ifelse
+/awcp {
+  exch {
+    false charpath
+    5 index 5 index rmoveto
+    6 index eq { 7 index 7 index rmoveto } if
+  } exch cshow2
+  6 {pop} repeat
+} def
+/Tj {
+  fCol
+  1 index stringwidth pdfTextMat idtransform pop
+  sub 1 index length dup 0 ne { div } { pop pop 0 } ifelse
+  pdfWordSpacing pdfHorizScaling mul 0 pdfTextMat dtransform 32
+  4 3 roll pdfCharSpacing pdfHorizScaling mul add 0
+  pdfTextMat dtransform
+  6 5 roll Tj1
+} def
+/Tj16 {
+  fCol
+  2 index stringwidth pdfTextMat idtransform pop
+  sub exch div
+  pdfWordSpacing pdfHorizScaling mul 0 pdfTextMat dtransform 32
+  4 3 roll pdfCharSpacing pdfHorizScaling mul add 0
+  pdfTextMat dtransform
+  6 5 roll Tj1
+} def
+/Tj16V {
+  fCol
+  2 index stringwidth pdfTextMat idtransform exch pop
+  sub exch div
+  0 pdfWordSpacing pdfTextMat dtransform 32
+  4 3 roll pdfCharSpacing add 0 exch
+  pdfTextMat dtransform
+  6 5 roll Tj1
+} def
+/Tj1 {
+  0 pdfTextRise pdfTextMat dtransform rmoveto
+  currentpoint 8 2 roll
+  pdfTextRender 1 and 0 eq {
+    6 copy awidthshow
+  } if
+  pdfTextRender 3 and dup 1 eq exch 2 eq or {
+    7 index 7 index moveto
+    6 copy
+    currentfont /FontType get 3 eq { fCol } { sCol } ifelse
+    false awcp currentpoint stroke moveto
+  } if
+  pdfTextRender 4 and 0 ne {
+    8 6 roll moveto
+    false awcp
+    /pdfTextClipPath [ pdfTextClipPath aload pop
+      {/moveto cvx}
+      {/lineto cvx}
+      {/curveto cvx}
+      {/closepath cvx}
+    pathforall ] def
+    currentpoint newpath moveto
+  } {
+    8 {pop} repeat
+  } ifelse
+  0 pdfTextRise neg pdfTextMat dtransform rmoveto
+} def
+/TJm { pdfFontSize 0.001 mul mul neg 0
+       pdfTextMat dtransform rmoveto } def
+/TJmV { pdfFontSize 0.001 mul mul neg 0 exch
+        pdfTextMat dtransform rmoveto } def
+/Tclip { pdfTextClipPath cvx exec clip newpath
+         /pdfTextClipPath [] def } def
+% Level 2 image operators
+/pdfImBuf 100 string def
+/pdfIm {
+  image
+  { currentfile pdfImBuf readline
+    not { pop exit } if
+    (%-EOD-) eq { exit } if } loop
+} def
+/pdfImM {
+  fCol imagemask
+  { currentfile pdfImBuf readline
+    not { pop exit } if
+    (%-EOD-) eq { exit } if } loop
+} def
+/pdfImClip {
+  gsave
+  0 2 4 index length 1 sub {
+    dup 4 index exch 2 copy
+    get 5 index div put
+    1 add 3 index exch 2 copy
+    get 3 index div put
+  } for
+  pop pop rectclip
+} def
+/pdfImClipEnd { grestore } def
+% shading operators
+/colordelta {
+  false 0 1 3 index length 1 sub {
+    dup 4 index exch get 3 index 3 2 roll get sub abs 0.004 gt {
+      pop true
+    } if
+  } for
+  exch pop exch pop
+} def
+/funcCol { func n array astore } def
+/funcSH {
+  dup 0 eq {
+    true
+  } {
+    dup 6 eq {
+      false
+    } {
+      4 index 4 index funcCol dup
+      6 index 4 index funcCol dup
+      3 1 roll colordelta 3 1 roll
+      5 index 5 index funcCol dup
+      3 1 roll colordelta 3 1 roll
+      6 index 8 index funcCol dup
+      3 1 roll colordelta 3 1 roll
+      colordelta or or or
+    } ifelse
+  } ifelse
+  {
+    1 add
+    4 index 3 index add 0.5 mul exch 4 index 3 index add 0.5 mul exch
+    6 index 6 index 4 index 4 index 4 index funcSH
+    2 index 6 index 6 index 4 index 4 index funcSH
+    6 index 2 index 4 index 6 index 4 index funcSH
+    5 3 roll 3 2 roll funcSH pop pop
+  } {
+    pop 3 index 2 index add 0.5 mul 3 index  2 index add 0.5 mul
+    funcCol sc
+    dup 4 index exch mat transform m
+    3 index 3 index mat transform l
+    1 index 3 index mat transform l
+    mat transform l pop pop h f*
+  } ifelse
+} def
+/axialCol {
+  dup 0 lt {
+    pop t0
+  } {
+    dup 1 gt {
+      pop t1
+    } {
+      dt mul t0 add
+    } ifelse
+  } ifelse
+  func n array astore
+} def
+/axialSH {
+  dup 0 eq {
+    true
+  } {
+    dup 8 eq {
+      false
+    } {
+      2 index axialCol 2 index axialCol colordelta
+    } ifelse
+  } ifelse
+  {
+    1 add 3 1 roll 2 copy add 0.5 mul
+    dup 4 3 roll exch 4 index axialSH
+    exch 3 2 roll axialSH
+  } {
+    pop 2 copy add 0.5 mul axialCol sc
+    exch dup dx mul x0 add exch dy mul y0 add
+    3 2 roll dup dx mul x0 add exch dy mul y0 add
+    dx abs dy abs ge {
+      2 copy yMin sub dy mul dx div add yMin m
+      yMax sub dy mul dx div add yMax l
+      2 copy yMax sub dy mul dx div add yMax l
+      yMin sub dy mul dx div add yMin l
+      h f*
+    } {
+      exch 2 copy xMin sub dx mul dy div add xMin exch m
+      xMax sub dx mul dy div add xMax exch l
+      exch 2 copy xMax sub dx mul dy div add xMax exch l
+      xMin sub dx mul dy div add xMin exch l
+      h f*
+    } ifelse
+  } ifelse
+} def
+/radialCol {
+  dup t0 lt {
+    pop t0
+  } {
+    dup t1 gt {
+      pop t1
+    } if
+  } ifelse
+  func n array astore
+} def
+/radialSH {
+  dup 0 eq {
+    true
+  } {
+    dup 8 eq {
+      false
+    } {
+      2 index dt mul t0 add radialCol
+      2 index dt mul t0 add radialCol colordelta
+    } ifelse
+  } ifelse
+  {
+    1 add 3 1 roll 2 copy add 0.5 mul
+    dup 4 3 roll exch 4 index radialSH
+    exch 3 2 roll radialSH
+  } {
+    pop 2 copy add 0.5 mul dt mul t0 add axialCol sc
+    exch dup dx mul x0 add exch dup dy mul y0 add exch dr mul r0 add
+    0 360 arc h
+    dup dx mul x0 add exch dup dy mul y0 add exch dr mul r0 add
+    0 360 arc h f*
+  } ifelse
+} def
+end
+/CIDInit /ProcSet findresource begin
+10 dict begin
+  begincmap
+  /CMapType 1 def
+  /CMapName /Identity-H def
+  /CIDSystemInfo 3 dict dup begin
+    /Registry (Adobe) def
+    /Ordering (Identity) def
+    /Supplement 0 def
+  end def
+  1 begincodespacerange
+    <0000> <ffff>
+  endcodespacerange
+  0 usefont
+  1 begincidrange
+    <0000> <ffff> 0
+  endcidrange
+  endcmap
+  currentdict CMapName exch /CMap defineresource pop
+end
+10 dict begin
+  begincmap
+  /CMapType 1 def
+  /CMapName /Identity-V def
+  /CIDSystemInfo 3 dict dup begin
+    /Registry (Adobe) def
+    /Ordering (Identity) def
+    /Supplement 0 def
+  end def
+  /WMode 1 def
+  1 begincodespacerange
+    <0000> <ffff>
+  endcodespacerange
+  0 usefont
+  1 begincidrange
+    <0000> <ffff> 0
+  endcidrange
+  endcmap
+  currentdict CMapName exch /CMap defineresource pop
+end
+end
+xpdf begin
+%!FontType1-1.0: Helvetica
+12 dict begin
+/FontInfo 10 dict dup begin
+/Notice (Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development; Cyrillic glyphs added by Valek Filippov (C) 2001-2004) readonly def
+/FullName (Nimbus Sans L Regular) readonly def
+/FamilyName (Nimbus Sans L) readonly def
+/isFixedPitch false def
+/ItalicAngle 0 def
+/UnderlinePosition 0 def
+/UnderlineThickness 0 def
+end readonly def
+/FontName /Helvetica def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/FontBBox [0 -218 762 741] readonly def
+/StrokeWidth 0 def
+/UniqueID 4095404 def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 32 /space put
+dup 40 /parenleft put
+dup 41 /parenright put
+dup 45 /minus put
+dup 46 /period put
+dup 48 /zero put
+dup 49 /one put
+dup 50 /two put
+dup 51 /three put
+dup 52 /four put
+dup 53 /five put
+dup 54 /six put
+dup 56 /eight put
+dup 67 /C put
+dup 69 /E put
+dup 73 /I put
+dup 80 /P put
+dup 83 /S put
+dup 84 /T put
+dup 86 /V put
+dup 97 /a put
+dup 99 /c put
+dup 101 /e put
+dup 102 /f put
+dup 103 /g put
+dup 105 /i put
+dup 108 /l put
+dup 109 /m put
+dup 110 /n put
+dup 111 /o put
+dup 112 /p put
+dup 114 /r put
+dup 115 /s put
+dup 116 /t put
+dup 118 /v put
+dup 119 /w put
+dup 120 /x put
+readonly def
+currentdict end
+currentfile eexec
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diff --git a/BookGPU/Chapters/chapter16/figures/ef_flow.fig b/BookGPU/Chapters/chapter16/figures/ef_flow.fig
new file mode 100644 (file)
index 0000000..8b2210e
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/figures/ef_flow.fig
@@ -0,0 +1,115 @@
+#FIG 3.2
+Landscape
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+4 0 1 50 -1 0 9 0.0000 2 156 2141 2028 3150 the LU factors of $J_i$\001
+-6
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+       2 1 1.00 60.00 120.00
+        1800 1875 1800 2100
+2 2 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 5
+        975 1650 2625 1650 2625 1875 975 1875 975 1650
+2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2
+       2 1 1.00 60.00 120.00
+        1800 1425 1800 1650
+2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2
+       2 1 1.00 60.00 120.00
+        1800 975 1800 1200
+2 2 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 5
+        1200 750 2475 750 2475 975 1200 975 1200 750
+2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2
+       2 1 1.00 60.00 120.00
+        1800 525 1800 750
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+       2 1 1.00 60.00 120.00
+        1800 3600 1800 3900
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 3
+       2 1 1.00 60.00 120.00
+        1800 4125 1800 4275 1800 4350
+2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2
+       2 1 1.00 60.00 120.00
+        1800 4575 1800 4800
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+       2 1 1.00 60.00 120.00
+        1800 5025 1800 5250
+2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 2
+        1350 5400 750 5400
+2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 2
+       2 1 1.00 60.00 120.00
+        1800 5475 1800 5700
+2 2 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 5
+        1350 5700 2250 5700 2250 5925 1350 5925 1350 5700
+2 4 1 2 0 7 50 -1 -1 4.000 0 0 7 0 0 5
+        2250 5475 1350 5475 1350 5250 2250 5250 2250 5475
+2 1 0 1 0 7 50 -1 -1 0.000 0 0 -1 1 0 5
+       2 1 1.00 60.00 120.00
+        1800 5925 1800 6000 3525 6000 3525 825 2475 825
+2 4 1 2 0 7 50 -1 -1 4.000 0 0 6 0 0 5
+        2400 3600 1200 3600 1200 3375 2400 3375 2400 3600
+2 2 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 5
+        600 300 3150 300 3150 525 600 525 600 300
+2 2 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 5
+        675 1200 3000 1200 3000 1425 675 1425 675 1200
+2 2 0 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 5
+        600 2625 1650 2625 1650 3075 600 3075 600 2625
+2 2 0 1 0 -1 50 -1 -1 0.000 0 0 -1 0 0 5
+        900 3900 2775 3900 2775 4125 900 4125 900 3900
+2 2 0 1 0 7 50 -1 -1 0.000 0 0 -1 0 0 5
+        750 4350 2925 4350 2925 4575 750 4575 750 4350
+3 0 3 1 1 7 50 -1 -1 4.000 0 1 0 2
+       2 0 1.00 60.00 120.00
+        1650 2850 1950 2850
+        0.000 0.000
+4 0 0 50 -1 0 9 0.0000 2 165 1440 1050 1800 set $t=(m+k-1)T$\001
+4 0 0 50 -1 0 9 0.0000 2 180 2610 750 1350 make a guess of $x((m+k-1)T)$\001
+4 0 0 50 -1 0 9 0.0000 2 165 1800 1275 900 select jump size $k$\001
+4 0 0 50 -1 0 9 0.0000 2 165 3240 675 450 circuit state $x(mT)$ at time $t=mT$\001
+4 0 0 50 -1 0 9 0.0000 2 120 270 750 3450 yes\001
+4 0 0 50 -1 0 9 0.0000 2 165 2520 825 4500 solve Newton update equation\001
+4 0 0 50 -1 0 9 0.0000 2 90 180 1950 3750 no\001
+4 0 0 50 -1 0 9 0.0000 2 150 2340 975 4050 compute sensitivity matrix\001
+4 0 0 50 -1 0 9 0.0000 2 165 990 1425 5400 converged ?\001
+4 0 0 50 -1 0 9 0.0000 2 90 180 1125 5325 no\001
+4 0 0 50 -1 0 9 0.0000 2 120 270 1950 5625 yes\001
+4 0 0 50 -1 0 9 0.0000 2 135 630 1425 5850 $m=m+k$\001
+4 0 0 50 -1 0 9 0.0000 2 165 1080 1275 3525 $t<(m+k)T$ ?\001
+4 0 0 50 -1 0 9 0.0000 2 165 900 675 3000 for $x(t)$\001
+4 0 0 50 -1 0 9 0.0000 2 165 1170 675 2775 integrate DAE\001

diff --git a/BookGPU/Chapters/chapter16/figures/ef_flow.pdf b/BookGPU/Chapters/chapter16/figures/ef_flow.pdf
new file mode 100644 (file)
index 0000000..d7dc1e8
Binary files /dev/null and b/BookGPU/Chapters/chapter16/figures/ef_flow.pdf differ

diff --git a/BookGPU/Chapters/chapter16/figures/ef_flow.pdf_t b/BookGPU/Chapters/chapter16/figures/ef_flow.pdf_t
new file mode 100644 (file)
index 0000000..7bd7063
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/figures/ef_flow.pdf_t
@@ -0,0 +1,53 @@
+\begin{picture}(0,0)%
+\includegraphics{./Chapters/chapter16/figures/ef_flow.pdf}%
+\end{picture}%
+\setlength{\unitlength}{3947sp}%
+%
+\begingroup\makeatletter\ifx\SetFigFont\undefined%
+\gdef\SetFigFont#1#2#3#4#5{%
+  \reset@font\fontsize{#1}{#2pt}%
+  \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
+  \selectfont}%
+\fi\endgroup%
+\begin{picture}(3174,5724)(364,-5173)
+\put(2029,-1861){\makebox(0,0)[lb]{\smash{{\SetFigFont{9}{10.8}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,1}save time steps $h_i$,}%
+}}}}
+\put(2029,-2086){\makebox(0,0)[lb]{\smash{{\SetFigFont{9}{10.8}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,1}matrices $C_i$, and}%
+}}}}
+\put(2029,-2311){\makebox(0,0)[lb]{\smash{{\SetFigFont{9}{10.8}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,1}the LU factors of $J_i$}%
+}}}}
+\put(1426,-1430){\makebox(0,0)[lb]{\smash{{\SetFigFont{9}{10.8}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$t=t+h_i$}%
+}}}}
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+\put(1276,-61){\makebox(0,0)[lb]{\smash{{\SetFigFont{9}{10.8}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}select jump size $k$}%
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diff --git a/BookGPU/Chapters/chapter16/figures/ef_intro.fig b/BookGPU/Chapters/chapter16/figures/ef_intro.fig
new file mode 100644 (file)
index 0000000..0c08a86
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/figures/ef_intro.fig
@@ -0,0 +1,79 @@
+#FIG 3.2
+Landscape
+Center
+Inches
+Letter  
+100.00
+Single
+-2
+1200 2
+6 2550 6450 4950 7650
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 4800 6600 75 75 4800 6600 4800 6675
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 4425 6600 75 75 4425 6600 4425 6675
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 4050 6675 75 75 4050 6675 4050 6750
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 3675 6750 75 75 3675 6750 3675 6825
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 2700 7500 75 75 2700 7500 2700 7575
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 3075 7350 75 75 3075 7350 3075 7425
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 3375 7050 75 75 3375 7050 3375 7125
+3 0 0 2 0 7 50 -1 -1 0.000 0 0 0 4
+        2700 7500 3300 7350 3450 6675 4800 6600
+        0.000 1.000 1.000 0.000
+-6
+6 450 7350 2850 8550
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 2700 7500 75 75 2700 7500 2700 7575
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 2325 7500 75 75 2325 7500 2325 7575
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 1950 7575 75 75 1950 7575 1950 7650
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 1575 7650 75 75 1575 7650 1575 7725
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 600 8400 75 75 600 8400 600 8475
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 975 8250 75 75 975 8250 975 8325
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 1275 7950 75 75 1275 7950 1275 8025
+3 0 0 2 0 7 50 -1 -1 0.000 0 0 0 4
+        600 8400 1200 8250 1350 7575 2700 7500
+        0.000 1.000 1.000 0.000
+-6
+6 8250 4050 10650 5250
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 10500 4200 75 75 10500 4200 10500 4275
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 10125 4200 75 75 10125 4200 10125 4275
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 9750 4275 75 75 9750 4275 9750 4350
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 9375 4350 75 75 9375 4350 9375 4425
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 8400 5100 75 75 8400 5100 8400 5175
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 8775 4950 75 75 8775 4950 8775 5025
+1 3 0 1 0 1 50 -1 20 0.000 1 0.0000 9075 4650 75 75 9075 4650 9075 4725
+3 0 0 2 0 7 50 -1 -1 0.000 0 0 0 4
+        8400 5100 9000 4950 9150 4275 10500 4200
+        0.000 1.000 1.000 0.000
+-6
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+       0 0 1.00 60.00 120.00
+       0 0 1.00 60.00 120.00
+        2700 5700 4800 5700
+2 1 1 2 0 7 50 -1 -1 4.000 0 0 7 0 0 2
+        2700 7500 10500 4200
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 7 0 0 2
+        2700 8100 2700 3900
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 7 0 0 2
+        4800 8100 4800 3900
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 7 0 0 2
+        8400 8100 8400 3900
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 7 0 0 2
+        10500 8100 10500 3900
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+       0 0 1.00 60.00 120.00
+       0 0 1.00 60.00 120.00
+        4800 3750 10500 3750
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+       0 0 1.00 60.00 120.00
+        1275 7350 1275 7725
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+       0 0 1.00 60.00 120.00
+        6600 6525 6300 6075
+2 4 0 1 0 7 50 -1 -1 4.000 0 0 7 0 0 5
+        2175 7350 2175 6600 750 6600 750 7350 2175 7350
+4 0 0 50 -1 0 20 0.0000 2 165 1350 2850 5550 clock cycle $T$\001
+4 0 0 50 -1 0 20 0.0000 2 165 1440 6600 3600 jump $k$ periods\001
+4 0 0 50 -1 0 20 0.0000 2 165 720 6225 6825 envelope\001
+4 0 0 50 -1 0 20 0.0000 2 120 810 900 6900 transient\001
+4 0 0 50 -1 0 20 0.0000 2 150 540 900 7200 points\001
+4 0 0 50 -1 0 20 0.0000 2 135 540 4425 8400 $t=mT$\001
+4 0 0 50 -1 0 20 0.0000 2 165 1080 6525 8400 $t=(m+k-1)T$\001
+4 0 0 50 -1 0 20 0.0000 2 165 900 9450 8400 $t=(m+k)T$\001

diff --git a/BookGPU/Chapters/chapter16/figures/ef_intro.pdf b/BookGPU/Chapters/chapter16/figures/ef_intro.pdf
new file mode 100644 (file)
index 0000000..5134824
Binary files /dev/null and b/BookGPU/Chapters/chapter16/figures/ef_intro.pdf differ

diff --git a/BookGPU/Chapters/chapter16/figures/ef_intro.pdf_t b/BookGPU/Chapters/chapter16/figures/ef_intro.pdf_t
new file mode 100644 (file)
index 0000000..dcee58d
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/figures/ef_intro.pdf_t
@@ -0,0 +1,29 @@
+\begin{picture}(0,0)%
+\includegraphics{./Chapters/chapter16/figures/ef_intro.pdf}%
+\end{picture}%
+\setlength{\unitlength}{3947sp}%
+%
+\begingroup\makeatletter\ifx\SetFigFont\undefined%
+\gdef\SetFigFont#1#2#3#4#5{%
+  \reset@font\fontsize{#1}{#2pt}%
+  \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
+  \selectfont}%
+\fi\endgroup%
+\begin{picture}(10066,5172)(518,-7666)
+\put(2851,-4711){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}clock cycle $T$}%
+}}}}
+\put(6601,-2761){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}jump $k$ periods}%
+}}}}
+\put(6226,-5986){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}envelope}%
+}}}}
+\put(901,-6061){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}transient}%
+}}}}
+\put(901,-6361){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}points}%
+}}}}
+\put(4426,-7561){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$t=mT$}%
+}}}}
+\put(6526,-7561){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$t=(m+k-1)T$}%
+}}}}
+\put(9451,-7561){\makebox(0,0)[lb]{\smash{{\SetFigFont{20}{24.0}{\familydefault}{\mddefault}{\updefault}{\color[rgb]{0,0,0}$t=(m+k)T$}%
+}}}}
+\end{picture}%

diff --git a/BookGPU/Chapters/chapter16/figures/envelope.fig b/BookGPU/Chapters/chapter16/figures/envelope.fig
new file mode 100644 (file)
index 0000000..6bae849
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/figures/envelope.fig
@@ -0,0 +1,248 @@
+#FIG 3.2
+Landscape
+Center
+Inches
+Letter  
+100.00
+Single
+-2
+1200 2
+0 32 #9c0000
+0 33 #8c8c8c
+0 34 #8c8c8c
+0 35 #424242
+0 36 #8c8c8c
+0 37 #424242
+0 38 #8c8c8c
+0 39 #424242
+0 40 #8c8c8c
+0 41 #424242
+0 42 #8c8c8c
+0 43 #424242
+0 44 #c2c2c2
+0 45 #6e6e6e
+0 46 #444444
+0 47 #8e8f8e
+0 48 #717171
+0 49 #aeaeae
+0 50 #333333
+0 51 #949395
+0 52 #747075
+0 53 #555555
+0 54 #b3b3b3
+0 55 #c3c3c3
+0 56 #6d6d6d
+0 57 #454545
+0 58 #8e8e8e
+0 59 #c6b797
+0 60 #eff8ff
+0 61 #dccba6
+0 62 #404040
+0 63 #808080
+0 64 #c0c0c0
+0 65 #e0e0e0
+0 66 #aaaaaa
+0 67 #d7d7d7
+0 68 #bebebe
+0 69 #515151
+0 70 #e7e3e7
+0 71 #000049
+0 72 #797979
+0 73 #303430
+0 74 #414141
+0 75 #c7b696
+0 76 #414541
+0 77 #dd9d93
+0 78 #f1ece0
+0 79 #e2c8a8
+0 80 #e1e1e1
+0 81 #d2d2d2
+0 82 #ededed
+0 83 #da7a1a
+0 84 #f1e41a
+0 85 #887dc2
+0 86 #d6d6d6
+0 87 #8c8ca5
+0 88 #4a4a4a
+0 89 #8c6b6b
+0 90 #5a5a5a
+0 91 #636363
+0 92 #b79b73
+0 93 #4193ff
+0 94 #bf703b
+0 95 #db7700
+0 96 #dab800
+0 97 #006400
+0 98 #5a6b3b
+0 99 #d3d3d3
+0 100 #8e8ea4
+0 101 #f3b95d
+0 102 #89996b
+0 103 #646464
+0 104 #b7e6ff
+0 105 #86c0ec
+0 106 #bdbdbd
+0 107 #d39552
+0 108 #98d2fe
+0 109 #8c9c6b
+0 110 #f76b00
+0 111 #5a6b39
+0 112 #8c9c6b
+0 113 #8c9c7b
+0 114 #184a18
+0 115 #adadad
+0 116 #f7bd5a
+0 117 #636b9c
+0 118 #f7f7f7
+0 119 #de0000
+0 120 #adadad
+0 121 #f7bd5a
+0 122 #adadad
+0 123 #f7bd5a
+0 124 #636b9c
+0 125 #526b29
+0 126 #949494
+0 127 #006300
+0 128 #00634a
+0 129 #7b844a
+0 130 #e7bd7b
+0 131 #a5b5c6
+0 132 #6b6b94
+0 133 #846b6b
+0 134 #529c4a
+0 135 #d6e7e7
+0 136 #526363
+0 137 #186b4a
+0 138 #9ca5b5
+0 139 #ff9400
+0 140 #ff9400
+0 141 #00634a
+0 142 #7b844a
+0 143 #63737b
+0 144 #e7bd7b
+0 145 #184a18
+0 146 #f7bd5a
+0 147 #dedede
+0 148 #f3eed3
+0 149 #f5ae5d
+0 150 #95ce99
+0 151 #b5157d
+0 152 #eeeeee
+0 153 #848484
+0 154 #7b7b7b
+0 155 #005a00
+0 156 #e77373
+0 157 #ffcb31
+0 158 #29794a
+0 159 #de2821
+0 160 #2159c6
+0 161 #f8f8f8
+0 162 #e6e6e6
+0 163 #21845a
+0 164 #c9c9c9
+0 165 #dfd8df
+0 166 #f7f3f7
+0 167 #cdcdcd
+0 168 #6c6c6c
+# Brace under
+6 1500 3975 2400 4425
+3 4 0 1 0 7 50 0 -1 0.000 0 0 0 7
+        1573 4291 1598 4358 1980 4218 2004 4285 1980 4218 2363 4079
+        2339 4012
+        0.000 1.000 1.000 0.000 1.000 1.000 0.000
+-6
+# Brace under
+6 3000 3600 3900 3825
+3 4 0 1 0 7 50 0 -1 0.000 0 0 0 7
+        3070 3747 3082 3818 3484 3747 3496 3816 3484 3747 3885 3676
+        3873 3606
+        0.000 1.000 1.000 0.000 1.000 1.000 0.000
+-6
+# Brace under
+6 4575 3525 5625 3900
+3 4 0 1 0 7 50 0 -1 0.000 0 0 0 7
+        4639 3531 4622 3600 5089 3734 5072 3802 5089 3734 5556 3868
+        5574 3799
+        0.000 1.000 1.000 0.000 1.000 1.000 0.000
+-6
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
+        7500 4800 7500 5100
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
+        6225 4200 6225 5100
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
+        3000 3825 3000 5100
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
+        900 4575 900 5100
+2 1 3 1 0 7 50 -1 -1 4.000 0 0 -1 0 0 2
+        1500 4350 1500 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+       0 0 1.00 60.00 120.00
+       0 0 1.00 60.00 120.00
+        900 5100 1500 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+       0 0 1.00 60.00 120.00
+       0 0 1.00 60.00 120.00
+        1500 5100 3000 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+       0 0 1.00 60.00 120.00
+       0 0 1.00 60.00 120.00
+        3075 5100 4575 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+       0 0 1.00 60.00 120.00
+       0 0 1.00 60.00 120.00
+        4575 5100 6225 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 1 2
+       0 0 1.00 60.00 120.00
+       0 0 1.00 60.00 120.00
+        6300 5100 7500 5100
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+       0 0 1.00 60.00 120.00
+        1500 3225 1050 4200
+2 1 0 1 0 7 50 -1 -1 4.000 0 0 -1 1 0 2
+       0 0 1.00 60.00 120.00
+        3150 3075 3900 3225
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diff --git a/BookGPU/Chapters/chapter16/figures/flyback_wave_emb-eps-converted-to.pdf b/BookGPU/Chapters/chapter16/figures/flyback_wave_emb-eps-converted-to.pdf
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diff --git a/BookGPU/Chapters/chapter16/figures/flyback_wave_emb.eps b/BookGPU/Chapters/chapter16/figures/flyback_wave_emb.eps
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+      neg 0 rlineto closepath } def
+/h { closepath } def
+% path painting operators
+/S { sCol stroke } def
+/Sf { fCol stroke } def
+/f { fCol fill } def
+/f* { fCol eofill } def
+% clipping operators
+/W { clip newpath } def
+/W* { eoclip newpath } def
+% text state operators
+/Tc { /pdfCharSpacing exch def } def
+/Tf { dup /pdfFontSize exch def
+      dup pdfHorizScaling mul exch matrix scale
+      pdfTextMat matrix concatmatrix dup 4 0 put dup 5 0 put
+      exch findfont exch makefont setfont } def
+/Tr { /pdfTextRender exch def } def
+/Ts { /pdfTextRise exch def } def
+/Tw { /pdfWordSpacing exch def } def
+/Tz { /pdfHorizScaling exch def } def
+% text positioning operators
+/Td { pdfTextMat transform moveto } def
+/Tm { /pdfTextMat exch def } def
+% text string operators
+/cshow where {
+  pop
+  /cshow2 {
+    dup {
+      pop pop
+      1 string dup 0 3 index put 3 index exec
+    } exch cshow
+    pop pop
+  } def
+}{
+  /cshow2 {
+    currentfont /FontType get 0 eq {
+      0 2 2 index length 1 sub {
+        2 copy get exch 1 add 2 index exch get
+        2 copy exch 256 mul add
+        2 string dup 0 6 5 roll put dup 1 5 4 roll put
+        3 index exec
+      } for
+    } {
+      dup {
+        1 string dup 0 3 index put 3 index exec
+      } forall
+    } ifelse
+    pop pop
+  } def
+} ifelse
+/awcp {
+  exch {
+    false charpath
+    5 index 5 index rmoveto
+    6 index eq { 7 index 7 index rmoveto } if
+  } exch cshow2
+  6 {pop} repeat
+} def
+/Tj {
+  fCol
+  1 index stringwidth pdfTextMat idtransform pop
+  sub 1 index length dup 0 ne { div } { pop pop 0 } ifelse
+  pdfWordSpacing pdfHorizScaling mul 0 pdfTextMat dtransform 32
+  4 3 roll pdfCharSpacing pdfHorizScaling mul add 0
+  pdfTextMat dtransform
+  6 5 roll Tj1
+} def
+/Tj16 {
+  fCol
+  2 index stringwidth pdfTextMat idtransform pop
+  sub exch div
+  pdfWordSpacing pdfHorizScaling mul 0 pdfTextMat dtransform 32
+  4 3 roll pdfCharSpacing pdfHorizScaling mul add 0
+  pdfTextMat dtransform
+  6 5 roll Tj1
+} def
+/Tj16V {
+  fCol
+  2 index stringwidth pdfTextMat idtransform exch pop
+  sub exch div
+  0 pdfWordSpacing pdfTextMat dtransform 32
+  4 3 roll pdfCharSpacing add 0 exch
+  pdfTextMat dtransform
+  6 5 roll Tj1
+} def
+/Tj1 {
+  0 pdfTextRise pdfTextMat dtransform rmoveto
+  currentpoint 8 2 roll
+  pdfTextRender 1 and 0 eq {
+    6 copy awidthshow
+  } if
+  pdfTextRender 3 and dup 1 eq exch 2 eq or {
+    7 index 7 index moveto
+    6 copy
+    currentfont /FontType get 3 eq { fCol } { sCol } ifelse
+    false awcp currentpoint stroke moveto
+  } if
+  pdfTextRender 4 and 0 ne {
+    8 6 roll moveto
+    false awcp
+    /pdfTextClipPath [ pdfTextClipPath aload pop
+      {/moveto cvx}
+      {/lineto cvx}
+      {/curveto cvx}
+      {/closepath cvx}
+    pathforall ] def
+    currentpoint newpath moveto
+  } {
+    8 {pop} repeat
+  } ifelse
+  0 pdfTextRise neg pdfTextMat dtransform rmoveto
+} def
+/TJm { pdfFontSize 0.001 mul mul neg 0
+       pdfTextMat dtransform rmoveto } def
+/TJmV { pdfFontSize 0.001 mul mul neg 0 exch
+        pdfTextMat dtransform rmoveto } def
+/Tclip { pdfTextClipPath cvx exec clip newpath
+         /pdfTextClipPath [] def } def
+% Level 2 image operators
+/pdfImBuf 100 string def
+/pdfIm {
+  image
+  { currentfile pdfImBuf readline
+    not { pop exit } if
+    (%-EOD-) eq { exit } if } loop
+} def
+/pdfImM {
+  fCol imagemask
+  { currentfile pdfImBuf readline
+    not { pop exit } if
+    (%-EOD-) eq { exit } if } loop
+} def
+/pdfImClip {
+  gsave
+  0 2 4 index length 1 sub {
+    dup 4 index exch 2 copy
+    get 5 index div put
+    1 add 3 index exch 2 copy
+    get 3 index div put
+  } for
+  pop pop rectclip
+} def
+/pdfImClipEnd { grestore } def
+% shading operators
+/colordelta {
+  false 0 1 3 index length 1 sub {
+    dup 4 index exch get 3 index 3 2 roll get sub abs 0.004 gt {
+      pop true
+    } if
+  } for
+  exch pop exch pop
+} def
+/funcCol { func n array astore } def
+/funcSH {
+  dup 0 eq {
+    true
+  } {
+    dup 6 eq {
+      false
+    } {
+      4 index 4 index funcCol dup
+      6 index 4 index funcCol dup
+      3 1 roll colordelta 3 1 roll
+      5 index 5 index funcCol dup
+      3 1 roll colordelta 3 1 roll
+      6 index 8 index funcCol dup
+      3 1 roll colordelta 3 1 roll
+      colordelta or or or
+    } ifelse
+  } ifelse
+  {
+    1 add
+    4 index 3 index add 0.5 mul exch 4 index 3 index add 0.5 mul exch
+    6 index 6 index 4 index 4 index 4 index funcSH
+    2 index 6 index 6 index 4 index 4 index funcSH
+    6 index 2 index 4 index 6 index 4 index funcSH
+    5 3 roll 3 2 roll funcSH pop pop
+  } {
+    pop 3 index 2 index add 0.5 mul 3 index  2 index add 0.5 mul
+    funcCol sc
+    dup 4 index exch mat transform m
+    3 index 3 index mat transform l
+    1 index 3 index mat transform l
+    mat transform l pop pop h f*
+  } ifelse
+} def
+/axialCol {
+  dup 0 lt {
+    pop t0
+  } {
+    dup 1 gt {
+      pop t1
+    } {
+      dt mul t0 add
+    } ifelse
+  } ifelse
+  func n array astore
+} def
+/axialSH {
+  dup 0 eq {
+    true
+  } {
+    dup 8 eq {
+      false
+    } {
+      2 index axialCol 2 index axialCol colordelta
+    } ifelse
+  } ifelse
+  {
+    1 add 3 1 roll 2 copy add 0.5 mul
+    dup 4 3 roll exch 4 index axialSH
+    exch 3 2 roll axialSH
+  } {
+    pop 2 copy add 0.5 mul axialCol sc
+    exch dup dx mul x0 add exch dy mul y0 add
+    3 2 roll dup dx mul x0 add exch dy mul y0 add
+    dx abs dy abs ge {
+      2 copy yMin sub dy mul dx div add yMin m
+      yMax sub dy mul dx div add yMax l
+      2 copy yMax sub dy mul dx div add yMax l
+      yMin sub dy mul dx div add yMin l
+      h f*
+    } {
+      exch 2 copy xMin sub dx mul dy div add xMin exch m
+      xMax sub dx mul dy div add xMax exch l
+      exch 2 copy xMax sub dx mul dy div add xMax exch l
+      xMin sub dx mul dy div add xMin exch l
+      h f*
+    } ifelse
+  } ifelse
+} def
+/radialCol {
+  dup t0 lt {
+    pop t0
+  } {
+    dup t1 gt {
+      pop t1
+    } if
+  } ifelse
+  func n array astore
+} def
+/radialSH {
+  dup 0 eq {
+    true
+  } {
+    dup 8 eq {
+      false
+    } {
+      2 index dt mul t0 add radialCol
+      2 index dt mul t0 add radialCol colordelta
+    } ifelse
+  } ifelse
+  {
+    1 add 3 1 roll 2 copy add 0.5 mul
+    dup 4 3 roll exch 4 index radialSH
+    exch 3 2 roll radialSH
+  } {
+    pop 2 copy add 0.5 mul dt mul t0 add axialCol sc
+    exch dup dx mul x0 add exch dup dy mul y0 add exch dr mul r0 add
+    0 360 arc h
+    dup dx mul x0 add exch dup dy mul y0 add exch dr mul r0 add
+    0 360 arc h f*
+  } ifelse
+} def
+end
+/CIDInit /ProcSet findresource begin
+10 dict begin
+  begincmap
+  /CMapType 1 def
+  /CMapName /Identity-H def
+  /CIDSystemInfo 3 dict dup begin
+    /Registry (Adobe) def
+    /Ordering (Identity) def
+    /Supplement 0 def
+  end def
+  1 begincodespacerange
+    <0000> <ffff>
+  endcodespacerange
+  0 usefont
+  1 begincidrange
+    <0000> <ffff> 0
+  endcidrange
+  endcmap
+  currentdict CMapName exch /CMap defineresource pop
+end
+10 dict begin
+  begincmap
+  /CMapType 1 def
+  /CMapName /Identity-V def
+  /CIDSystemInfo 3 dict dup begin
+    /Registry (Adobe) def
+    /Ordering (Identity) def
+    /Supplement 0 def
+  end def
+  /WMode 1 def
+  1 begincodespacerange
+    <0000> <ffff>
+  endcodespacerange
+  0 usefont
+  1 begincidrange
+    <0000> <ffff> 0
+  endcidrange
+  endcmap
+  currentdict CMapName exch /CMap defineresource pop
+end
+end
+xpdf begin
+%!FontType1-1.0: Helvetica
+12 dict begin
+/FontInfo 10 dict dup begin
+/Notice (Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development; Cyrillic glyphs added by Valek Filippov (C) 2001-2004) readonly def
+/FullName (Nimbus Sans L Regular) readonly def
+/FamilyName (Nimbus Sans L) readonly def
+/isFixedPitch false def
+/ItalicAngle 0 def
+/UnderlinePosition 0 def
+/UnderlineThickness 0 def
+end readonly def
+/FontName /Helvetica def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/FontBBox [0 -218 762 741] readonly def
+/StrokeWidth 0 def
+/UniqueID 4095404 def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 32 /space put
+dup 40 /parenleft put
+dup 41 /parenright put
+dup 45 /minus put
+dup 46 /period put
+dup 48 /zero put
+dup 49 /one put
+dup 50 /two put
+dup 51 /three put
+dup 52 /four put
+dup 67 /C put
+dup 69 /E put
+dup 73 /I put
+dup 80 /P put
+dup 83 /S put
+dup 84 /T put
+dup 86 /V put
+dup 97 /a put
+dup 99 /c put
+dup 101 /e put
+dup 102 /f put
+dup 103 /g put
+dup 105 /i put
+dup 108 /l put
+dup 109 /m put
+dup 110 /n put
+dup 111 /o put
+dup 112 /p put
+dup 114 /r put
+dup 115 /s put
+dup 116 /t put
+dup 118 /v put
+dup 119 /w put
+dup 120 /x put
+readonly def
+currentdict end
+currentfile eexec
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+  currentpoint 8 2 roll
+  pdfTextRender 1 and 0 eq {
+    6 copy awidthshow
+  } if
+  pdfTextRender 3 and dup 1 eq exch 2 eq or {
+    7 index 7 index moveto
+    6 copy
+    currentfont /FontType get 3 eq { fCol } { sCol } ifelse
+    false awcp currentpoint stroke moveto
+  } if
+  pdfTextRender 4 and 0 ne {
+    8 6 roll moveto
+    false awcp
+    /pdfTextClipPath [ pdfTextClipPath aload pop
+      {/moveto cvx}
+      {/lineto cvx}
+      {/curveto cvx}
+      {/closepath cvx}
+    pathforall ] def
+    currentpoint newpath moveto
+  } {
+    8 {pop} repeat
+  } ifelse
+  0 pdfTextRise neg pdfTextMat dtransform rmoveto
+} def
+/TJm { pdfFontSize 0.001 mul mul neg 0
+       pdfTextMat dtransform rmoveto } def
+/TJmV { pdfFontSize 0.001 mul mul neg 0 exch
+        pdfTextMat dtransform rmoveto } def
+/Tclip { pdfTextClipPath cvx exec clip newpath
+         /pdfTextClipPath [] def } def
+% Level 2 image operators
+/pdfImBuf 100 string def
+/pdfIm {
+  image
+  { currentfile pdfImBuf readline
+    not { pop exit } if
+    (%-EOD-) eq { exit } if } loop
+} def
+/pdfImM {
+  fCol imagemask
+  { currentfile pdfImBuf readline
+    not { pop exit } if
+    (%-EOD-) eq { exit } if } loop
+} def
+/pdfImClip {
+  gsave
+  0 2 4 index length 1 sub {
+    dup 4 index exch 2 copy
+    get 5 index div put
+    1 add 3 index exch 2 copy
+    get 3 index div put
+  } for
+  pop pop rectclip
+} def
+/pdfImClipEnd { grestore } def
+% shading operators
+/colordelta {
+  false 0 1 3 index length 1 sub {
+    dup 4 index exch get 3 index 3 2 roll get sub abs 0.004 gt {
+      pop true
+    } if
+  } for
+  exch pop exch pop
+} def
+/funcCol { func n array astore } def
+/funcSH {
+  dup 0 eq {
+    true
+  } {
+    dup 6 eq {
+      false
+    } {
+      4 index 4 index funcCol dup
+      6 index 4 index funcCol dup
+      3 1 roll colordelta 3 1 roll
+      5 index 5 index funcCol dup
+      3 1 roll colordelta 3 1 roll
+      6 index 8 index funcCol dup
+      3 1 roll colordelta 3 1 roll
+      colordelta or or or
+    } ifelse
+  } ifelse
+  {
+    1 add
+    4 index 3 index add 0.5 mul exch 4 index 3 index add 0.5 mul exch
+    6 index 6 index 4 index 4 index 4 index funcSH
+    2 index 6 index 6 index 4 index 4 index funcSH
+    6 index 2 index 4 index 6 index 4 index funcSH
+    5 3 roll 3 2 roll funcSH pop pop
+  } {
+    pop 3 index 2 index add 0.5 mul 3 index  2 index add 0.5 mul
+    funcCol sc
+    dup 4 index exch mat transform m
+    3 index 3 index mat transform l
+    1 index 3 index mat transform l
+    mat transform l pop pop h f*
+  } ifelse
+} def
+/axialCol {
+  dup 0 lt {
+    pop t0
+  } {
+    dup 1 gt {
+      pop t1
+    } {
+      dt mul t0 add
+    } ifelse
+  } ifelse
+  func n array astore
+} def
+/axialSH {
+  dup 0 eq {
+    true
+  } {
+    dup 8 eq {
+      false
+    } {
+      2 index axialCol 2 index axialCol colordelta
+    } ifelse
+  } ifelse
+  {
+    1 add 3 1 roll 2 copy add 0.5 mul
+    dup 4 3 roll exch 4 index axialSH
+    exch 3 2 roll axialSH
+  } {
+    pop 2 copy add 0.5 mul axialCol sc
+    exch dup dx mul x0 add exch dy mul y0 add
+    3 2 roll dup dx mul x0 add exch dy mul y0 add
+    dx abs dy abs ge {
+      2 copy yMin sub dy mul dx div add yMin m
+      yMax sub dy mul dx div add yMax l
+      2 copy yMax sub dy mul dx div add yMax l
+      yMin sub dy mul dx div add yMin l
+      h f*
+    } {
+      exch 2 copy xMin sub dx mul dy div add xMin exch m
+      xMax sub dx mul dy div add xMax exch l
+      exch 2 copy xMax sub dx mul dy div add xMax exch l
+      xMin sub dx mul dy div add xMin exch l
+      h f*
+    } ifelse
+  } ifelse
+} def
+/radialCol {
+  dup t0 lt {
+    pop t0
+  } {
+    dup t1 gt {
+      pop t1
+    } if
+  } ifelse
+  func n array astore
+} def
+/radialSH {
+  dup 0 eq {
+    true
+  } {
+    dup 8 eq {
+      false
+    } {
+      2 index dt mul t0 add radialCol
+      2 index dt mul t0 add radialCol colordelta
+    } ifelse
+  } ifelse
+  {
+    1 add 3 1 roll 2 copy add 0.5 mul
+    dup 4 3 roll exch 4 index radialSH
+    exch 3 2 roll radialSH
+  } {
+    pop 2 copy add 0.5 mul dt mul t0 add axialCol sc
+    exch dup dx mul x0 add exch dup dy mul y0 add exch dr mul r0 add
+    0 360 arc h
+    dup dx mul x0 add exch dup dy mul y0 add exch dr mul r0 add
+    0 360 arc h f*
+  } ifelse
+} def
+end
+/CIDInit /ProcSet findresource begin
+10 dict begin
+  begincmap
+  /CMapType 1 def
+  /CMapName /Identity-H def
+  /CIDSystemInfo 3 dict dup begin
+    /Registry (Adobe) def
+    /Ordering (Identity) def
+    /Supplement 0 def
+  end def
+  1 begincodespacerange
+    <0000> <ffff>
+  endcodespacerange
+  0 usefont
+  1 begincidrange
+    <0000> <ffff> 0
+  endcidrange
+  endcmap
+  currentdict CMapName exch /CMap defineresource pop
+end
+10 dict begin
+  begincmap
+  /CMapType 1 def
+  /CMapName /Identity-V def
+  /CIDSystemInfo 3 dict dup begin
+    /Registry (Adobe) def
+    /Ordering (Identity) def
+    /Supplement 0 def
+  end def
+  /WMode 1 def
+  1 begincodespacerange
+    <0000> <ffff>
+  endcodespacerange
+  0 usefont
+  1 begincidrange
+    <0000> <ffff> 0
+  endcidrange
+  endcmap
+  currentdict CMapName exch /CMap defineresource pop
+end
+end
+xpdf begin
+%!FontType1-1.0: Helvetica
+12 dict begin
+/FontInfo 10 dict dup begin
+/Notice (Copyright (URW)++,Copyright 1999 by (URW)++ Design & Development; Cyrillic glyphs added by Valek Filippov (C) 2001-2004) readonly def
+/FullName (Nimbus Sans L Regular) readonly def
+/FamilyName (Nimbus Sans L) readonly def
+/isFixedPitch false def
+/ItalicAngle 0 def
+/UnderlinePosition 0 def
+/UnderlineThickness 0 def
+end readonly def
+/FontName /Helvetica def
+/PaintType 0 def
+/FontType 1 def
+/FontMatrix [0.001 0 0 0.001 0 0] readonly def
+/FontBBox [0 -218 762 741] readonly def
+/StrokeWidth 0 def
+/UniqueID 4095404 def
+/Encoding 256 array
+0 1 255 {1 index exch /.notdef put} for
+dup 32 /space put
+dup 40 /parenleft put
+dup 41 /parenright put
+dup 45 /minus put
+dup 46 /period put
+dup 48 /zero put
+dup 49 /one put
+dup 50 /two put
+dup 51 /three put
+dup 52 /four put
+dup 53 /five put
+dup 54 /six put
+dup 55 /seven put
+dup 56 /eight put
+dup 67 /C put
+dup 69 /E put
+dup 73 /I put
+dup 80 /P put
+dup 83 /S put
+dup 86 /V put
+dup 97 /a put
+dup 99 /c put
+dup 101 /e put
+dup 102 /f put
+dup 103 /g put
+dup 105 /i put
+dup 108 /l put
+dup 109 /m put
+dup 110 /n put
+dup 111 /o put
+dup 112 /p put
+dup 114 /r put
+dup 115 /s put
+dup 116 /t put
+dup 118 /v put
+dup 119 /w put
+dup 120 /x put
+readonly def
+currentdict end
+currentfile eexec
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+  /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef
+  /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef
+  /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef
+  /space/exclam/quotedbl/numbersign/dollar/percent/ampersand/quotesingle
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+4 0 0 50 -1 0 12 0.0000 2 135 1890 1200 5175 residual below $tol$?\001

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diff --git a/BookGPU/Chapters/chapter16/figures/pgMesh.pdf b/BookGPU/Chapters/chapter16/figures/pgMesh.pdf
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index 0000000..a4914d6
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diff --git a/BookGPU/Chapters/chapter16/figures/pgMesh.pdf_t b/BookGPU/Chapters/chapter16/figures/pgMesh.pdf_t
new file mode 100644 (file)
index 0000000..c50b40d
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/figures/pgMesh.pdf_t
@@ -0,0 +1,13 @@
+\begin{picture}(0,0)%
+\includegraphics{./Chapters/chapter16/figures/pgMesh.pdf}%
+\end{picture}%
+\setlength{\unitlength}{3947sp}%
+%
+\begingroup\makeatletter\ifx\SetFigFont\undefined%
+\gdef\SetFigFont#1#2#3#4#5{%
+  \reset@font\fontsize{#1}{#2pt}%
+  \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
+  \selectfont}%
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+\begin{picture}(5636,4372)(2249,-4348)
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diff --git a/BookGPU/Chapters/chapter16/figures/resonant_flyback.fig b/BookGPU/Chapters/chapter16/figures/resonant_flyback.fig
new file mode 100644 (file)
index 0000000..f83b4e0
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/figures/resonant_flyback.fig
@@ -0,0 +1,678 @@
+#FIG 3.2
+Landscape
+Center
+Inches
+Letter  
+100.00
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+        3450 4762 3750 4762
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+2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 0 0 2
+        3600 4950 3600 4838
+2 1 0 1 0 7 100 0 -1 0.000 0 0 -1 0 0 2
+        3600 4650 3600 4762
+-6
+-6
+# Air core transformer
+6 3525 3210 3975 3975
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+5 1 0 1 -1 -1 0 0 -1 0.000 0 0 0 0 3877.500 3652.500 3930 3720 3795 3630 3930 3585
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+5 1 0 1 -1 -1 0 0 -1 0.000 0 1 0 0 3873.750 3393.750 3885 3315 3795 3405 3930 3450
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+2 1 0 1 -1 -1 0 0 -1 0.000 0 0 -1 0 0 1
+        3885 3315
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+        3885 3315 3885 3225
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+-6
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+5 1 0 1 0 7 100 0 -1 0.000 0 1 0 0 3011.460 3476.000 3000 3538 3075 3476 3000 3413
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+-6
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+-6
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+-6
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+-6
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+        4379 3138 4379 2862
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+4 0 1 50 -1 0 12 0.0000 2 165 1080 5250 3525 power/ground\001
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diff --git a/BookGPU/Chapters/chapter16/figures/resonant_flyback.pdf b/BookGPU/Chapters/chapter16/figures/resonant_flyback.pdf
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diff --git a/BookGPU/Chapters/chapter16/figures/resonant_flyback.pdf_t b/BookGPU/Chapters/chapter16/figures/resonant_flyback.pdf_t
new file mode 100644 (file)
index 0000000..bace9de
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/figures/resonant_flyback.pdf_t
@@ -0,0 +1,25 @@
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diff --git a/BookGPU/Chapters/chapter16/gpu.tex b/BookGPU/Chapters/chapter16/gpu.tex
new file mode 100644 (file)
index 0000000..bcfc686
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/gpu.tex
@@ -0,0 +1,212 @@
+\section{New parallel envelope-following method}
+\label{sec:gmres}
+
+In this section, we explain how to efficiently
+use matrix-free GMRES to solve
+the Newton update problems with implicit sensitivity calculation,
+i.e., the steps enclosed by the double dashed block
+in Fig.~\ref{fig:ef_flow}.
+Then implementation issues of GPU acceleration
+will be discussed in detail. 
+Finally,  the Gear-2 integration is briefly introduced.
+
+\subsection{GMRES solver for Newton update equation}
+
+\underline{G}eneralized \underline{M}inimum \underline{Res}idual,
+or GMRES method is an iterative method for solving
+systems of linear equations ($A x=b$) with dense matrix $A$.
+The standard GMRES\index{GMRES} is given in Algorithm~\ref{alg:GMRES}.
+It constructs a Krylov subspace\index{Krylov subspace} with order $m$,
+\[ \mathcal{K}_m = \mathrm{span}( b, A^{} b, A^2 b,\ldots, A^{m-1} b ),\]
+where the approximate solution $x_m$ resides.
+In practice, an orthonormal basis $V_m$ that spans the
+subspace $\mathcal{K}_{m}$ can be generated by
+the Arnoldi iteration\index{Arnoldi iterations}.
+The goal of GMRES is to search for an optimal coefficient $y$
+such that the linear combination $x_m = V_m y$ will minimize
+its residual $\| b-Ax_m \|_2$.
+The Arnoldi iteration also creates a by-product,
+an upper Hessenberg matrix\index{Hessenberg matrix} $\tilde{H}_m \in R^{(m+1)\times m}$.
+Thus, the projection of $A$ on the orthonormal basis $V_m$ is described by
+the Arnoldi decomposition
+$
+ AV_m = V_{m+1} \tilde{H}_{m},
+$
+which is useful to check the residual at each iteration
+without forming $x_m$,
+and to solve for coefficient $y$ when residual is smaller than
+a preset tolerance~\cite{Golub:Book'96}.
+
+% Givens rotations are generated to triangularize
+% the Hessenberg matrix $\tilde{H}_m$ in order to solve the overdetermined system.
+% These same Givens rotations are also applied to the right-hand-side $h_{1,0}e_1$,
+% whose last element provides residual information on whether the tolerance
+% has been met in the current iteration.
+
+%$\left\| h_{1,0}e_1-\tilde{H}_m y_m \right\|_2$.
+
+
+%% \begin{algorithm}
+%% \caption{Standard GMRES\index{GMRES} algorithm.} \label{alg:GMRES}
+%% \begin{algorithmic}[1]
+%%   \REQUIRE $ A \in \mathbb{R}^{N \times N}$, $b \in \mathbb{R}^N$,
+%%       and initial guess $x_0 \in \mathbb{R}^N$
+%%   \ENSURE $x \in \mathbb{R}^N$: $\| b - A x\|_2 < tol$
+
+%%   \STATE $r = b - A x_0$
+%%   \STATE $h_{1,0}=\left \| r \right \|_2$
+%%   \STATE $m=0$
+
+%%   \WHILE{$m < max\_iter$}
+%%     \STATE $m = m+1$
+%%     \STATE $v_{m} = r / h_{m,m-1}$
+%%     \STATE \label{line:mvp} $r = A v_m$
+%%     \FOR{$i = 1\ldots m$}
+%%       \STATE $h_{i,m} = \langle v_i, r \rangle$
+%%       \STATE $r = r - h_{i,m} v_i$
+%%     \ENDFOR
+%%     \STATE $h_{m+1,m} = \left\| r \right\|_2$\label{line:newnorm}
+%%     %\STATE Generate Givens rotations to triangularize $\tilde{H}_m$
+%%     %\STATE Apply Givens rotations on $h_{1,0}e_1$ to get residual $\epsilon$
+%%     \STATE Compute the residual $\epsilon$
+%%     \IF{$\epsilon < tol$}
+%%     \STATE Solve the problem: minimize $\|b-Ax_m\|_2$
+%%         \STATE Return $x_m = x_0 + V_m y_m$
+%%     \ENDIF
+%%   \ENDWHILE
+%% \end{algorithmic}
+%% \end{algorithm}
+
+At a first glance, the cost of using standard GMRES directly to
+solve the Newton update in Eq.~\eqref{eq:Newton}
+seems to come mainly from two parts: the
+formulation of the sensitivity matrix $J = \ud x_M/\ud x_0$
+in Eq.~\eqref{eq:sensM} in Section~\ref{sec:gear},
+and the iteration inside the standard GMRES,
+especially the matrix-vector multiplication and the orthonormal
+basis construction (Line~\ref{line:mvp} through
+Line~\ref{line:newnorm} in Algorithm~\ref{alg:GMRES}).
+Based on the observation that only the matrix-vector product is
+required in GMRES, the work in~\cite{Telichevesky:DAC'95} introduces
+an efficient matrix-free algorithm in the shooting-Newton method,
+where the equation solving part also involves with a sensitivity
+matrix.
+The matrix-free method does not take an explicit matrix as
+input, but directly passes the saved capacitance matrices $C_i$ and
+the LU factorizations of $J_i$, $i=0,\ldots,M$, into the Arnoldi
+iteration for Krylov subspace generation.
+Therefore, it avoids the
+cost of forming the dense sensitivity matrix and focuses on
+subspace construction.
+Briefly speaking, Line~\ref{line:mvp} will be replaced
+by a procedure without explict $A$, and we will talk
+about the flow of matrix-free generation of new basis vectors
+in later sections.
+
+
+\subsection{Parallelization on GPU platforms}
+\label{sec:gpu}
+% In this subsection, we describe how to parallelize
+% the proposed matrix-free GMRES solver with Gear-2 sensitivity
+% on GPU platforms.
+There exist many GPU-friendly computing operations
+in GMRES, such as the vector addition (\verb|axpy|),
+2-norm of vector (\verb|nrm2|), and
+sparse matrix-vector multiplication (\verb|csrmv|),
+which have been parallelized in the CUDA Basic Linear
+Algebra Subroutine (CUBLAS) Library and the CUDA Sparse Linear Algebra
+Library (CUSPARSE)~\cite{CUDASDK}.
+
+GPU programming is typically limited by the data transfer bandwidth as
+GPU favors computationally intensive algorithms~\cite{Kirk:Book'10}.
+So how to efficiently transfer the data and wisely partition
+the data between CPU memory and GPU memory
+is crucial for GPU programming.
+In the following, we discuss these two issues in our implementation.
+
+As noted in Section~\ref{sec:ef}, the envelope-following method
+requires the matrices gathered from all the time steps in a
+period in order to solve a Newton update.
+At each time step, SPICE\index{SPICE} has
+to linearize device models, stamp matrix elements
+into MNA (short for modified nodal analysis\index{modified nodal analysis, or MNA}) matrices,
+and solve circuit equations in its inner Newton iteration\index{Newton iteration}.
+When convergence is attained,
+circuit states are saved and then next time step begins.
+This is also the time when we store the needed matrices
+for the envelope-following computation.
+Since these data are involved in the calculation of Gear-2 sensitivity
+matrix in the generation of Krylov subspace vectors in
+Algorithm~\ref{alg:mf_Gear}, it is desirable that all of these
+matrices are transferred to GPU for its data parallel capability.
+
+To efficiently transfer those large data, we explore asynchronous
+memory copy between host and device in the recent GPUs (Fermi architecture),
+so that the copying overlaps with the host's computing of
+the next time step's circuit state. The implementation of asynchronous matrices
+copy includes two parts: allocating page-locked memory, also known as
+pinned memory, where we save matrices for one time step, and using
+asynchronous memory transfer to copy these data to GPU memory.  While
+it is known that page-locked host memory is a scarce resource and
+should not be overused, the demand of memory size of the data
+generated at one time step can be generously accommodated by today's
+mainstream server configurations.
+
+\begin{figure}[htbp]
+  \centering
+  \resizebox{.9\textwidth}{!}{\input{./Chapters/chapter16/figures/gmres_flow.pdf_t}}
+  \caption{GPU parallel solver for envelope-following update.}
+  \label{fig:gmres}
+\end{figure}
+
+
+%When SPICE simulation reaches the end of the period, all data needed by
+%Newton update equation are ready in GPU memory, and the matrix-free
+%GMRES can be applied directly on them to solve for the update vector
+%in \eqref{eq:Newton}. 
+
+% Knowing the dimensions or the sizes of the
+%matrices in a computation helps to boost GPU computing efficiency.
+The second issue is to decide the location of data between CPU and GPU memories.
+Therefore let us first make a rough sketch of the quantities in the
+GMRES Algorithm~\ref{alg:GMRES}.  Although GMRES tends to converge
+quickly for most circuit examples, i.e., the iteration number $m \ll
+N$, the space for storing all the subspace basis $V_m$ of $N$-by-$m$,
+i.e., $m$ column vectors with $N$-length, is still big.  In addition,
+every newly generated matrix-vector product needs to be orthogonalized
+with respect to all its previous basis vectors in
+the Arnoldi iterations\index{Arnoldi iterations}.
+Hence, keeping all the vectors of $V_m$ in GPU global
+memory allows GPU to handle those operations, such as inner-product of
+basis vectors (\verb|dot|) and vector subtraction (\verb|axpy|), in
+parallel.  
+
+On the other hand, it is better to keep the Hessenberg matrix
+$\tilde{H}$, where intermediate results of
+the orthogonalization are stored, at the host side.
+This comes with the following reasons.
+First, its size is $(m+1)$-by-$m$ at most, rather small
+if compared to circuit matrices and Krylov basis.
+Besides, it is also necessary to triangularize $\tilde{H}$
+and check the residual regularly in
+each iteration so the GMRES can return the approximate solution as
+soon as the residual is below a preset tolerance.
+Hence, in consideration of the serial nature of the trianularization,
+the small size of Hessenberg matrix,
+and the frequent inspection of values by host, it is
+preferable to allocate $\tilde{H}$ in CPU (host) memory.
+As shown in Fig.~\ref{fig:gmres}, the memory copy from device to host
+is called each time when Arnoldi iteration generates a new vector
+and the orthogonalization produces the vector $h$.
+
+% \begin{figure}[!tb]
+% \centering
+% \resizebox{.45\textwidth}{!}{\input{./matlab/drawing.ps_tex}}
+% \caption{The flow of envelope-following method.}
+% \label{fig:gmres}
+% %\end{center}
+% \end{figure}
+
+% talk about gear 2 method here.
+\input{Chapters/chapter16/bdf.tex}
+

diff --git a/BookGPU/Chapters/chapter16/intro.tex b/BookGPU/Chapters/chapter16/intro.tex
new file mode 100644 (file)
index 0000000..cbed054
--- /dev/null
+++ b/BookGPU/Chapters/chapter16/intro.tex
@@ -0,0 +1,163 @@
+\section{Introduction}
+
+
+Power converters\index{power converters} have seen a surge of new trends and novel
+applications due to their widespread use in renewable energy systems
+and emerging hybrid and purely-electric vehicles. More efficient
+simulation techniques for power converters are urgently needed to meet
+more design constraints.  In this chapter, we present a novel
+envelope-following parallel transient analysis method for the general
+switching power converters. The new method first exploits the
+parallelisim in the envelope-following method and parallelize the
+Newton update solving part, which is the most computational expensive,
+in GPU platforms to boost the simulation performance.  To further
+speed up the iterative GMRES\index{GMRES} solving for Newton update equation in the
+envelope-following method, we apply the matrix-free\index{matrix-free}
+Krylov subspace\index{Krylov subspace} basis
+generation technique, which was previously used for RF simulation.
+Last, the new method also applies more robust Gear-2 integration to
+compute the sensitivity matrix instead of traditional integration
+methods.
+
+Over the past decades, power electronics\index{power electronics} and especially switching
+power converters have seen a surge of new trends and novel applications
+--- from the growing significance of PWM (pulse-width modulation)
+rectifiers and multilevel inverters to their widespread use in
+renewable (like solar and wind) energy systems, smart grids and emerging
+electric and hybrid vehicles~\cite{Trzynadlowski:Book'10}.
+
+This requires more efficient simulation techniques for power
+electronics to meet the new application and demanding design
+constraints. To facilitate the design of typical power electronics
+circuits, many special-purpose simulation algorithms and
+tools were developed.  Among them is the envelope-following
+method~\cite{Kundert:ICCAD'88,White:TPE'91,Feldmann:ICCAD'96}, %,Bose:TIE'09
+which is able to calculate the slowly changing contour, or
+envelope, of a carrier waveform with a much higher
+switching frequency\index{switching frequency}.
+This is the case for switching power converters,
+which have fast switching currents to convert powers
+from one level to another level.
+In those switching power converters, it is the envelope,
+which is the power voltage delivered,
+not the fast switching waves in every cycle,
+that is of interest to the designers.
+As shown in Fig.~\ref{fig:ef1}, the solid line is
+the waveform of the output node in a Buck
+converter~\cite{Krein:book'97}, the dots are the simulation points
+of SPICE\index{SPICE}, and the appended dash line is the envelope.
+
+Obviously, traditional SPICE will incur an extraordinarily high
+simulation time for this task, since it has to integrate the
+circuit's differential equation with many time points in every
+clock cycle to get the accurate details of the carrier.  For
+switching power converters, the waveform of the carrier in
+consequent cycles does not change much, envelope-following method
+is an approximation analysis method, which skips over several
+cycles (the dash line in Fig.~\ref{fig:ef2}), the so called
+envelope step, without simulating them, and then carries out a
+correction, which usually contains a sensitivity-based Newton
+iteration or shooting until convergence, in order to begin the
+next envelope step.
+
+% \begin{figure}[!b]
+%   \resizebox{.5\textwidth}{!}{\input{./matlab/ef_intro.pstex_t}}
+%   %\includegraphics[width=.45\textwidth]{./matlab/ef_intro_ppt.eps}
+%   \caption{Transient envelope-following analysis.
+%     (This figure reflects a backward-Euler style envelope-following.)}
+%   \label{fig:ef_intro}
+% \end{figure}
+\begin{figure}
+  \centering
+  \subfigure[Illustration of one envelope skip.]
+       {\resizebox{.8\textwidth}{!}{\input{./Chapters/chapter16/figures/ef_intro.pdf_t}}
+            \label{fig:ef1} }
+  \subfigure[The envelope changes in a slow time scale.]
+       {\resizebox{.9\textwidth}{!}{\input{./Chapters/chapter16/figures/envelope.pdf_t}}
+            \label{fig:ef2} }
+  \caption{Transient envelope-following\index{envelope-following} analysis.
+    (Both two figures reflect backward-Euler\index{backward-Euler} style envelope-following.)}
+  \label{fig:ef_intro}
+\end{figure}
+
+%\IEEEpubidadjcol
+
+Also, iterative GMRES\index{GMRES} solver is typically used in the
+envelope-following method to compute the solution of Newton update
+due to its efficiency compared to direct LU\index{LU} method.
+However, as the Jacobian matrix\index{Jacobian matrix}
+or the sensitivity matrix\index{sensitivity matrix}
+in the equation to be solved is dense,
+explicit computing of the Jacobian is a very expense process.
+Recently, the matrix-free\index{matrix-free} GMRES
+was proposed~\cite{Telichevesky:DAC'95}
+for RF shooting based simulation.
+The new method leads to significant
+savings due to its implicit calculation of new basis vectors
+without the explicit formulation of the sensitivity matrix.
+
+
+Modern computer architecture has shifted towards designs that employ
+so called multi-core processor or chip-multiprocessors (CMP)~\cite{IntelMC'06,AMDMC'06}.
+The family of graphic processing units (GPU) are among the most
+powerful many-core computing systems in mass-market use~\cite{nvidia}.
+For instance, the state-of-the-art NVIDIA Tesla T10 chip has a peak
+performance of over 1~TFLOPS versus about 80--100~GFLOPS of Intel i5
+series Quad-core CPUs~\cite{Kirk:Book'10}.
+In addition to the primary
+use of GPUs in accelerating graphics rendering operations, there has
+been considerable interest in exploiting GPUs for general purpose
+computation (GPGPU)~\cite{gpgpu}. The introduction of new parallel
+programming interfaces for general purpose computation, such as
+Computer Unified Device Architecture (CUDA),
+Stream SDK, and OpenCL~\cite{cuda,streamSDK,openCL},
+has made GPUs an attractive choice
+for developing high-performance scientific computation tools
+and solving practical engineering problems.
+Hence, the applications with GPGPUs are rapidly growing in a broad
+variety of parallel numerical computation works~\cite{CUDA_Zone}.
+
+%For this correction to be efficient and precise, it is critical
+%for the Newton iteration to converge fast and yield accurate
+%result.
+
+%Switching power converters and RF (radio-frequency) circuits are
+%typically stiff systems where
+%the fast signals (switching signals or carrier signals) and
+%the slow moving signals (the output voltage waveforms and user signals)
+%coexist.
+%However, it is challenging to simulate the stiff systems
+%using the traditional integration method such as
+%Backward Euler and Trapezoidal method~\cite{Iserles:Book'96}.
+%It has been shown that Gear-2 is more suitable
+%for many practical problems such as stiff problems
+%where one have different time constants
+%(fast ones with large poles and slow ones with small poles)~\cite{Vlach:Book'94}.
+
+%GMRES is a Krylov-subspace based method for solving linear
+%equation, which guarantees convergence and is efficient if the
+%eigenvalues are tightly clustered together~\cite{Golub:Book'96}.
+%Fortunately, most problems arising from the shooting method and
+%the envelope-following method satisfy naturally the convergence
+%criterion of GMRES~\cite{Nastov:ProcIEEE'07} (or after
+%preconditioners applied if not).
+
+
+%\item Newton update equation for envelope-following is formed with
+%    the sensitivity matrix derived from the Gear-2 integration rule for
+%    better stability.
+%   \item Detailed mathematical derivations to show
+%      the relationship between the sensitivity matrix and
+%      the information extracted from transient steps.
+
+After the basic algorithm of envelope-following is briefly reviewed
+and Newton update equation is derived in Section~\ref{sec:ef}.  In
+Section~\ref{sec:gmres} presents the new parallel envelope method
+where the CPU parallelization, matrix-free GMRES and Gear-2
+integration will be discussed. We will show the parallelization of the
+Newton update, which is the most time consuming step, in the
+envelop-follow method, in GPU platforms.  We will also present the new
+GMRES solver using matrix-free Jacobian-vector multiplication and the
+Gear-2 integration for sensitivity based Newton update equation.
+Numerical examples are shown in Section~\ref{sec:exp}, and finally,
+this chapter is summarized in Section~\ref{sec:summary}.

diff --git a/BookGPU/Makefile b/BookGPU/Makefile
index 9280d0bfa0d5337258a87149ac2aee56bd14ed78..31bb997020509eaa2002f8e5a38e241dcfe22850 100644 (file)
--- a/BookGPU/Makefile
+++ b/BookGPU/Makefile
@@ -5,16 +5,18 @@ all:
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