new

[book_gpu.git] / BookGPU / Chapters / chapter16 / ef.tex

diff --git a/BookGPU/Chapters/chapter16/ef.tex b/BookGPU/Chapters/chapter16/ef.tex
index b0967a93f751da1e3af2fad6661bec783b57d0d4..1cc747bac95516040c9d0f4b092524b7f2b0ceca 100644 (file)
--- a/BookGPU/Chapters/chapter16/ef.tex
+++ b/BookGPU/Chapters/chapter16/ef.tex
@@ -52,7 +52,7 @@ 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 $x((m+k)T)$,
 % 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 $x((m-1)T)$,
+a forward Euler\index{Euler!forward Euler} style jumping relies only on $x(mT)$ and $x((m-1)T)$,

 i.e.,
 \[
    x((m+k)T)
 i.e.,
 \[
    x((m+k)T)


@@ -61,14 +61,14 @@ i.e.,
 \]
 However, this approach is inefficient due to its restriction on
 envelope step $k$, since larger $k$ usually causes instability.
 \]
 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,
+Instead, backward Euler\index{Euler!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
 %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},
+by Newton iteration\index{iterative method!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.
 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.


@@ -107,7 +107,7 @@ 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,
 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,
+if the DAE is integrated with backward Euler rule,

 the sensitivity is
 \[
    J = \frac{\ud x_M}{\ud x_0}
 the sensitivity is
 \[
    J = \frac{\ud x_M}{\ud x_0}