diff --git a/Interpolation/doc_tex/Interpolation/interpolation.tex b/Interpolation/doc_tex/Interpolation/interpolation.tex index ab88ae2ab6b..94ffc4c0670 100644 --- a/Interpolation/doc_tex/Interpolation/interpolation.tex +++ b/Interpolation/doc_tex/Interpolation/interpolation.tex @@ -7,11 +7,11 @@ re-produces linear functions exactly. The interpolation of $\Phi(\mathbf{x})$ is given as the linear combination of the neighbors' function values weighted by the coordinates: \begin{displaymath} - Z^0(\mathbf{x}) = \sum_i \lambda_i(\mathbf{x}) z_i. + Z^0(\mathbf{x}) = \ccSum{i}{}{ \lambda_i(\mathbf{x}) z_i}. \end{displaymath} Indeed, if $z_i=a + \mathbf{b}^t \mathbf{p_i}$ for all natural neighbors of $\mathbf{x}$, we have -\[ Z^0(\mathbf{x}) = \sum_i \lambda_i(\mathbf{x}) (a + \mathbf{b}^t\mathbf{p_i}) = a+\mathbf{b}^t \mathbf{x}\] +\[ Z^0(\mathbf{x}) = \ccSum{i}{}{ \lambda_i(\mathbf{x}) (a + \mathbf{b}^t\mathbf{p_i})} = a+\mathbf{b}^t \mathbf{x}\] by the barycentric coordinate property. The first example in Subsection~\ref{subsec:interpol_examples} shows how the function is called. @@ -31,20 +31,20 @@ Sibson's $Z^1$ interpolant is a combination of the linear interpolant $Z^0$ and an interpolant $\xi$ which is the weighted sum of the first degree functions $$\xi_i(\mathbf{x}) = z_i -+\mathbf{g_i}^t(\mathbf{x}-\mathbf{p_i}),\qquad \xi(\mathbf{x})= \frac{\sum_i \frac{\lambda_i(\mathbf{x})} - {\|\mathbf{x}-\mathbf{p_i}\|}\xi_i(\mathbf{x}) }{\sum_i - \frac{\lambda_i(\mathbf{x})}{\|\mathbf{x}-\mathbf{p_i}\|}}.$$ ++\mathbf{g_i}^t(\mathbf{x}-\mathbf{p_i}),\qquad \xi(\mathbf{x})= \frac{\ccSum{i}{}{ \frac{\lambda_i(\mathbf{x})} + {\|\mathbf{x}-\mathbf{p_i}\|}\xi_i(\mathbf{x}) } }{\ccSum{i}{}{ + \frac{\lambda_i(\mathbf{x})}{\|\mathbf{x}-\mathbf{p_i}\|}}}.$$ Sibson observed that the combination of $Z^0$ and $\xi$ reconstructs exactly a spherical quadric if they are mixed as follows: $$ Z^1(\mathbf{x}) = \frac{\alpha(\mathbf{x}) Z^0(\mathbf{x}) + \beta(\mathbf{x}) \xi(\mathbf{x})}{\alpha(\mathbf{x}) + - \beta(\mathbf{x})} \textrm{ where } \alpha(\mathbf{x}) = -\frac{\sum_i \lambda_i(\mathbf{x}) \frac{\|\mathbf{x} - - \mathbf{p_i}\|^2}{f(\|\mathbf{x} - \mathbf{p_i}\|)}}{\sum_i - \frac{\lambda_i(\mathbf{x})} {f(\|\mathbf{x} - \mathbf{p_i}\|)}} -\textrm{ and } \beta(\mathbf{x})= \sum_i \lambda_i(\mathbf{x}) -\|\mathbf{x} - \mathbf{p_i}\|^2,$$ + \beta(\mathbf{x})} \mbox{ where } \alpha(\mathbf{x}) = +\frac{\ccSum{i}{}{ \lambda_i(\mathbf{x}) \frac{\|\mathbf{x} - + \mathbf{p_i}\|^2}{f(\|\mathbf{x} - \mathbf{p_i}\|)}}}{\ccSum{i}{}{ + \frac{\lambda_i(\mathbf{x})} {f(\|\mathbf{x} - \mathbf{p_i}\|)}}} +\mbox{ and } \beta(\mathbf{x})= \ccSum{i}{}{ \lambda_i(\mathbf{x}) +\|\mathbf{x} - \mathbf{p_i}\|^2},$$ where in Sibson's original work, $f(\|\mathbf{x} - \mathbf{p_i}\|) = \|\mathbf{x} - \mathbf{p_i}\|$. @@ -77,8 +77,8 @@ Knowing the gradient $\mathbf{g_i}$ for all $\mathbf{p_i} \in exactly quadratic functions. This interpolant is not $C^1$ continuous in general. It is defined as follows: \begin{displaymath} - I^1(\mathbf{x}) = \sum_i \lambda_i(\mathbf{x}) - (z_i + \frac{1}{2} \mathbf{g_i}^t (\mathbf{x} - \mathbf{p_i})) + I^1(\mathbf{x}) = \ccSum{i}{}{ \lambda_i(\mathbf{x}) + (z_i + \frac{1}{2} \mathbf{g_i}^t (\mathbf{x} - \mathbf{p_i}))} \end{displaymath} @@ -89,9 +89,9 @@ $f$ from the function values on the data sites. For the data point $\mathbf{p_i}$, we determine $$\mathbf{g_i} = \min_{\mathbf{g}} -\sum_j +\ccSum{j}{}{ \frac{\lambda_j(\mathbf{p_i})}{\|\mathbf{p_i} - \mathbf{p_j}\|^2} -\left( z_j - (z_i + \mathbf{g}^t (\mathbf{p_j} -\mathbf{p_i})) \right), +\left( z_j - (z_i + \mathbf{g}^t (\mathbf{p_j} -\mathbf{p_i})) \right)}, $$ where $\lambda_j(\mathbf{p_i})$ is the natural neighbor coordinate of $\mathbf{p_i}$ with respect to $\mathbf{p_i}$ associated to diff --git a/QP_solver/doc_tex/QP_solver/main.tex b/QP_solver/doc_tex/QP_solver/main.tex index 6345edce7e0..726a7646fb5 100644 --- a/QP_solver/doc_tex/QP_solver/main.tex +++ b/QP_solver/doc_tex/QP_solver/main.tex @@ -514,7 +514,7 @@ the case if and only if there are real coefficients $\lambda_1,\ldots,\lambda_n$ such that $p$ is a convex combination of $p_1,\ldots,p_n$: \[ -p = \sum_{j=1}^{n}~\lambda_j~p_j, \quad \sum_{j=1}^{n}~\lambda_j = 1, +p = \ccSum{j=1}{n}{~\lambda_j~p_j}, \quad \ccSum{j=1}{n}{~\lambda_j} = 1, \quad \lambda_j \geq 0 \mbox{~for all $j$.} \] The problem of testing the existence of such $\lambda_j$ can @@ -526,11 +526,11 @@ $h_1,\ldots,h_n,h$, we have \[q_j = h_j \cdot (p_j \mid 1) \mbox{~for all $j$, and~} q = h \cdot (p\mid 1).\] Now, nonnegative $\lambda_1,\ldots,\lambda_n$ are suitable coefficients for a convex combination if and only if -\[\sum_{j=1}^n~ \lambda_j(p_j \mid 1) = (p\mid 1), \] +\[\ccSum{j=1}{n}{~ \lambda_j(p_j \mid 1)} = (p\mid 1), \] equivalently, if there are $\mu_1,\ldots,\mu_n$ (with $\mu_j = \lambda_j \cdot h/{h_j}$ for all $j$) such that \[ -\sum_{j=1}^n~\mu_j~q_j = q, \quad \mu_j \geq 0\mbox{~for all $j$}. +\ccSum{j=1}{n}{~\mu_j~q_j} = q, \quad \mu_j \geq 0\mbox{~for all $j$}. \] The linear program now tests for the existence of nonnegative $\mu_j$ diff --git a/QP_solver/doc_tex/QP_solver_ref/Quadratic_program_solution.tex b/QP_solver/doc_tex/QP_solver_ref/Quadratic_program_solution.tex index 436b319a8fc..63c7c914360 100644 --- a/QP_solver/doc_tex/QP_solver_ref/Quadratic_program_solution.tex +++ b/QP_solver/doc_tex/QP_solver_ref/Quadratic_program_solution.tex @@ -349,8 +349,8 @@ then $\lambda_i\geq 0$ ($\lambda_i\leq 0$, respectively). &&\leq 0 & \mbox{if $l_j=-\infty$.} \end{array} \] -\item \[\qplambda^T\qpb \quad<\quad \sum_{j: \qplambda^TA_j <0} \qplambda^TA_j u_j -\quad+\quad \sum_{j: \qplambda^TA_j >0} \qplambda^TA_j l_j.\] +\item \[\qplambda^T\qpb \quad<\quad \ccSum{j: \qplambda^TA_j <0}{}{ \qplambda^TA_j u_j } +\quad+\quad \ccSum{j: \qplambda^TA_j >0}{}{ \qplambda^TA_j l_j}.\] \end{enumerate} {\bf Proof:} Let us assume for the purpose of obtaining a contradiction @@ -358,10 +358,10 @@ that there is a feasible solution $\qpx$. Then we get \[ \begin{array}{lcll} 0 &\geq& \qplambda^T(A\qpx -\qpb) & \mbox{(by $A\qpx\qprel \qpb$ and 1.)} \\ - &=& \sum_{j: \qplambda^TA_j <0} \qplambda^TA_j x_j -\quad+\quad \sum_{j: \qplambda^TA_j >0} \qplambda^TA_j x_j - \qplambda^T \qpb \\ - &\geq& \sum_{j: \qplambda^TA_j <0} \qplambda^TA_j u_j -\quad+\quad \sum_{j: \qplambda^TA_j >0} \qplambda^TA_j l_j - \qplambda^T \qpb & + &=& \ccSum{j: \qplambda^TA_j <0}{}{ \qplambda^TA_j x_j } +\quad+\quad \ccSum{j: \qplambda^TA_j >0}{}{ \qplambda^TA_j x_j} - \qplambda^T \qpb \\ + &\geq& \ccSum{j: \qplambda^TA_j <0}{}{ \qplambda^TA_j u_j } +\quad+\quad \ccSum{j: \qplambda^TA_j >0}{}{ \qplambda^TA_j l_j} - \qplambda^T \qpb & \mbox{(by $\qpl\leq \qpx \leq \qpu$ and 2.)} \\ &>& 0 & \mbox{(by 3.)}, \end{array}