Updates to A2, now includes selection of polynomial order

Assignments/A2/A2.tex

@@ -279,15 +279,15 @@ Questions - rewrite isInBounds not assuming ascending
 
 \newpage
 
-\section* {Curve Interpolation ADT Module}
+\section* {Curve ADT Module}
 
 \subsection*{Template Module}
 
-CurveInterpADT
+CurveADT
 
 \subsection* {Uses}
 
-SeqServicesModule for isAscending, interp1d and index
+SeqServicesModule
 
 \subsection* {Syntax}
 
@@ -297,7 +297,7 @@ MAX\_ORDER = 2
 
 \subsubsection* {Exported Types}
 
-CurveInterpT = ?
+CurveT = ?
 
 \subsubsection* {Exported Access Programs}
 
@@ -305,8 +305,8 @@ CurveInterpT = ?
 \hline
 \textbf{Routine name} & \textbf{In} & \textbf{Out} & \textbf{Exceptions}\\
 \hline
-new CurveInterpT & $X: \mathbb{R}^n$, $Y: \mathbb{R}^n$, $i: \mathbb{N}$ &
-                                                                           CurveInterpT & IndepVarNotAscending,\newline InvalidInterpOrder\\
+new CurveT & $X: \mathbb{R}^n$, $Y: \mathbb{R}^n$, $i: \mathbb{N}$ &
+                                                                           CurveT & IndepVarNotAscending,\newline InvalidInterpOrder\\
 \hline
 minD & ~ & $\mathbb{R}$ & ~\\
 \hline
@@ -333,14 +333,15 @@ None
 
 \subsubsection* {Assumptions}
 
-None
+The user will not request function evaluations that would cause the
+interpolation to select index values outside of where the function is defined.
 
 \subsubsection* {Access Routine Semantics}
 
-\noindent new CurveInterpT($X, Y$):
+\noindent new CurveT($X, Y$):
 \begin{itemize}
 \item transition: $\mbox{minx}, \mbox{maxx}, o, f := X_0, X_{|X|-1}, i, (\lambda v:
-  \mbox{interp}(X, Y, v))$
+  \mbox{interp}(X, Y, o, v))$
 
 \item output: $out := \mbox{self}$
 \item exception: $(\neg \mbox{isAscending}(X) \Rightarrow
@@ -373,11 +374,11 @@ None
 
 \subsection*{Local Functions}
 
-interp: $\mathbb{R}^n \times \mathbb{R}^n \times \mathbb{R} \times \mathbb{N}
+interp: $\mathbb{R}^n \times \mathbb{R}^n \times \mathbb{N} \times \mathbb{R}
 \rightarrow \mathbb{R}$\\
 
-\noindent interp($X$, $Y$, $v$, $o$) $$\equiv (o=1 \Rightarrow \mbox{interpLin}(X_i, Y_i,
-X_{i+1}, Y_{i+1}, v) | o = 2 \Rightarrow \mbox{interpQuad}(X_i, Y_i, X_{i+1},
+\noindent interp($X$, $Y$, $o$, $v$) $$\equiv (o=1 \Rightarrow \mbox{interpLin}(X_i, Y_i,
+X_{i+1}, Y_{i+1}, v) | o = 2 \Rightarrow \mbox{interpQuad}(X_{i-1}, Y_{i-1}, X_i, Y_i, X_{i+1},
 Y_{i+1}, v) )$$  where $i = \mbox{index}(X, v)$\\
 
 \newpage
@@ -390,7 +391,7 @@ DataModule
 
 \subsection* {Uses}
 
-CurveInterpT, SeqServicesModule for interp1d and index
+CurveT, SeqServicesModule for interpLin and index
 
 \subsection* {Syntax}
 
@@ -406,11 +407,11 @@ MAX\_SIZE = 10
 \hline
 Data\_init & ~ & ~ & ~\\
 \hline
-Data\_add & c: CurveInterpT, $z: \mathbb{R}$ & ~ & Full, IndepVarNotAscending\\
+Data\_add & c: CurveT, $z: \mathbb{R}$ & ~ & Full, IndepVarNotAscending\\
 \hline
 eval & $x: \mathbb{R}, z: \mathbb{R}$ & ~ & ~\\
 \hline
-slice & $x: \mathbb{R}$ & CurveInterpT & ~\\
+slice & $x: \mathbb{R}$ & CurveT & ~\\
 \hline
 
 \end{tabular}
@@ -419,7 +420,7 @@ slice & $x: \mathbb{R}$ & CurveInterpT & ~\\
 
 \subsubsection* {State Variables}
 
-$s$: sequence of CurveInterpT\\
+$s$: sequence of CurveT\\
 $Z$: sequence of $\mathbb{R}$
 
 \subsubsection* {State Invariant}
@@ -448,14 +449,14 @@ Data\_init():
 
 \noindent Data\_eval($x, z$):
 \begin{itemize}
-\item output: $out := \mbox{interp1d}(z_j, s_j.\mbox{eval}(x), z_{j+1},
+\item output: $out := \mbox{interpLin}(z_j, s_j.\mbox{eval}(x), z_{j+1},
   s_{j+1}.\mbox{eval}(x), z)$, where $j = \mbox{index}(Z, z)$
 \item exception:  $exc := (\neg \mbox{isInBounds}(Z, z) \Rightarrow \mbox{OutOfDomain})$
 \end{itemize}
 
 \noindent Data\_slice($x$):
 \begin{itemize}
-\item output: $out := \mbox{CurveInterpT}(Z, Y)$, where
+\item output: $out := \mbox{CurveT}(Z, Y)$, where
   $Y = ||(i: \mathbb{N}| i \in [0 .. |Z|-1] : \langle s_i.\mbox{eval}(x) \rangle
   )$ or $Y = \mbox{ map } \mbox{eval}(x) \mbox{ } s$ \textit{\# in both cases
     there is an abuse of notation}
@@ -490,7 +491,7 @@ isAscending & $x: \mathbb{R}^n$ & $\mathbb{B}$ & ~\\
 \hline
 isInBounds & $x: \mathbb{R}^n, \mathbb{R}$ & $\mathbb{B}$ & ~\\
 \hline
-interp1d & $x_1: \mathbb{R}, y_1: \mathbb{R}, x_2: \mathbb{R}, y_2: \mathbb{R},
+interpLin & $x_1: \mathbb{R}, y_1: \mathbb{R}, x_2: \mathbb{R}, y_2: \mathbb{R},
            x: \mathbb{R}$ & $\mathbb{R}$ & ~\\
 \hline
 index & $X: \mathbb{R}^n, x: \mathbb{R}$ & $\mathbb{N}$ & ~\\
@@ -533,9 +534,10 @@ None, unless noted with a particular access program
 \item exception: none
 \end{itemize}
 
-\noindent interpQuad($x_1, y_1, x_2, y_2, x$) \textit{\# assuming isAscending is True}
+\noindent interpQuad($x_0, y_0, x_1, y_1, x_2, y_2, x$) \textit{\# assuming isAscending is True}
 \begin{itemize}
-\item output: $out := \frac{(y_2 - y_1)}{(x_2-x_1)} (x - x_1) + y_1$
+\item output: $out := y_1 + \frac{y_2 - y_0}{x_2-x_0} (x - x_1) + \frac{y_2 - 2 y_1 + y_0}{2
+  (x_2-x_1)^2} (x - x_1)^2$
 \item exception: none
 \end{itemize}