From 0bc82beaa8850c94a9d1a2829ff35cbdaabde210 Mon Sep 17 00:00:00 2001 From: Manas Rachh Date: Tue, 20 Feb 2024 11:01:39 -0500 Subject: [PATCH] fixing typos in math --- docs/conf.py | 4 +-- docs/math.rst | 90 ++++++++++++++++++++++++++++++--------------------- 2 files changed, 55 insertions(+), 39 deletions(-) diff --git a/docs/conf.py b/docs/conf.py index 366b53f..d7180f2 100644 --- a/docs/conf.py +++ b/docs/conf.py @@ -58,8 +58,8 @@ # General information about the project. project = u'fmm2d' -copyright = u'2018-2019 The Simons Foundation, Inc. - All Rights Reserved' -author = u"Zydrunas Gimbutas, Leslie Greengard, Mike O'Neil, Manas Rachh, and Vladimir Rokhlin" +copyright = u'2018-2023 The Simons Foundation, Inc. - All Rights Reserved' +author = u"Travis Askham, Zydrunas Gimbutas, Leslie Greengard, Libin Lu, Mike O'Neil, Manas Rachh, and Vladimir Rokhlin" # The version info for the project you're documenting, acts as replacement for # |version| and |release|, also used in various other places throughout the diff --git a/docs/math.rst b/docs/math.rst index d339bda..944c8fb 100644 --- a/docs/math.rst +++ b/docs/math.rst @@ -1,22 +1,23 @@ Definitions =========== Let $x_{j} = (x_{j,1}, x_{j,2}) \in \mathbb{R}^{2}$, $i=1,2,\ldots N$, denote a collection -of source locations and let $t_{i} = (t_{i,1}, t_{i,2}) \in \mathbb{R}^{2}$ denote a collection -of target locations. - +of source locations and let $x = (x_{1}, x_{2}) \in \mathbb{R}^{2}$ denote a +target location. Unless stated otherwise, the gradients are with +respect to the target variable $x$. Laplace FMM ************* The Laplace FMM comes in three varieties * rfmm2d: Charges, dipole strengths, potentials, their - gradients, and hessians are all double precision + gradients, and Hessians are all double precision * lfmm2d: All of the above quantities are double complex * cfmm2d: Same as lfmm2d except that there are no dipole orientation vectors, - and the dipole kernel is given by $1/(z_{i} - \xi_{j})$ where $z_{i}, \xi_{j}$ - are the complex versions of the source and target locations given by - $z_{i} = t_{i,1} + i\cdot t_{i,2}$, and $\xi_{j} = x_{j,1} + i \cdot x_{j,2}$. - Moreover, the gradients, and hessians + and the dipole kernel is given by $1/(z - \xi_{j})$ where $z, \xi_{j}$ + are the target, and source locations viewed as points in the complex + plane given by + $z = x_{1} + i\cdot x_{2}$, and $\xi_{j} = x_{j,1} + i \cdot x_{j,2}$. + Moreover, the gradients, and Hessians in this case are derivatives with respect to $z$, and in a slight abuse of notation, we set $\frac{\mathrm{d}}{\mathrm{d} z} \log(\|z\|) = 1/z$ @@ -31,8 +32,7 @@ denote a collection of dipole strengths, and $d_{j} \in \mathbb{R}^{2}$, $j=1,2,\ldots N$, denote the corresponding dipole orientation vectors. The Laplace FMM (rfmm2d) computes -the potential $u(x) \in \mathbb{R}$ and its gradient $\nabla u(x) \in -\mathbb{R}^{2}$ +the potential $u(x) \in \mathbb{R}$ given by .. math:: @@ -40,6 +40,7 @@ given by u(x) = \sum_{j=1}^{N} c_{j} \log{(\|x-x_{j}\|)} - v_{j} d_{j}\cdot \nabla \log{(\|x-x_{j}\|)} \, , +and its gradient $\nabla u(x) \in \mathbb{R}^{2}$ at the source and target locations. When $x=x_{j}$, the term corresponding to $x_{j}$ is dropped from the sum. @@ -54,8 +55,7 @@ denote a collection of dipole strengths, and $d_{j} \in \mathbb{R}^{2}$, $j=1,2,\ldots N$, denote the corresponding dipole orientation vectors. The Laplace FMM (rfmm2d) computes -the potential $u(x) \in \mathbb{C}$ and its gradient $\nabla u(x) \in -\mathbb{C}^{2}$ +the potential $u(x) \in \mathbb{C}$ given by .. math:: @@ -63,6 +63,7 @@ given by u(x) = \sum_{j=1}^{N} c_{j} \log{(\|x-x_{j}\|)} - v_{j} d_{j}\cdot \nabla \log{(\|x-x_{j}\|)} \, , +and its gradient $\nabla u(x) \in \mathbb{C}^{2}$ at the source and target locations. When $x=x_{j}$, the term corresponding to $x_{j}$ is dropped from the sum. @@ -76,8 +77,7 @@ $j=1,2,\ldots N$, denote a collection of dipole strengths. The Laplace FMM (cfmm2d) computes -the potential $u(z) \in \mathbb{C}$ and its gradient $\frac{\mathrm{d}}{\mathrm{d}z} u(z) \in -\mathbb{C}$ +the potential $u(z) \in \mathbb{C}$ given by .. math:: @@ -85,9 +85,11 @@ given by u(z) = \sum_{j=1}^{N} c_{j} \log{(\|z-\xi_{j}\|)} - \frac{v_{j}}{z-\xi_{j}} \, , +and its gradient $\frac{\mathrm{d}}{\mathrm{d}z} u(z) \in +\mathbb{C}$ at the source and target locations. When $z=\xi_{j}$, the term corresponding to $\xi_{j}$ is dropped from the sum. Here -$z = x(1) + i x(2)$, and $\xi_{j} = x_{j}(1) + ix_{j}(2)$, are the complex +$z = x(1) + i x(2)$, and $\xi_{j} = x_{j,1} + ix_{j,2}$, are the complex numbers corresponding to $x$ and $x_{j}$ respectively. @@ -103,7 +105,7 @@ Let $k\in\mathbb{C}$ denote the wave number or the Helmholtz parameter. The Helmholtz FMM computes -the potential $u(x)$ and its gradient $\nabla u(x)$ +the potential $u(x) \in \mathbb{C}$ given by .. math:: @@ -111,6 +113,7 @@ given by u(x) = \sum_{j=1}^{N} c_{j} H_{0}^{(1)}(k\|x-x_{j}\|) - v_{j} d_{j}\cdot \nabla H_{0}^{(1)}(k\|x-x_{j}\|) \, , +and its gradient $\nabla u(x) \in \mathbb{C}^{2}$ at the source and target locations, where $H_{0}^{(1)}$ is the Hankel function of the first kind of order $0$. When $x=x_{j}$, the term corresponding to $x_{j}$ is dropped from the sum. @@ -126,7 +129,8 @@ $j=1,2,\ldots N$, denote a collection of dipole strengths. The Biharmonic FMM computes -the potential $u(x)$ and its `gradient` = $(P_{z} \frac{\mathrm{d}}{\mathrm{d}z}, P_{\overline{z}} \frac{\mathrm{d}}{\mathrm{d}z}, \frac{\mathrm{d}}{\mathrm{d}\overline{z}})$ +the potential $u(x)$ and its `gradient` = +$(P_{z} \frac{\mathrm{d}}{\mathrm{d}z}, P_{\overline{z}} \frac{\mathrm{d}}{\mathrm{d}z}, \frac{\mathrm{d}}{\mathrm{d}\overline{z}})$ given by .. math:: @@ -148,12 +152,19 @@ given by \frac{v_{j,3}}{(\overline{z-\xi_{j}})^2} - 2v_{j,2} \frac{z - \xi_{j}}{(\overline{z-\xi_{j}})^3} \, , \\ -at the source and target locations. When $z=\xi_{j}$, the term +at the source and target locations. +When $z=\xi_{j}$, the term corresponding to $\xi_{j}$ is dropped from the sum. Here $z = x(1) + i x(2)$, and $\xi_{j} = x_{j}(1) + ix_{j}(2)$, are the complex numbers corresponding to $x$ and $x_{j}$ respectively. When $x=x_{j}$, the term corresponding to $x_{j}$ is dropped from the sum. +In the above, $(P_{z} \frac{\mathrm{d}}{\mathrm{d}z}$ +is the part of the gradient $\frac{\mathrm{d}}{\mathrm{d} z}$ +which is purely a function of $z$, and similarly +$(P_{\overline{z}} \frac{\mathrm{d}}{\mathrm{d}z}$ +is the part of the gradient $\frac{\mathrm{d}}{\mathrm{d} z}$ +which is purely a function of $\overline{z}$. Modified Biharmonic FMM @@ -169,23 +180,23 @@ where $K_{0}$ is the modified Bessel function of order $0$, and $\beta$ is the Modified Biharmonic wavenumber. -Let $c_{j} \in \mathbb{C}$, +Let $c_{j} \in \mathbb{R}$, denote a collection of charge strengths, -$v_{j} \in \mathbb{C}$, +$v_{j} \in \mathbb{R}$, denote a collection of dipole strengths, $d_{j} = (d_{j,1}, d_{j,2})$ denote a collection of dipole vectors, -$q_{j} \in \mathbb{C}$ denote a collection of +$q_{j} \in \mathbb{R}$ denote a collection of quadrupole strengths, -$w_{j} = (w_{j,1}, w_{j,2}, w_{j,3}) \in \mathbb{C}^{3}$, +$w_{j} = (w_{j,1}, w_{j,2}, w_{j,3}) \in \mathbb{R}^{3}$, denote a collection of quadrupole orientation vectors, $o_{j}$ denote a collection of octopole strengths, and -$p_{j} = (p_{j,1}, p_{j,2}, p_{j,3}, p_{j,4}) \in \mathbb{C}^{4}$, -denote a collceetion of octopole strengths. For all the vectors, +$p_{j} = (p_{j,1}, p_{j,2}, p_{j,3}, p_{j,4}) \in \mathbb{R}^{4}$, +denote a collection of octopole strengths. For all the vectors, $j=1,2,\ldots N$. The Modified Biharmonic FMM computes -the potential $u(x)$ and its gradients, and hessians, +the potential $u(x)\in \mathbb{R}$ given by .. math:: @@ -201,6 +212,7 @@ given by &p_{j,3} \partial_{y_{1},y_{2},y_{2}} G^{\textrm{mbh}}(x,x_{j}) + p_{j,4} \partial_{y_{2},y_{2},y_{2}} G^{\textrm{mbh}}(x,x_{j}) \big) \, , +and its gradients $\nabla u(x) \in \mathbb{R}^{2}$ at the source and target locations. When $x=x_{j}$, the term corresponding to $x_{j}$ is dropped from the sum. @@ -236,10 +248,10 @@ is given by (x_{1}-y_{1})(x_{2}-y_{2}) & (x_{2} - y_{2})^2 \end{bmatrix} \, , -and finally let $\Pi^{\textrm{stok}} (x,y)$ denote its assocaited pressure tensor given by +and finally let $\Pi^{\textrm{stok}} (x,y)$ denote its associated pressure tensor given by .. math:: - v\cdot \Pi(x,y)^{textrm{stok}} = -\frac{v}{\|x-y \|^2} + \frac{2 v \cdot(x-y)}{\|x-y \|^4} + v\cdot \Pi(x,y)^{\textrm{stok}} = -\frac{v}{\|x-y \|^2} + \frac{2 v \cdot(x-y)}{\|x-y \|^4} \begin{bmatrix} (x_{1}-y_{1}) \\ (x_{2}-y_{2}) @@ -247,10 +259,10 @@ and finally let $\Pi^{\textrm{stok}} (x,y)$ denote its assocaited pressure tenso Let $c_{j} \in \mathbb{R}^2$, $j=1,2,\ldots N$, -denote a collection of stokeslet strengths, $v_{j} \in \mathbb{R}^2$, +denote a collection of Stokeslet strengths, $v_{j} \in \mathbb{R}^2$, $j=1,2,\ldots N$, -denote a collection of stresslet strengths, and $d_{j} \in \mathbb{R}^{2}$, -$j=1,2,\ldots N$, denote the corresponding stresslet orientation vectors. +denote a collection of Stresslet strengths, and $d_{j} \in \mathbb{R}^{2}$, +$j=1,2,\ldots N$, denote the corresponding Stresslet orientation vectors. The Stokes FMM computes the potential $u(x) \in \mathbb{R}^2$, its gradient $\nabla u(x) \in @@ -269,24 +281,28 @@ corresponding to $x_{j}$ is dropped from the sum. Vectorized versions ******************* -The vectorized versions of the Helmholtz FMM, -computes repeated FMMs for new charge and dipole strengths -located at the same source locations, where the potential and its -gradient are evaluated at the same set of target locations. +The vectorized versions of the FMMs compute the same sums +as above for a set of problems in which the source and target +locations are constant but multiple values of the charges, dipoles, etc +are specified at each source. Given a set of problems with this structure, +the vectorized versions are faster than calling the standard +FMM multiple times in sequence. For example, let $c_{\ell,j}\in\mathbb{C}$, $j=1,2,\ldots N$, $\ell=1,2,\ldots n_{d}$ denote a collection of $n_{d}$ charge strengths, and let $v_{\ell,j} \in \mathbb{C}$, $d_{\ell,j} \in \mathbb{R}^2$ denote a collection of $n_{d}$ dipole strengths and orientation vectors. -Then the vectorized Helmholtz FMM computes the potentials $u_{\ell}(x)$ -and its gradients $\nabla u_{\ell}(x)$ defined by the formula +Then the vectorized Helmholtz FMM computes the potentials $u_{\ell}(x) \in +\mathbb{C}$ +given by .. math:: :label: helm_nbody_vec - u_{\ell}(x) = \sum_{j=1}^{N} c_{\ell,j} H_{0}^{(1)}(k\|x-x_{j}\|) - v_{\ell,j} d_{j}\cdot \nabla H_{0}^{(1)}(k\|x-x_{j}\|) \, , + u_{\ell}(x) = \sum_{j=1}^{N} c_{\ell,j} H_{0}^{(1)}(k\|x-x_{j}\|) - v_{\ell,j} d_{\ell,j}\cdot \nabla H_{0}^{(1)}(k\|x-x_{j}\|) \, , +and its gradients $\nabla u_{\ell}(x) \in \mathbb{C}^{2}$ at the source and target locations. .. note::