# Very long equation manipulation with arrows

I was trying to fit this equation manipulation in a page as shown in the MWE:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{geometry} \geometry{a4paper,top=2.5cm,bottom=2.5cm,left=2cm,right=2cm,heightrounded,bindingoffset=5mm}

\usepackage{amsmath,cancel,witharrows}

\begin{document}

\noindent
some words
\begin{DispWithArrows*}
P(t|t)&=E\left[\tilde{e}_x(t)\tilde{e}_x^T(t)\right]\\
&=E\left[\big((I-K(t)C)e_x(t)-K(t)v(t)\big)\big((I-K(t)C)e_x(t)-K(t)v(t)\big)^T\right] \Arrow[i]{$e_x$ is uncorrelated with $v(t)$\\collect $E[e_x(t)e_x^T(t)]$ and $E[v(t)v^T(t)]$}\\
&=\big(I-K(t)C\big)E[e_x(t)e_x^T(t)]\big(I-K(t)C\big)^T+K(t)E[v(t)v^T(t)]K(t)\\
&=\big(I-K(t)C\big)P(t|t-1)\big(I-K(t)C\big)^T+K(t)R_vK^T(t)\\
&\begin{split}
=P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\\
+K(t)CP(t|t-1)C^TK^T(t)+K(t)R_vK^T(t)
\end{split}
\Arrow[i]{in the 4th and 5th terms,\\factor out $K(t)$}\\
&\begin{split}
=P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\\
+K(t)\big(CP(t|t-1)C^T+R_v\big)K^T(t)
\end{split}
\Arrow[i]{\eqref{eqn_kalman_filter_gain}}\\
&\begin{split}=P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\\
+\underbrace{P(t|t-1)C^T\big(CP(t|t-1)C^T+R_v\big)^{-1}}_{=K(t)}\big(CP(t|t-1)C^T+R_v\big)K^T(t)\end{split}\\
&=P(t|t-1)\cancel{-P(t|t-1)C^T K^T(t)}-K(t)CP(t|t-1)\cancel{+P(t|t-1)C^TK^T(t)}\\
&=P(t|t-1)-K(t)CP(t|t-1)\Arrow[i]{factor out $P(t|t-1)$}\\
P(t|t)&=\big(I-K(t)C\big)P(t|t-1)
\end{DispWithArrows*}
some other words

\end{document}


Problem 1: The text displayed with the package witharrows and with the command \Arrow[i]{...} falls off the right edge of the sheet, while the equation doesn't start immediately at left but in the center-left of the sheet, so it doesn't exploit all the available space.

Problem 2: I tried to use the split environment but it doesn't work very well because some equations don't start aligned with the others (I think it's because I don't use it properly) and because its efforts are useless due to problem 1.

If my description is not clear, the output of the MWE will be clearer:

How can I solve these problems?

• Use the options fleqn and mathindent=0pt, i.e., start the equation with \begin{DispWithArrows*}[fleqn,mathindent=0pt]. This typesets the equation flush left . Aug 2 at 17:45

Solution to problem 1: Start the equation with \begin{DispWithArrows*}[fleqn,mathindent=0pt]. This typesets the equation flushed left. See the documentation of the package witharrows for an explanation.

Solution to problem 2: Precede the lines of the split environment with &. The first column of a split environment is right-aligned, the second one left aligned. Moreover, add \quad in lines that do not start with a relation symbol.

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{geometry} \geometry{a4paper,top=2.5cm,bottom=2.5cm,left=2cm,right=2cm,heightrounded,bindingoffset=5mm}

\usepackage{amsmath,cancel,witharrows}

\begin{document}

\noindent
some words
\begin{DispWithArrows*}[fleqn,mathindent=0pt]
P(t|t)&=E\left[\tilde{e}_x(t)\tilde{e}_x^T(t)\right]\\
&=E\left[\big((I-K(t)C)e_x(t)-K(t)v(t)\big)\big((I-K(t)C)e_x(t)-K(t)v(t)\big)^T\right] \Arrow[i]{$e_x$ is uncorrelated with $v(t)$\\collect $E[e_x(t)e_x^T(t)]$ and $E[v(t)v^T(t)]$}\\
&=\big(I-K(t)C\big)E[e_x(t)e_x^T(t)]\big(I-K(t)C\big)^T+K(t)E[v(t)v^T(t)]K(t)\\
&=\big(I-K(t)C\big)P(t|t-1)\big(I-K(t)C\big)^T+K(t)R_vK^T(t)\\
&\begin{split}
&=P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\\
\end{split}
\Arrow[i]{in the 4th and 5th terms,\\factor out $K(t)$}\\
&\begin{split}
&=P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\\
\end{split}
\Arrow[i]{\eqref{eqn_kalman_filter_gain}}\\
&\begin{split}
&=P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\\
&=P(t|t-1)\cancel{-P(t|t-1)C^T K^T(t)}-K(t)CP(t|t-1)\cancel{+P(t|t-1)C^TK^T(t)}\\
&=P(t|t-1)-K(t)CP(t|t-1)\Arrow[i]{factor out $P(t|t-1)$}\\
P(t|t)&=\big(I-K(t)C\big)P(t|t-1)
\end{DispWithArrows*}
some other words

\end{document}

• Very neat solution, thank you! Aug 4 at 8:18
• ahahah amazing bio XD Aug 4 at 9:59

Ok, here is my analysis and suggestion for solution.

First I introduced some abbreviations via the \def macro to better see your structure. Already with adaptions described below it reads:

% --- formulas ----------------------------------------------------
\def\ZA{E\left[\tilde{e}_x(t)\tilde{e}_x^T(t)\right]}
\def\ZB{E\left[\big((I-K(t)C)e_x(t)-K(t)v(t)\big)\big((I-K(t)C)e_x(t)-K(t)v(t)\big)^T\right]}
\def\ZCa{\big(I-K(t)C\big)E[e_x(t)e_x^T(t)]\big(I-K(t)C\big)^T+K(t)E[v(t)v^T(t)]K(t)}
\def\ZCb{\big(I-K(t)C\big)P(t|t-1)\big(I-K(t)C\big)^T+K(t)R_vK^T(t)}
\def\ZD{P(t|t-1)\cancel{-P(t|t-1)C^T K^T(t)}-K(t)CP(t|t-1)\cancel{+P(t|t-1)C^TK^T(t)}}
\def\AB{P(t|t-1)-K(t)CP(t|t-1)\Arrow[i]{factor out $P(t|t-1)$}}
\def\AA{P(t|t)&=\big(I-K(t)C\big)P(t|t-1)}
% --- split groups -------------------------------------------------------
\def\SG{P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\\
+K(t)CP(t|t-1)C^TK^T(t)+K(t)R_vK^T(t)}
\def\SH{P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\\
+K(t)\big(CP(t|t-1)C^T+R_v\big)K^T(t)}
\def\SK{P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\\
+\underbrace{P(t|t-1)C^T\big(CP(t|t-1)C^T+R_v\big)^{-1}}_{=K(t)}\times \\
\times\big(CP(t|t-1)C^T+R_v\big)K^T(t)}
% --- arrow texts --------------------------------------------------------
\def\RA{$e_x$ is uncorrelated with\\ $v(t)$ collect\\ $E[e_x(t)e_x^T(t)]$ and\\ $E[v(t)v^T(t)]$}
\def\RB{in the 4th and 5th terms,\\factor out $K(t)$}
\def\RC{\eqref{eqn_kalman_filter_gain}}
% ------------------------------------


There is no particular naming. ZA..AB just are your formulas, SG, SH and SK are inside your split-environemnts, RA..RC are the arrow texts; all of them free from &, besides the \underbrace. With this your set of equations and hence your structure reads:

\begin{DispWithArrows*}[format = ll]
P(t|t)  &=\ZA\\
&=\ZB \Arrow[i]{\RA}\\% \RA is too long
&=\ZCa\\
&=\ZCb\\
&\begin{split}% line #5, continuing with '+'
=\SG
\end{split}
\Arrow[i]{\RB}\\
&\begin{split}
=\SH
\end{split}
\Arrow[i]{\RC}\\% unreferenced eqn, yields ??
&\begin{split}
=\SK
\end{split}\\
&=\ZD\\
&=\AB\\
\AA
\end{DispWithArrows*}


A solution to the first arrow text can be obtained by just entering some '\' linebreaks manually, see \def\RA.

For problem 2 I suggest to split the line with \times as indicated in \def\SK, so that the remaining centered part is less of a distraction, hopefully.

As you can see I also entered the [format = ll] statement at the beginning of the DispWithArrows* environment, to make sure alignement as intended. So %\noindent becomes obsolete ... (done)

Some new problems remain, like the touching arrows #2 and #3.

Result after 2 compile runs, at least:

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{geometry} \geometry{a4paper,top=2.5cm,bottom=2.5cm,left=2cm,right=2cm,heightrounded,bindingoffset=5mm}

\usepackage{amsmath,cancel,witharrows}

% --- formulas ----------------------------------------------------
\def\ZA{E\left[\tilde{e}_x(t)\tilde{e}_x^T(t)\right]}
\def\ZB{E\left[\big((I-K(t)C)e_x(t)-K(t)v(t)\big)\big((I-K(t)C)e_x(t)-K(t)v(t)\big)^T\right]}
\def\ZCa{\big(I-K(t)C\big)E[e_x(t)e_x^T(t)]\big(I-K(t)C\big)^T+K(t)E[v(t)v^T(t)]K(t)}
\def\ZCb{\big(I-K(t)C\big)P(t|t-1)\big(I-K(t)C\big)^T+K(t)R_vK^T(t)}
\def\ZD{P(t|t-1)\cancel{-P(t|t-1)C^T K^T(t)}-K(t)CP(t|t-1)\cancel{+P(t|t-1)C^TK^T(t)}}
\def\AB{P(t|t-1)-K(t)CP(t|t-1)\Arrow[i]{factor out $P(t|t-1)$}}
\def\AA{P(t|t)&=\big(I-K(t)C\big)P(t|t-1)}
% --- split groups -------------------------------------------------------
\def\SG{P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\\
+K(t)CP(t|t-1)C^TK^T(t)+K(t)R_vK^T(t)}
\def\SH{P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\\
+K(t)\big(CP(t|t-1)C^T+R_v\big)K^T(t)}
\def\SK{P(t|t-1)-P(t|t-1)C^TK^T(t)-K(t)CP(t|t-1)+\\
+\underbrace{P(t|t-1)C^T\big(CP(t|t-1)C^T+R_v\big)^{-1}}_{=K(t)}\times \\
\times\big(CP(t|t-1)C^T+R_v\big)K^T(t)}
% --- arrow texts --------------------------------------------------------
\def\RA{$e_x$ is uncorrelated with\\ $v(t)$ collect\\ $E[e_x(t)e_x^T(t)]$ and\\ $E[v(t)v^T(t)]$}
\def\RB{in the 4th and 5th terms,\\factor out $K(t)$}
\def\RC{\eqref{eqn_kalman_filter_gain}}
% ------------------------------------

\begin{document}
%\noindent
some words
\begin{DispWithArrows*}[format = ll]
P(t|t)  &=\ZA\\
&=\ZB \Arrow[i]{\RA}\\% \RA is too long
&=\ZCa\\
&=\ZCb\\
&\begin{split}% line #5, continuing with '+'
=\SG
\end{split}
\Arrow[i]{\RB}\\
&\begin{split}
=\SH
\end{split}
\Arrow[i]{\RC}\\% unreferenced eqn, yields ??
&\begin{split}
=\SK
\end{split}\\
&=\ZD\\
&=\AB\\
\AA
\end{DispWithArrows*}
some other words

\end{document}

• Thank you! Your modular way to write the equation is pretty cool, but it's not so feasible in my case. I don't understand the meaning of [format = ll] and "to make sure alignment as intended". Aug 4 at 8:29
• @Stefano, Fine :) // I refer to page 6 in mirror.marwan.ma/ctan/macros/generic/witharrows/witharrows.pdf , where using this format is explained. It's intended to work as in tables, where you could align individual columns left (l), center (c) or right (r). Sometimes this is a source of error, so I tried to exert some control ;-) // Turned out that the line I split via \times was causing the "wrong alignment". Aug 4 at 8:54
• @Stefano, about the \def parts, the purpose was more for myself to better see the structure of your table-like part. It might have been that some & got misplaced hidden inside your formulas ... which wasn't :) // If it's useful in a regular document may depend. However, I myself strive for a clear coding here, because it's so easy to do sth wrong AND being unable to find it. // Perhaps it's a good idea to define such substitutes for smaller entities, which occure often, like K(t)CP(t|t-1). Then you'd use the \newcommand-approach, not TeX as I did ;-) Aug 4 at 8:58
• Ok, thanks! I was misled by the double l :) As you said, the \def parts are very useful in this case but since I'm working on a document of 100+ pages with a lot of equations, it would take too long to create pieces of the equation and organize them in another .tex file. Moreover this a document containing the notes of an exam I have already given :))) There would be no need to change it, but I like to "tinker" with LaTeX. In the future, maybe for a project report or my master thesis, I will definitely use this kind of approach. Thank you again! Aug 4 at 9:40

I propose this code, not perfect, but using the fleqn option and setting the value of mathindent.

I added some simplification of the code, and replaced the | for conditional probabilities with \mid which has a better spacing. Also, the pairs \big( … \big) with \bigl(… \bigr) for the same reason. Last, I replaced the split environment, which produced warnings messages, with aligned.

\documentclass[fleqn]{article}
\usepackage[a4paper, vmargin=2.5cm, hmargin=2cm, heightrounded, bindingoffset=5mm, showframe]{geometry}%
\usepackage{amsmath, cancel, witharrows}

\begin{document}

\noindent
some words
\begin{DispWithArrows*}[fleqn, mathindent=1em, wrap-lines, show-nodes]
P(t\mid t)&=E\left[\tilde{e}_x(t)\tilde{e}_x^T(t)\right]\\
&=E\left[\bigl((I-K(t)C)e_x(t)-K(t)v(t)\bigr)\bigl((I-K(t)C)e_x(t)-K(t)v(t)\bigr)^T\right] \Arrow[i]{{$e_x$ is uncorrelated with $v(t)$\\ collect $E[e_x(t)e_x^T(t)]$ and $E[v(t)v^T(t)]$}}\\
&=\bigl(I-K(t)C\bigr)E[e_x(t)e_x^T(t)]\bigl(I-K(t)C\bigr)^T+K(t)E[v(t)v^T(t)]K(t)\\
&=\bigl(I-K(t)C\bigr)P(t\mid t-1)\bigl(I-K(t)C\bigr)^T+K(t)R_vK^T(t)\\
&=\begin{aligned}[t]P(t\mid t-1) & -P(t\mid t-1)C^TK^T(t)-K(t)CP(t\mid t-1)+{\,}\\
& +K(t)CP(t\mid t-1)C^TK^T(t)+K(t)R_vK^T(t)
\end{aligned}
\Arrow[i]{in the 4th and 5th terms,\\factor out $K(t)$}\\
&=\begin{aligned}[t]P(t\mid t-1) & -P(t\mid t-1)C^TK^T(t)-K(t)CP(t\mid t-1)+{}\\
& +K(t)\big(CP(t|t-1)C^T+R_v\big)K^T(t)
\end{aligned}
\Arrow[i]{\eqref{eqn_kalman_filter_gain}}\\
&=\begin{aligned}[t]P(t &\mid t-1) -P(t\mid t-1)C^TK^T(t)-K(t)CP(t\mid t-1)+{}\\
& +\underbrace{P(t\mid t-1)C^T\bigl(CP(t\mid t-1)C^T+R_v\bigr)^{-1}}_{=K(t)}\bigl(CP(t\mid t-1)C^T+R_v\bigr)K^T(t)
\end{aligned}\\
&=P(t\mid t-1)\cancel{-P(t\mid t-1)C^T K^T(t)}-K(t)CP(t\mid t-1)\cancel{+P(t\mid t-1)C^TK^T(t)}\\
&=P(t\mid t-1)-K(t)CP(t\mid t-1)\Arrow[i]{factor out $P(t\mid t-1)$}\\
P(t\mid t)&=\bigl(I-K(t)C\bigr)P(t\mid t-1)
\end{DispWithArrows*}
some other words

\end{document}


• Thank you Bernard! I will immediately change the | character with \mid, it looks much better. Up to now I wrote all my conditional probabilities like that (^_^;). Same stuff for the parentheses. The aligned environment is neat, but is it possible to align the 2nd row to the first? Aug 4 at 8:34
• @Stefano: What do you means exactly? All rows are aligned on the = sign. Could you explain more explicitly? Aug 4 at 9:01
• Yes, for example, look at the 6th line (right below the 5th equal sign). Is there a way to shift the line so that the plus sign is slightly to the right with respect to the equal sign? Aug 4 at 9:49
• I see. I had thought it looked fine to align the + sign with the – above (via the aligned environment. But instead, you may put the & in the line above just before the P at the very beginning of the line above. Aug 4 at 9:56
• Oops, right, I didn't notice the other & sign Aug 4 at 9:58