Yingzhou Li

Probability bound in factoring to order-finding

2022-06-09T00:00:00+00:00

Given a positive integer $N$ and an integer $x$ ($1\leq x < N$) co-prime to $N$. The order of $x$ modulo $N$ is the smallest positive integer $r$ such that $x^r \equiv 1 \pmod{N}$. Order-finding problem then aims to solve $r$ given $N$ and $x$. A probalistic result, Theorem A4.13, is given in Appendix 4.3 of (Nielsen and Chuang 2010)¹.

As mentioned by Yihan Guo, the bound in (Nielsen and Chuang 2010)¹ could be incorrect. After a detailed checking, I agree with Yihan and a modified version of Theorem A4.13 should be as follows.

Probability of Random Choosing an Even Order Number

Theorem Suppose $N = p_1^{\alpha_1} \cdots p_m^{\alpha_m}$ is the prime factorization of an odd composite positive integer. Let $x$ be chosen uniformly at random from $\mathbb{Z}_N^*$, and let $r$ be the order of $x$, modulo $N$. Then
\[\mathbb{P} \left(r \text{ is even and } x^{r/2} \not\equiv -1 \pmod{N} \right) \geq 1 - \frac{1}{2^{m-1}}.\]

In the above Theorem, $\mathbb{Z}_N^*$ is the set of all elements in $\mathbb{Z}_N$, which are co-prime to $N$. The size of $\mathbb{Z}_N^*$ is $\varphi(N)$, which is known as Euler $\varphi$ function. The proof of Theorem relies on the Chinese remainder theorem and Lemma A4.12 in (Nielsen and Chuang 2010)¹. We cite the statement of Lemma A4.12 in this post.

Lemma A4.12 Let $p$ be an odd prime. Let $2^D$ be the largest power of 2 dividing $\varphi(p^\alpha)$. Then with probability exactly one-half $2^D$ divides the order modulo $p^{\alpha}$ of a randomly chosen element of $\mathbb{Z}_{p^\alpha}^*$.

Proof of Theorem. We prove that $\begin{equation} \label{eq:ineq} \mathbb{P} \left(r \text{ is odd or } x^{r/2} \equiv -1 \pmod{N} \right) \leq \frac{1}{2^{m-1}}. \end{equation}$

Given an integer $x \in \mathbb{Z}_N^*$ with $r$ being its order modulo $N$. Since $p_i^{\alpha_i}$ is a factor of $N$, we have,

\[\begin{equation*} x^r \equiv 1 \pmod{N} \Longrightarrow x^r \equiv 1 \pmod{p_i^{\alpha_i}} \quad i = 1, 2, \dots, m. \end{equation*}\]

We define ${x_i}_{i=1}^m$ as, $\begin{equation} \label{eq:xieqx} x_i \equiv x \pmod{p_i^{\alpha_i}} \quad i = 1, 2, \dots, m, \end{equation}$ with $r_i$ being their orders modulo $p_i^{\alpha_i}$ respectively, i.e.,

\[\begin{equation*} x_i^{r_i} \equiv 1 \pmod{p_i^{\alpha_i}}. \end{equation*}\]

We take to $r$-th power of \eqref{eq:xieqx} and obtain, $\begin{equation} \label{eq:mod1} x_i^r \equiv x^r \equiv 1 \pmod{p_i^{\alpha_i}}. \end{equation}$ Hence we conclude that $r_i$ divides $r$, i.e., $r_i \mid r$.

We further denote $d$ and $d_i$ be the largest power of 2 divide $r$ and $r_i$, respectively.

Now we consider the cases in the probability statement \eqref{eq:ineq}: 1) $r$ is odd; and 2) $r$ is even and $x^{r/2} \equiv -1 \pmod{N}$.

When $r$ is odd, we have that all $r_i$ are odd. Hence we have $d = d_1 = \cdots = d_m = 0$.

When $r$ is even and $x^{r/2} \equiv -1 \pmod{N}$, we have similar modulo congruence formula, $\begin{equation} \label{eq:modn1} x_i^{r/2} \equiv x^{r/2} \equiv -1 \pmod{p_i^{\alpha_i}}. \end{equation}$ Comparing \eqref{eq:mod1} and \eqref{eq:mod1}, we know that $r_i \nmid \frac{r}{2}$. Together with $r_i \mid r$, we know that $r$ and $r_i$ have the same power of 2 factor, i.e., $d_i = d$ for $i=1, 2, \dots, m$.

Now we are ready to bound the probability \eqref{eq:ineq}. By Chinese remainder theorem, uniform sampling from $\mathbb{Z}_N^*$ is equivalent to uniform sampling from $\mathbb{Z}_{p_i^{\alpha_i}}^*$ independetly. Further we denote the the largest power of 2 dividing $\varphi(p_i^{\alpha_i})$ as $D_i$. By Lemma A4.12, we know that

\[\begin{equation*} \mathbb{P}\left(r_i = 2^{D_i}\{odd\}\right) = \frac{1}{2}. \end{equation*}\]

The above analysis on $r$ shows that the statement in the probability must obeys that $d = d_1 = \cdots = d_m$. Hence we could rewrite \eqref{eq:ineq} as,

\[\begin{equation*} \begin{split} & \mathbb{P} \left(r \text{ is odd or } x^{r/2} \equiv -1 \pmod{N} \right) \\ \leq & \sum_{d = 0}^{\min_i{D_i}} \prod_{i=1}^m \mathbb{P} \left(r_i = 2^{d}\{odd\}\right) \\ \leq & \sum_{d = 0}^{\min_i{D_i}-1} \prod_{i=1}^m \mathbb{P} \left(r_i = 2^{d}\{odd\} \right) + \frac{1}{2^m}\\ \leq & \prod_{i=1}^m \sum_{d = 0}^{\min_i{D_i}-1} \mathbb{P} \left(r_i = 2^{d}\{odd\} \right) + \frac{1}{2^m} \\ \leq & \prod_{i=1}^m (1-\frac{1}{2}) + \frac{1}{2^m} \\ = & \frac{1}{2^{m-1}}. \end{split} \end{equation*}\]

Hence we proved Theorem.

Revisit the Bound

Through the above proof, we find that the original bound $1-\frac{1}{2^m}$ as in Theorem A4.13 in (Nielsen and Chuang 2010)¹ could still hold if we change the assumption in the statement. Hence we give the following Corollar.

Corollary Suppose $N = p_1^{\alpha_1} \cdots p_m^{\alpha_m}$ is the prime factorization of an odd composite positive integer, and $D_i$ is the largest power of 2 dividing $\varphi(p_i^{\alpha_i})$. Assume $D_i$ are not all equal to each other. Let $x$ be chosen uniformly at random from $\mathbb{Z}_N^*$, and let $r$ be the order of $x$, modulo $N$. Then
\[\mathbb{P} \left(r \text{ is even and } x^{r/2} \not\equiv -1 \pmod{N}\right) \geq 1 - \frac{1}{2^m}.\]

Proof of Corollary.

All other parts of the proof remain the same as that of Theorem. We would rederive the bound on the probability. Without loss of generality, we assume $D_1$ is greater than the minimum of $D_i$, i.e., $D_1 > \min_i D_i$. By Lemma A4.12, we have

\[\sum_{d=0}^{\min_i D_i} \mathbb{P} \left(r_1 = 2^d\{odd\} \right) \leq \frac{1}{2}.\]

Then we could rewrite \eqref{eq:ineq} as,

\[\begin{equation*} \begin{split} & \mathbb{P}\left(r \text{ is odd or } x^{r/2} \equiv -1 \pmod{N} \right) \\ \leq & \sum_{d = 0}^{\min_i{D_i}} \prod_{i=1}^m \mathbb{P} \left(r_i = 2^{d}\{odd\} \right) \\ \leq & \left( \sum_{d = 0}^{\min_i{D_i}} \mathbb{P}\left( r_1 = 2^{d}\{odd\} \right) \right) \prod_{i=2}^m \frac{1}{2}\\ \leq & \frac{1}{2^m}. \end{split} \end{equation*}\]

Hence we proved Corollary.

Example 1 ($N=3$)

Through the proof of Theorem, we find that the Theorem should also hold for odd prime numbers. We take $N = 3$ as an example (as provided by Yihan Guo).

Number	Order $r$	$x^{r/2}$ mod $N$	Check
1	1		:x:
2	2	-1	:x:

The probability is 0 and the bound is also 0, i.e.,

\[0 = \mathbb{P} \left(r \text{ is even and } x^{r/2} \not\equiv -1 \pmod{3} \right) \geq 1 - \frac{1}{2^{1-1}} = 0.\]

Example 2 ($N=15$)

We take $N = 15$ as another example.

Number	Order $r$	$x^{r/2}$ mod $N$	Check
1	1		:x:
2	4	4	:heavy_check_mark:
4	2	4	:heavy_check_mark:
7	4	4	:heavy_check_mark:
8	4	4	:heavy_check_mark:
11	2	11	:heavy_check_mark:
13	4	4	:heavy_check_mark:
14	2	-1	:x:

The probability is $\frac{6}{8}$ and the bound is $\frac{1}{2}$, i.e.,

\[\frac{3}{4} = \mathbb{P} \left(r \text{ is even and } x^{r/2} \not\equiv -1 \pmod{3} \right) \geq 1 - \frac{1}{2^{2-1}} = \frac{1}{2}.\]

In this case, the original Theorem A4.13 in (Nielsen and Chuang 2010)¹ also holds. By the argument around Corollary, we have $D_1 = 1$ and $D_2 = 2$, which are not equal to each other. Hence Corollary holds and so is the bound in Theorem A4.13.

Example 3 ($N=63$)

We take $N = 63 = 3^2 \cdot 7$ as the last example.

Number	Order $r$	$x^{r/2}$ mod $N$	Check
1	1		:x:
2	6	8	:heavy_check_mark:
4	3		:x:
5	6	-1	:x:
8	2	8	:heavy_check_mark:
10	6	55	:heavy_check_mark:
11	6	8	:heavy_check_mark:
13	6	55	:heavy_check_mark:
16	3		:x:
17	6	-1	:x:
19	6	55	:heavy_check_mark:
20	6	-1	:x:
22	3		:x:
23	6	8	:heavy_check_mark:
25	3		:x:
26	6	-1	:x:
29	6	8	:heavy_check_mark:
31	6	55	:heavy_check_mark:
32	6	8	:heavy_check_mark:
34	6	55	:heavy_check_mark:
37	3		:x:
38	6	-1	:x:
40	6	55	:heavy_check_mark:
41	6	-1	:x:
43	3		:x:
44	6	8	:heavy_check_mark:
46	3		:x:
47	6	-1	:x:
50	6	8	:heavy_check_mark:
52	6	55	:heavy_check_mark:
53	6	8	:heavy_check_mark:
55	2	55	:heavy_check_mark:
58	3		:x:
59	6	-1	:x:
61	6	55	:heavy_check_mark:
62	2	-1	:x:

The probability is $\frac{18}{36}$ and the bound is $\frac{1}{2}$, i.e.,

\[\frac{1}{2} = \mathbb{P} \left(r \text{ is even and } x^{r/2} \not\equiv -1 \pmod{3} \right) \geq 1 - \frac{1}{2^{2-1}} = \frac{1}{2}.\]

In this case, the original Theorem A4.13 in (Nielsen and Chuang 2010)¹ does not hold.

Yingzhou Li

Reference

Nielsen, M. A., & Chuang, I. L. (2010). Quantum computation and quantum information (10th Anniversary Edition ed.). Cambridge University Press. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶

Fudan Library Access Button

2021-01-15T00:00:00+00:00

The content in this post is inspired by Duke library access button as in link. I wrote a bookmarklet to access Fudan library resources via Fudan WebVPN. Although the final functionality is nearly identical to that of Duke, the implementation is different.

We will not discuss the actual implementation in this post. Instead, we simply give the installation and usage instructions.

javascript: (function() {
    a = document.createElement('script');
    a.src = 'https://webvpn.fudan.edu.cn/wengine-vpn/js/aes-js.js';
    a.async = false;
    document.body.appendChild(a);
    e = document.createElement('script');
    e.src = 'https://webvpn.fudan.edu.cn/wengine-vpn/js/js/portal.js';
    e.async = false;
    e.onload = function() {
        wrdvpnIV = 'wrdvpnisthebest!';
        wrdvpnKey = 'wrdvpnisthebest!';
        location.href = 'https://webvpn.fudan.edu.cn' + encrypUrl(parseProtocol(location.href), location.href);
    };
    document.body.appendChild(e);
})();

After installation of the access button, you will have a button on your browser’s bookmark toolbar. You can gain access to web content that requires Fudan VPN service through one-click of the button (UIS login is still needed, at least for the first time).

Installation Instruction

Mozilla Firefox

Fudan Library Access Button

Right click on the above link to add the button to your Bookmarks bar. From the dropdown, select Bookmark this Link and choose your folder.

Google Chrome, Edge or Safari

Fudan Library Access Button

Drag the above link to your Bookmarks Toolbar.

Internet Explorer

Fudan Library Access Button

Right click on the above link to add the button to your Bookmarks bar. From the dropdown, select Add to Favorites.
Probably, there will be a safety warning. If you are not confident, you can read through the detail of the link below.

Mobile Users

javascript:(function(){a=document.createElement('script');a.src='https://webvpn.fudan.edu.cn/wengine-vpn/js/aes-js.js';a.async=false;document.body.appendChild(a);e=document.createElement('script');e.src='https://webvpn.fudan.edu.cn/wengine-vpn/js/js/portal.js';e.async=false;e.onload=function(){wrdvpnIV='wrdvpnisthebest!';wrdvpnKey='wrdvpnisthebest!';location.href='https://webvpn.fudan.edu.cn'+encrypUrl(parseProtocol(location.href),location.href);};document.body.appendChild(e);})();

Copy the single line text above.
Open the browser and create a new bookmark of any page. Then tap Edit and replace the name as Fudan Library Access Button(you can change the name as you like) and replace the bookmark link by the copied text.

Usage Instruction

Our usage instruction will be demonstrated through an example.

Go to a Fudan library resource link, for example, link. If you are not connected to Fudan VPN or other library VPNs, you should not have access to the PDF file.
After the webpage is fully loaded, click the button. You will be navigated to Fudan WebVPN homepage and asked to input your UIS credentials.
Login your UIS account.
You will be redirected back to the link you have opened. Try to open the PDF file.

If you encounter any issue, please leave a message below.

Known Issue

Recently, Fudan WebVPN cannot redirect to the link after login. Temporary alternative solution is to login WebVPN first and then use the button.

Yingzhou Li

Von Neumann's Trace Inequalities

2020-07-18T00:00:00+00:00

Trace inequality is important for matrix analysis. In this post we first discuss Von Neumann’s trace inequality and another related one, which can be viewed as a slightly more detailed version of the trace inequality Wiki page. Then we prove a related trace inequality for general Hermitian matrices.

In 1937, Von Neumann proved the following trace inequality ¹, where the trace of the product of two general matrices is upper bounded by the sum of product of their singular values in order.

Theorem (Von Neumann’s trace inequality). If $A, B$ are complex $n \times n$ matrices with singular values,
\[\alpha_1 \geq \cdots \geq \alpha_n, \quad \beta_1 \geq \cdots \geq \beta_n,\]
respectively, then
\[\lvert \mathrm{tr}(AB) \rvert \leq \sum_{i=1}^n \alpha_i \beta_i.\]

Later in 1970, Ruhe proved a related trace inequality for the product of positive semidefinite matrices ². Importantly, both the upper and the lower bounds are provided. The original statement is for $A^* A$ and $B^* B$ for general complex matrices $A \in \mathbb{C}^{m\times n}$ and $B \in \mathbb{C}^{n\times p}$. Here we restate the lemma with positive semidefinite Hermitian matrices.

Lemma 1 (Ruhe’s trace inequality). If $A, B$ are $n \times n$ positive semidefinite Hermitian matrices with eigenvalues,
\[a_1 \geq \cdots \geq a_n \geq 0, \quad b_1 \geq \cdots \geq b_n \geq 0,\]
respectively, then
\[\sum_{i=1}^n a_i b_{n-i+1} \leq \mathrm{tr}(AB) \leq \sum_{i=1}^n a_i b_i.\]

Generalized version of both Theorem and Lemma 1 are proved by Marshall et al. ³. In the following, we use $\lambda_i(\cdot)$ and $\sigma_i(\cdot)$ to denote the $i$-th eigenvalue and singular value of a matrix. Both are of descending order.

Lemma 2 If $A, B$ are complex $n \times n$ matrices, then for any $k = 1, \dots, n$,
\[\sum_{i=1}^k \sigma_i(AB) \leq \sum_{i=1}^k \sigma_i(A) \sigma_i(B).\]

Lemma 3 If $A, B$ are $n \times n$ positive semidefinite Hermitian matrices, then for any $k = 1, \dots, n$,
\[\sum_{i=1}^k \lambda_i(A) \lambda_{n-i+1}(B) \leq \sum_{i=1}^k \lambda_i(AB)\]

Although Lemma 3 only generalizes lower bound part of Lemma 1, the other inequality also holds under the same generalization.

Lemma 1 for general Hermitian matrices

In (Marshall et al. 2011) ³, it is mentioned the upper bound part of Lemma 1 can be generalized to general Hermitian matrices. Here we rigorously prove that the positive semidefinite assumption can be removed and both inequalities in Lemma 1 still hold.

Lemma 4 (Generalized Ruhe’s trace inequality). If $A, B$ are $n \times n$ Hermitian matrices, then
\[\sum_{i=1}^n \lambda_i(A) \lambda_{n-i+1}(B) \leq \mathrm{tr}(AB) \leq \sum_{i=1}^n \lambda_i(A) \lambda_i(B).\]

Proof. Since Hermitian matrices have unitary eigenvectors, we denote the eigen-decomposition of $A$ as $A = Q \Lambda Q^\star$ where $Q$ is a unitary matrix. The trace of $AB$ is then, $\begin{equation} \mathrm{tr}(AB) = \mathrm{tr}(Q \Lambda Q^\star B) = \mathrm{tr}(\Lambda Q^\star BQ), \end{equation}$ where $Q^\star BQ$ and $B$ share the same eigenvalues.

Therefore, without loss of generality, we can assume that $A$ is a diagonal matrix with diagonal entries being eigenvalues, $\lambda_1(A), \dots, \lambda_n(A)$. Then we have,

\[\begin{equation} \label{eq:trace} \mathrm{tr}(AB) = \sum_{i=1}^n \lambda_i(A) B_{ii}. \end{equation}\]

We use induction over $n$ to prove the lemma. For $n=1$, both inequalities become equality and hold. Now we assume that both inequalities are true for all $(n-1) \times (n-1)$ matrices.

We reorganize the trace \eqref{eq:trace} as follows,

\[\begin{split} \mathrm{tr}(AB) = & \sum_{i=1}^{n-1} (\lambda_i(A) - \lambda_n(A)) B_{ii} + \sum_{i=1}^n \lambda_n(A) B_{ii}\\ = & \mathrm{tr}(\tilde{\Lambda} \tilde{B}) + \lambda_n(A) \mathrm{tr}(B)\\ \end{split} ,\]

where $\tilde{\Lambda}$ is an $(n-1)\times (n-1)$ diagonal matrix with diagonal entries being $\lambda_i(A) - \lambda_n(A)$, and $\tilde{B}$ is the first $(n-1) \times (n-1)$ principal submatrix of $B$.

By Cauchy Interlacing Theorem, we have the following relations for eigenvalues of $B$ and $\tilde{B}$,

\[\begin{equation} \label{eq:interlacing} \lambda_{i+1}(B) \leq \lambda_i(\tilde{B}) \leq \lambda_i(B). \end{equation}\]

Applying the induction hypothesis on $\mathrm{tr}(\tilde{\Lambda} \tilde{B})$, we first show the upper bound part,

\[\begin{split} \mathrm{tr}(AB) \leq & \sum_{i=1}^{n-1} (\lambda_i(A) - \lambda_n(A)) \lambda_i(\tilde{B}) + \lambda_n(A) \sum_{i=1}^n \lambda_i(B)\\ \leq & \sum_{i=1}^{n-1} (\lambda_i(A) - \lambda_n(A)) \lambda_i(B) + \lambda_n(A) \sum_{i=1}^n \lambda_i(B)\\ = & \sum_{i=1}^n \lambda_i(A) \lambda_i(B) \end{split} ,\]

where the second inequality adopts positivity of $\lambda_i(A) - \lambda_n(A)$ and the interlacing theorem \eqref{eq:interlacing}.

Similarly, we can show the lower bound part as,

\[\begin{split} \mathrm{tr}(AB) \geq & \sum_{i=1}^{n-1} (\lambda_i(A) - \lambda_n(A)) \lambda_{n-i}(\tilde{B}) + \lambda_n(A) \sum_{i=1}^{n} \lambda_{n-i+1}(B)\\ \geq & \sum_{i=1}^{n-1} (\lambda_i(A) - \lambda_n(A)) \lambda_{n-i+1}(B) + \lambda_n(A) \sum_{i=1}^{n} \lambda_{n-i+1}(B)\\ = & \sum_{i=1}^n \lambda_i(A) \lambda_{n-i+1}(B) \end{split} .\]

Hence we proved the lemma. $\tag*{$\blacksquare$}$

The conditions for the equalities in Lemma 4 are very restrictive. First, $A$ and $B$ must commute so that they have the same eigenvectors and the trace of product becomes the sum of products of eigenvalues. In addition to the commutative property, the ordering of eigenvalues also matters. When the eigenvalues of $A$ are decreasingly ordered, the corresponding eigenvalues of $B$ must be also of decreasing order to satisfy the equality of the upper bound part, and of increasing order to satisfy the equality of the lower bound part.

Yingzhou Li

Reference

Von Neumann, J., Some matrix-inequalities and metrization of matrix-space, Tomsk Univ. Rev. 1, 286-300 (1937). Reprinted in Collected Works (Pergamon Press, 1962), iv, 205-219. ↩
Ruhe, A., Perturbation bounds for means of eigenvalues and invariant subspaces, Nordisk Tidskr. Informationsbehandling (BIT) 10, 343-354 (1970). ↩
Marshall, A. W.; Olkin, I.; Arnold, B., Inequalities: Theory of Majorization and Its Applications (2nd ed.). New York: Springer. p. 340-341 (2011). ↩ ↩²

Setup mex with MSVC on Windows

2020-06-30T00:00:00+00:00

Recently, while I was trying to support metismex on Windows, the mex setup on Windows confused me for a very long time when my MSVC was installed through BuildTools. The direct setup in Matlab for mex, i.e., mex -setup C fails to locate compiler path. It turned out that the Matlab search path only include ‘Community’, ‘Professional’, and ‘Enterprise’ versions, while the ‘BuildTools’ version is left out. However, ‘BuildTools’ version is much lighter in disk cost than other versions and is more preferred if IDE is not needed.

When I ran mex -setup, the following error message appeared,

   >> mex -v -setup
   Verbose mode is on.

   %
   % many lines are removed
   %

   ... Looking for compiler 'Microsoft Visual C++ 2019 (C)' ...
   ... Looking for environment variable 'ProgramFiles(x86)' ...Yes ('C:\Program Files (x86)').
   ... Looking for file 'C:\Program Files (x86)\Microsoft Visual Studio\Installer\vswhere.exe' ...Yes.
   ... Executing command '"C:\Program Files (x86)\Microsoft Visual Studio\Installer\vswhere.exe" -version "[16.0,17.0)" -products Microsoft.VisualStudio.Product.Enterprise -property installationPath -format value' ...No.
   ... Looking for environment variable 'ProgramFiles(x86)' ...Yes ('C:\Program Files (x86)').
   ... Looking for file 'C:\Program Files (x86)\Microsoft Visual Studio\Installer\vswhere.exe' ...Yes.
   ... Executing command '"C:\Program Files (x86)\Microsoft Visual Studio\Installer\vswhere.exe" -version "[16.0,17.0)" -products Microsoft.VisualStudio.Product.Professional -property installationPath -format value' ...No.
   ... Looking for environment variable 'ProgramFiles(x86)' ...Yes ('C:\Program Files (x86)').
   ... Looking for file 'C:\Program Files (x86)\Microsoft Visual Studio\Installer\vswhere.exe' ...Yes.
   ... Executing command '"C:\Program Files (x86)\Microsoft Visual Studio\Installer\vswhere.exe" -version "[16.0,17.0)" -products Microsoft.VisualStudio.Product.Community -property installationPath -format value' ...No.
   Did not find installed compiler 'Microsoft Visual C++ 2019 (C)'.
   Error using mex
   No supported compiler or SDK was found. For options, visit http://www.mathworks.com/support/compilers/R2016a/maci64.html.

To be more precise, my MATLAB version is R2020a and MSVC version is 2019. Although MATLAB with versions earlier than R2019b does not support Visual Studio 2019 version 16, adding the following xml files to folder MATLABROOT/bin/win64/mexopts/ could resolve the issue. If any reader succeed, please let me know.

Here, we list a few steps in the modification.

Change the editing permission of the following two xml files.

MATLABROOT/bin/win64/mexopts/msvc2019.xml
MATLABROOT/bin/win64/mexopts/msvcpp2019.xml

Edit both xml files in the following way.

Find the block of lines as

<and>
    <envVarExists name="ProgramFiles(x86)" />
    <fileExists name="$$\Microsoft Visual Studio\Installer\vswhere.exe" />
    <cmdReturns name="&quot;$$\\vswhere.exe&quot; -version &quot;[16.0,17.0)&quot; -products Microsoft.VisualStudio.Product.Community -property installationPath -format value" />
    <cmdReturns name="set &quot;vcroot=$$&quot;&amp;for /f &quot;delims= &quot; %a in ('type &quot;$$\VC\Auxiliary\Build\Microsoft.VCToolsVersion.default.txt&quot;') do @if exist &quot;$$\VC\Tools\MSVC\%a\bin\HostX64\x64\cl.exe&quot; call echo %vcroot%" />
</and>

and attach the following block right below it

<and>
    <envVarExists name="ProgramFiles(x86)" />
    <fileExists name="$$\Microsoft Visual Studio\Installer\vswhere.exe" />
    <cmdReturns name="&quot;$$\\vswhere.exe&quot; -version &quot;[16.0,17.0)&quot; -products Microsoft.VisualStudio.Product.BuildTools -property installationPath -format value" />
    <cmdReturns name="set &quot;vcroot=$$&quot;&amp;for /f &quot;delims= &quot; %a in ('type &quot;$$\VC\Auxiliary\Build\Microsoft.VCToolsVersion.default.txt&quot;') do @if exist &quot;$$\VC\Tools\MSVC\%a\bin\HostX64\x64\cl.exe&quot; call echo %vcroot%" />
</and>

Find the block of lines as

<and>
    <envVarExists name="ProgramFiles(x86)" />
    <fileExists name="$$\Microsoft Visual Studio\Installer\vswhere.exe" />
    <cmdReturns name="&quot;$$\\vswhere.exe&quot; -version &quot;[16.0,17.0)&quot; -products Microsoft.VisualStudio.Product.Community -property installationPath -format value" />
</and>

and attach the following block right below it

<and>
    <envVarExists name="ProgramFiles(x86)" />
    <fileExists name="$$\Microsoft Visual Studio\Installer\vswhere.exe" />
    <cmdReturns name="&quot;$$\\vswhere.exe&quot; -version &quot;[16.0,17.0)&quot; -products Microsoft.VisualStudio.Product.BuildTools -property installationPath -format value" />
</and>

Find the block of lines as

<and>
    <envVarExists name="ProgramFiles(x86)" />
    <fileExists name="$$\Microsoft Visual Studio\Installer\vswhere.exe" />
    <cmdReturns name="&quot;$$\\vswhere.exe&quot; -version &quot;[16.0,17.0)&quot; -products Microsoft.VisualStudio.Product.Community -property installationPath -format value" />
    <cmdReturns name="set &quot;vcroot=$$&quot;&amp;for /f &quot;delims= &quot; %a in ('type &quot;$$\VC\Auxiliary\Build\Microsoft.VCToolsVersion.default.txt&quot;') do @if exist &quot;$$\VC\Tools\MSVC\%a\bin\HostX64\x64\cl.exe&quot; call echo %vcroot%\VC\Tools\MSVC\%a" />
</and>

and attach the following block right below it

<and>
    <envVarExists name="ProgramFiles(x86)" />
    <fileExists name="$$\Microsoft Visual Studio\Installer\vswhere.exe" />
    <cmdReturns name="&quot;$$\\vswhere.exe&quot; -version &quot;[16.0,17.0)&quot; -products Microsoft.VisualStudio.Product.BuildTools -property installationPath -format value" />
    <cmdReturns name="set &quot;vcroot=$$&quot;&amp;for /f &quot;delims= &quot; %a in ('type &quot;$$\VC\Auxiliary\Build\Microsoft.VCToolsVersion.default.txt&quot;') do @if exist &quot;$$\VC\Tools\MSVC\%a\bin\HostX64\x64\cl.exe&quot; call echo %vcroot%\VC\Tools\MSVC\%a" />
</and>

Find the block of lines as

<and>
    <envVarExists name="ProgramFiles(x86)" />
    <fileExists name="$$\Microsoft Visual Studio\Installer\vswhere.exe" />
    <cmdReturns name="&quot;$$\\vswhere.exe&quot; -version &quot;[16.0,17.0)&quot; -products Microsoft.VisualStudio.Product.Community -property installationPath -format value" />
    <fileExists name="$$\VC\Auxiliary\Build\vcvarsall.bat" />
    <dirExists name="$$"/>
</and>

and attach the following block right below it

<and>
    <envVarExists name="ProgramFiles(x86)" />
    <fileExists name="$$\Microsoft Visual Studio\Installer\vswhere.exe" />
    <cmdReturns name="&quot;$$\\vswhere.exe&quot; -version &quot;[16.0,17.0)&quot; -products Microsoft.VisualStudio.Product.BuildTools -property installationPath -format value" />
    <fileExists name="$$\VC\Auxiliary\Build\vcvarsall.bat" />
    <dirExists name="$$"/>
</and>

For convenience, I list two updated xml files for download: msvc2019.xml and msvcpp2019.xml.

The above updates should be valid across different Windows systems and most MATLAB versions. If there is major difference, and you find it impossible to update accordingly, please comment below or send me an Email.

Yingzhou Li

Reducible Singular Value Decomposition

2018-07-04T00:00:00+00:00

While reading the following review article on density-matrix renormalization group (DMRG), I encountered a linear algebra excise based on the last paragraph of page 125.

U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96-192 (2011). [journal]

The paper concludes the result of the excise in one sentence, “they are reduced basis transformations to orthonormal subspaces, hence the largest singular value must be less or equal to that of $S$”. However, I found that the proof is beyond such a single sentence. Therefore, it worth to post the excise together with the solution here.

Preliminary

We first briefly introduce the singular value decomposition (SVD) and another matrix decomposition called reducible SVD as preliminary for the excise.

Let $A \in \mathbb{R}^{n \times n}$ be a real $n$ by $n$ full rank matrix. The SVD of $A$ is $A = U\Sigma V^\top$, where $U \in \mathbb{R}^{n \times n}$ is a unitary matrix being the left singular vectors, $\Sigma \in \mathbb{R}^{n \times n}$ is a diagonal matrix with $A$’s singular values in decreasing order, and $V \in \mathbb{R}^{n \times n}$ is another unitary matrix being the right singular vectors. It is well-known that the SVD is unique up to a complex sign (if the singular vectors are restricted to real matrices, complex sign is simply $\pm$).

We now define another decomposition which has the same structure as the SVD,

\[A = P S Q^\top\]

where $P \in \mathbb{R}^{n \times m}$ satisfies $PP^\top = I$, $S \in \mathbb{R}^{m \times m}$ is a diagonal matrix with positive diagonal in decreasing order, $Q \in \mathbb{R}^{n \times m}$ satisfies $QQ^\top = I$, and $I$ denotes the identity matrix. This is very different from the usual case that $m$ denotes the numerical rank which is upper bounded by $n$. In our excise, we have $m \geq n$. In this post, we call such a decomposition reducible SVD.

If $m$ is strictly larger than $n$, through a trivial example,

\[1 = \begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} \cdot \begin{bmatrix} 1 & \\ & 1 \end{bmatrix} \cdot \begin{bmatrix} \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} \end{bmatrix} = \begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{bmatrix} \cdot \begin{bmatrix} 1.5 & \\ & 0.5 \end{bmatrix} \cdot \begin{bmatrix} \frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} \end{bmatrix},\]

we are convinced that such a decomposition is not unique.

Excise

With these definitions, the excise can be summarized as follows. Let $P S Q^\top$ be any reducible SVD of $A$. The following statement holds,

\[\max_i \sigma_i \leq \max_i s_{i},\]

where $\sigma_i$ denotes the $i$-th singular value of $A$ and $s_{i}$ is the $i$-th diagonal entry of $S$. The statement is equivalent to say that the largest singular value is smaller or equal to largest reducible singular value.

Solution

Now we will solve the excise. Since both SVD and reducible SVD are decomposition of $A$, we have,

\[A = U \Sigma V^\top = P S Q^\top \Leftrightarrow \Sigma = U^\top P S Q^\top V = G S H^\top,\]

where $G = U^\top P$ satisfies $GG^\top = I$, and $H = V^\top Q$ satisfies $HH^\top = I$. In terms of $G,S$, and $H$, the largest singular value of $A$ can be written as,

\[\sigma_1 = \sum_i G_{1,i} s_i H_{1,i},\]

where $G_{1,i}$ denotes the $(1,i)$-th entry of $G$, and $H_{1,i}$ is defined analogously. The conditions on $G$ and $H$ led to,

\[\sum_i G_{1,i}^2 = 1 \quad \text{and} \quad \sum_i H_{1,i}^2 = 1.\]

Then we have,

\[\sigma_1^2 = \left( \sum_i G_{1,i} s_i H_{1,i} \right)^2 \leq \left( \sum_i G_{1,i}^2 \right) \cdot \left( \sum_i s_i^2 H_{1,i}^2 \right) \leq \left( \sum_i H_{1,i}^2 \right) \cdot \max_i s_i^2 = s_1^2,\]

where the first inequality is due to Cauchy-Schwartz inequality and the second inequality is due to Hölder’s inequality with $p=1$ and $q=\infty$. Finally, since both $\sigma_1$ and $s_1$ are positive, we conclude $\max_i \sigma_i = \sigma_1 \leq s_1 = \max_i s_i$.

Yingzhou Li

Setup mex with GFORTRAN on Mac

2017-05-08T00:00:00+00:00

I was struggling with mex setup on Mac for a very long time since there was error with mex -setup C for Matlab on new MacOS. It turned out that the Matlab search path did not include the newest Xcode SDK “MacOSX10.12.sdk” in its search path. Once it was added, the problem with mex -setup C was resolved.

However, today, I encountered another mex issue in mex -setup FORTRAN. When I ran mex -setup -v FORTRAN, the following error message appeared,

   >> mex -setup -v FORTRAN
   Verbose mode is on.
   ... Looking for compiler 'gfortran' ...
   ... Looking for environment variable 'DEVELOPER_DIR' ...No.
   ... Executing command 'which gfortran' ...Yes ('/usr/local/bin/gfortran').
   ... Looking for folder '/usr/local/bin' ...Yes.
   ... Executing command 'which gfortran' ...Yes ('/usr/local/bin/gfortran').
   ... Executing command '/usr/local/bin/gfortran -print-file-name=libgfortran.dylib' ...Yes ('/usr/local/Cellar/gcc/6.3.0_1/lib/gcc/6/libgfortran.dylib').
   ... Looking for folder '/usr/local/Cellar/gcc/6.3.0_1/lib/gcc/6' ...Yes.
   ... Executing command 'which gfortran' ...Yes ('/usr/local/bin/gfortran').
   ... Executing command '/usr/local/bin/gfortran -print-file-name=libgfortranbegin.a' ...Yes ('libgfortranbegin.a').
   ... Looking for folder 'libgfortranbegin.a' ...No.
   Did not find installed compiler 'gfortran'.
   ... Looking for compiler 'Intel Fortran' ...
   ... Looking for environment variable 'IFORT_COMPILER14' ...No.
   ... Looking for environment variable 'IFORT_COMPILER13' ...No.
   ... Executing command 'which ifort' ...Yes ('/usr/local/bin/ifort').
   ... Looking for folder '/usr/local/bin' ...Yes.
   ... Looking for environment variable 'IFORT_COMPILER14' ...No.
   ... Looking for environment variable 'IFORT_COMPILER13' ...No.
   ... Executing command 'which ifort' ...Yes ('/usr/local/bin/ifort').
   ... Looking for folder '/usr/compiler/lib' ...No.
   Did not find installed compiler 'Intel Fortran'.
   Error using mex
   No supported compiler or SDK was found. For options, visit http://www.mathworks.com/support/compilers/R2016a/maci64.html.

To be more specific, my Fortran version is GFORTRAN (GCC 6.3.0_1) and my MATLAB version is R2016a. The reason causing the error above is due to the change of the Fortran library in the new GNU version. In recent versions, GNU removed the library libgfortranbegin.a and moved these piece of code into libgfortran.a. Therefore, if we need to link the mex with new GFORTRAN version, the searching in MATLAB must be modified.

Here, we list a few steps in the modification.

Copy a xml file to local start up folder.

cp /Applications/MATLAB_R2016a.app/bin/maci64/mexopts/gfortran.xml ~/.matlab/R2016a/mex_FORTRAN_maci64.xml

Edit mex_FORTRAN_maci64.xml with sudo permission.

Find the line with

LINKLIBS="-L&quot;$MATLABROOT/bin/maci64&quot; -lmx -lmex -lmat -L&quot;$GFORTRAN_LIBDIR&quot; -lgfortran -L&quot;$GFORTRANBEGIN_LIBDIR&quot; -lgfortranbegin"

and change it to

LINKLIBS="-L&quot;$MATLABROOT/bin/maci64&quot; -lmx -lmex -lmat -L&quot;$GFORTRAN_LIBDIR&quot; -lgfortran -lgcc_s.1"

Find the lines as

<cmdReturns name="$$/gfortran -print-file-name=libgfortranbegin.a    " />
<dirExists name="$$" />

and change it to

<cmdReturns name="$$/gfortran -print-file-name=libgfortranbegin.a    " />
<dirExists name="/" />

Find all lines with

<dirExists name="$$/Platforms/MacOSX.platform/Developer/SDKs/MacOSX10.11.sdk" />

and attach the following line below in every place

<dirExists name="$$/Platforms/MacOSX.platform/Developer/SDKs/MacOSX10.12.sdk" />

Find all lines with

<cmdReturns name="find $$ -name MacOSX10.11.sdk" />

and attach the following line below in every place

<cmdReturns name="find $$ -name MacOSX10.12.sdk" />

If there is minor difference between your system and mine, please update the corresponding path. If the difference is major, and you find it impossible to update accordingly, please comment below or send me an Email.

Yingzhou Li

Paper Corrections for A multiscale butterfly algorithm for multidimensional Fourier integral operators

2017-04-22T00:00:00+00:00

This post lists corrections for the following paper.

Y. Li, H. Yang, and L. Ying, A multiscale butterfly algorithm for multidimensional Fourier integral operators, Multiscale Modeling and Simulation 13:2, 614-631 (2015). [PDF, journal]

So far, I have found the following mistakes or typos.

Page 629, Equation (36), “$\Phi(x,\xi) = x \cdot \xi + c(x)\sqrt{\xi_1^2+\xi_2^2}$” should be “$\Phi(x,\xi) = x \cdot \xi + c(x) \sqrt{\xi_1^2+\xi_2^2+\xi_3^2}$”

Yingzhou Li

Paper Corrections for Butterfly Factorization

2017-04-22T00:00:00+00:00

This post lists corrections for the following paper.

Y. Li, H. Yang, E. Martin, K. Ho and L. Ying, Butterfly Factorization, Multiscale Modeling and Simulation 13:2, 714-732 (2015). [PDF, journal]

So far, I have found the following mistakes or typos.

Page 719, Section 2.2, last sentence of the first paragraph “… hence requires only $O(r(m+n))$ operations and memory.” should be “… hence requires only $O(r^2(m+n))$ operations and $O(r(m+n))$ memory …”
Page 718, Section 2.1, SVD via random matrix-vector multiplication. The formula for the middle matrix $ M = Q^\star_{col} Z Q_{row} $ in the third step should not be used in the middle level factorization directly, since the matrix Z could not be applied to basis $ Q_{row} $ or $ Q_{col} $ again. The actual way to implement is that $ M = \left( R^\star_{row}Q_{col} \right)^\dagger R^\star_{row} Z R_{col} \left( Q^\star_{row} R_{col} \right)^\dagger $. Such the case $ Z R_{col} $ can be reused and all the computations are in the local block.

Yingzhou Li

乔布斯传

2016-06-18T00:00:00+00:00

这次花了比较长的时间才读完《乔布斯传》。全书基本上按照时间顺序讲述了乔布斯的一生，从一出生被领养，跳到大学时期，再详细的介绍了从创立苹果之后的点点滴滴。

由于阅读的是中文译本，所以对于文笔不再做任何评论。但是对于人物传记而言，事件记录还算客观，作者对乔布斯的评价不能算客观，但是还是给读者留出了足够的客观事实，从而使得读者可以给出自己对乔布斯的评价。这里我们先零零散散的讲几处印象比较深刻的地方，最后再说说我个人对乔布斯的评价，以及对自身的反省。

第一个给我留下深刻印象的是乔布斯服用致幻剂。任何一个在中国长大的人应该都无法直接接受服用致幻剂这样一个事实。但是全书中多次出现乔布斯强调服用致幻剂给他带来的积极影响，以及后来设计了一些类似于幻觉艺术的屏保。由此我们也可以联想到很多服用致幻剂的艺术家。或者往轻一点的方向说，李白斗酒诗百篇，酒醉的状态也是许多大家在艺术创作时的追求。这种状态或许就是理性推理和艺术创作之间的不那么鲜明的界限。

第二个深刻的印象就是乔布斯的“扭曲现实力场”。作者或者乔布斯身边的人对此既爱又恨。一部分是因为扭曲现实力场让很多人做到了超越自己，而另外一方面就是扭曲现实力场也给人以绝望。这可能就是每天都需要面对绝境求生的挑战。每当完成一次挑战以后，其实每个人都会得到一个自身的提高以及对未来的自信。虽然乔布斯经常强调自己招募的都是一流人才，抵制二流人才，由此他的团队都是精英；而且他还强调精英也只会和精英合作。我承认后面一点，也就是精英只会和精英合作。不过与其说精英和精英合作，倒不如说精英和非精英在一起的时候，人的惰性就把他也变成了非精英。这也就是著名的短板效应。至于前面一点，我就不敢苟同了。虽然我并没有针对每个书中提到的员工做一番研究，但是显而易见的是那么大一个苹果公司，不可能所有的员工甚至说所有的核心员工都是一流人才。但是乔布斯的扭曲现实力场致使所有员工都超水平发挥，从而达到了一流的水平甚至超一流的水平。其实每个人对自己或多或少都有扭曲现实力场的作用，例如有很多人异想天开，很多人都有梦想。但是当面对周围的人的时候，人们往往都希望给别人留下美好的印象，而且不希望被人讥笑。由此对于任何扭曲现实的想法，也就胎死腹中了。乔布斯的与众不同之处就在于，他凭借着自己的任性，还能登顶人才金字塔。这往往是大多数人做不到的。在通往成功的路上，很多人都不愿意得罪周围的人，得罪金字塔顶端的人。乔布斯却杀出了一条血路，这或许也是一条不可复制的成功之路。

再一个深刻的印象就是乔布斯的禅修经历。

Yingzhou Li

艽野尘梦

2016-05-17T00:00:00+00:00

第一次听说《艽野尘梦》是快读到大冰的《他们最幸福》最后的时候看到的。大冰提到了想去徒步行走羌塘无人区正是由于受到了艽野尘梦的影响，并且他也在书中简要的介绍了陈渠珍的这本著作。出于好奇，我也拿起了这本也是我第一本阅读的文言体的书。过去初高中时期因为苦读文言片段而积累的单薄的功底也发挥了一次远离考试的作用，余心甚慰。

《艽(jiao)野尘梦》是民国时期陈渠珍所著的笔记。其中“艽野”一词出自《诗·小雅·小明》的“我征租西，至于艽野”，意思是“远荒之地”，而这里指的就是西藏。全书记录了陈渠珍在清末辛亥革命前后出入西藏的刻骨铭心的记忆。

作者陈渠珍(1882-1952)是著名的“湘西王”，一生历经三朝官场。晚晴时他先毕业于湖南武备学堂，而后在新军任三营管带；民国时期他统一湘西，并经营湘西数十载；在新中国成立以后，1950年，他赴京参加了全国政协会议。在丰富的人生经历背后，他还是一个爱书之人。陈渠珍的书记，也是后来的著名作家沈从文先生回忆他时提到，“平时极爱读书，以曾国藩、王守仁自许，看书与治事时间几乎各占一半。在他的军部会议室里，放置了五个大楠木橱柜，柜里藏有百来幅自宋及明清绘画，几十件铜器古瓷，十来箱书籍，一大批碑帖，和一套《四部丛书》”。

而《艽野尘梦》也曾于1940-1942年间在《康导月刊》连载。而后藏学家任乃强先生为其做注。于1982年和1999年分别由重庆出版社和西藏人民出版社出版。我阅读的版本是2011年由朝华出版社出的kindle版。

介绍完此书及其作者的基本背景，再来聊聊我对此书的些许印象。《艽野尘梦》着墨于大量的切身相关的历史事件。从最初的入藏，到在西藏波密剿匪，而后哥老会夺权，以致陈渠珍寻思返乡，再误入羌塘无人区，最后抵达长安止。但是诸多描写中，出作者自身所见之历史事件和相关人物外并无刻意描写某一个人物角色。唯有其藏妻西原，经常不经意间提起，确让读者对她的品质印象极深。大量的后记，书评都对其称赞有加着重评论。我也不免俗的要对她称赞一番。西原原是藏地营官加瓜彭错的侄女。一日加瓜彭错宴请陈渠珍，并让西原及其他在宴会上表演骑马把杆。而西原连拔五竿引起了陈渠珍的注意。在觥筹交错中，加瓜彭错许诺讲西原许配给陈渠珍，而陈渠珍也是醉意朦胧中答应。几日后，加瓜彭错竟然真的将西原送至陈渠珍的府邸，陈渠珍非常惊讶，但是还是欣然接受了这桩婚事，其中原由不可深究。而后在平定波密的战事中，西原也是横刀立马，与将士一起出站，更解围困之危。至此我们见识到了西原的女将之风。随着内地革命，藏地的军队政权发生更迭，陈渠珍处于自身安危的考虑，让西原及100多人的小部队横跨羌塘返回内地。西原带着母亲给的红珊瑚饰品泪别母亲，追随陈渠珍前往死地。由于大雪隆冬，又没有GPS的定位，一个喇嘛向导带领着这支队伍向死亡前进。不出几日，粮尽，又几日，牲畜都吃完了。在没有下雪的日子里，大家就打野兔、野羊为食，但是遇到连续大雪天，没有野味，出现了吞食死去同伴的尸体。即使在这样的情况下，西原仍旧为陈渠珍省下自己的口粮，并称自己可以挨饿，但是陈渠珍必须要吃饱指挥队伍。为了避免同伴杀死随行藏童来吃，西原独自提枪出猎。

在描述羌塘惨状时，有两个时刻分外精彩。一个是最后的几根火柴。另一个是截杀喇嘛团队。虽然这两个故事都与西原无关，但是却必须要在这里强行提到。

在大家只剩最后20来根火柴的时候，就集中保管到了陈渠珍手上，由他来分配使用。每次使用的时候，大家都先立成人墙，有点像足球里的人墙，但是这里必须要密不透风，然后另一人在人强中用火柴点燃牲畜粪便，等到有一点烟的时候，再令人墙分开一道缝让微风把火堆吹旺。众人齐心协力，为了能成功的点燃每一次火。

然而在后来遇到喇嘛团的时候，期初相安无事。喇嘛团也给了他们食物，还许诺要在分别时给他们再留一些食物。然而在分别前一天，陈渠珍发现了士兵有截杀喇嘛团的企图，但是他害怕士兵会集体谋害自己(也可能是他自己也是想的，却不想写在书中)，没有当众阻止。第二日，喇嘛团帮士兵们整理好食物，并且连同几批骆驼一起留给他们，在分别时，6个士兵却突然开枪相向，而喇嘛团发觉后也立马开枪回应。打死两三个喇嘛后，其余喇嘛骑着骆驼逃走了，而原来留给陈渠珍他们的骆驼受惊于枪声也背着食物逃走了。至于这六个士兵也是5死1重伤不能行。而余下的人却不愿携带重伤员前行，就把他遗弃在荒原等死。六个人虽无人性遭天谴，然而余下的人在人性上也未必比他们高级。

再回到西原的爱情故事，最终连带西原和陈渠珍，仅有7人逃出无人区回到长安。上天不公，西原在长安时却染上天花，病逝在长安，而陈渠珍当时穷困的无法厚葬西原。全书以“述至此，肝肠寸断矣。余书亦从此辍笔矣”做结。我不喜陈渠珍的行为和心性，却深深的被西原感染。

一个藏族女子，为其夫做到如此，感天动地。我并不确定是爱情的力量，或者是她的心性让西原如此。我更相信这是西原的心性。但是无论是什么，在死亡面前，任然处处为夫君考虑，愿意做一切，也愿意死在夫君之前，是值得赞叹的大女子。

Yingzhou Li

写在书架边缘

2016-05-13T00:00:00+00:00

这是一篇写在书架边缘的小文。

由于搬家的缘故，我每天都要坐一小时的校车横跨旧金山湾去学校，这给我提供了大量的固定的时间。于是，在几个月前我买了Kindle聊以慰藉路途的空虚。不知不觉中，三天一小本，半月一大本的也读了不少闲书。对于这些书，我并不觉得自己理解的多么深刻，也不觉得提高了多少自身修养，却仍旧感到收获颇丰。

俗话说，好记性不如烂笔头，更何况我这烂记性，可能连好笔头都救不回来。经常只记得自己读过哪本书，或记得书名，或记得作者，但是对于书中的内容，故事情节，主要人物等等都淡忘如书的简介，更甚者对应简介都无法回想。导致我经常要质问自己是否真的读过这本书。

迫于学术研究的需要，我架设了英文版个人主页，为了方便自己看综艺节目，又增加了中文页面。这里仅仅能称之为中文页面，而不能叫做中文版本，除了姓名是翻译以外，其余一切都未曾转换成中文。这周早些时候下午回家路上，由于忘带Kindle，就看看跨湾大桥两侧的广袤滩涂和稀疏的水草荡。顺带着也回忆回忆最近读过的书。忽地发现，能记住梗概的也只能是最近的几本了。而数月前阅读的书，情节，脉络都淡忘了许多。于是突发奇想，为什么不给自己新阅读的每本书籍写一篇小文，既能总结概括一下书的内容，偶尔还能犯犯矫情。

这就是个书架栏的诞生。

然而，我自幼不是一个好读书写字的人。更多的时间花在了摆弄数字和电脑。不流畅的语言和混乱的逻辑关系，希望不幸读到这里的人原谅。或许随着不断的添加新的文字，未来的某一天，我也可以向苏先生等等老一辈学者看齐。在摆弄符号之余，也有一定的文字功底。但是根据目前的状况，我阅读过的书可能都还放不满一整个书架。所以姑且称为书架。待日后，阅读量增加，笔尖可以流出点墨水而不是苦水的时候，兴许可以改之为“书斋”。

Yingzhou Li

Command Line Cheat Sheet

2016-01-30T00:00:00+00:00

Recently, I was trying to auto-launch multiple terminals with one command and got different terminals into different directories. A typical output of the solution on Mac is like:

Mac Desktop

In the process of searching for solution, I found the following command line cheat sheets useful for both Linux and Mac user. Sharing them in my post is to keep a record for myself, the credits go to the original authors, DavidChild and Onno Schwanen.

If you prefer to have a more complete list of basic commends, please refer to PCWDLD page.

Bash Shortcuts

Mac	Linux	Description
Ctrl + C	Ctrl + C	Kill current command
Ctrl + Z	Ctrl + Z	Puts whatever you are running into a suspended background process. fg restores it.
Ctrl + A	Ctrl + A	Go to the beginning of the line you are currently typing on
Ctrl + E	Ctrl + E	Go to the end of the line you are currently typing on
Ctrl + U	Ctrl + U	Clears the line before the cursor position. If you are at the end of the line, clears the entire line.
Ctrl + K	Ctrl + K	Clear the line after the cursor
Ctrl + L	Ctrl + L	Clears the Screen
Command + K		Reset the Terminal display
Ctrl + H	Ctrl + H	Same as backspace
Ctrl + R	Ctrl + R	Lets you search through previously used commands
Ctrl + D	Ctrl + D	Exit the current shell
Ctrl + W	Ctrl + W	Delete the word before the cursor
Ctrl + T	Ctrl + T	Swap the last two characters before the cursor
Ctrl + F	Ctrl + F	Move cursor one character forward
Ctrl + B	Ctrl + B	Move cursor one character backward
Esc + F	Esc + F	Move cursor one word forward
Esc + B	Esc + B	Move cursor one word backward
Esc + T	Esc + T	Swap the last two words before the cursor
Tab	Tab	Auto-complete files and folder names

Core Commands

Mac	Linux	Description
cd	cd	Home directory
cd [folder]	cd [folder]	Change directory
cd ~	cd ~	Home directory, e.g. ‘cd ~/folder/’
cd /	cd /	Root of drive
ls	ls	Short listing
ls -l	ls -l	Long listing
ls -a	ls -a	Listing all files
ls -lh	ls -lh	Long listing with Human readable file sizes
ls -R	ls -R	Entire content of folder recursively
sudo [cmd]	sudo [cmd]	Run command with the security privileges of the superuser (Super User DO)
open [file]	open [file]	Opens a file ( as if you double clicked it )
top	top	Displays active processes. Press q to quit
nano [file]	nano [file]	Opens the Terminal its editor
vi [file]	vi [file]	Opens the editor VIM
exit	exit	Exit
clear	clear	Clear screen
reset	reset	Reset the Terminal display

Command History

Mac	Linux	Description
history n	history n	Shows the stuff typed - add a number to limit the last n items
![value]	![value]	Execute the last command typed that starts with ‘value’
!!	!!	Execute the last command typed

File Management

Mac	Linux	Description
touch [file]	touch [file]	Create new file
cat	cat	Concatenate to screen
cat [file1] [file2]	cat [file1] [file2]	Concatenate files and output
less [file]	less [file]	View and paginate file
file [file]	file [file]	Get type of file
rm [file]	rm [file]	Remove a file
rm -i [file]	rm -i [file]	Remove with confirmation
rm -r [dir]	rm -r [dir]	Remove a directory and contents
rm -f [file]	rm -f [file]	Force removal without confirmation
cp [file1] [file2]	cp [file1] [file2]	Copy file1 to file2
cp [file] [dir]	cp [file] [dir]	Copy file to directory
mv [file1] [file2]	mv [file1] [file2]	Move/Rename file1 to file2
head [file]	head [file]	Show first 10 lines of file
tail [file]	tail [file]	Show last 10 lines of file
tail -F [file]	tail -F [file]	Show last lines of file as it changes

Directory Management

Mac	Linux	Description
pwd	pwd	Full path to working directory
..	..	Parent/enclosing directory
.	.	Current folder
mkdir [dir]	mkdir [dir]	Create new directory
mkdir -p [dir]/[dir]	mkdir -p [dir]/[dir]	Create nested directories
rmdir [dir]	rmdir [dir]	Remove directory ( only operates on empty directories )
rm -R [dir]	rm -R [dir]	Remove directory and contents

Pipes - Allows to combine multiple commands that generate output

Mac	Linux	Description
more	more	Output content delivered in screensize chunks
> [file]	> [file]	Push output to file, keep in mind it will get overwritten
» [file]	» [file]	Append output to existing file
<	<	Tell command to read content from a file

Help

Mac	Linux	Description
[cmd] -h	[cmd] -h	Offers help
[cmd] –help	[cmd] –help	Offers help
[cmd] help	[cmd] help	Offers help
man [cmd]	man [cmd]	Show the help for ‘command’
whatis [cmd]	whatis [cmd]	Gives a one-line description of ‘command’

Yingzhou Li

User compilers installation

2015-11-19T00:00:00+00:00

Installation of GNU compilers

There are many ways to install GNU compilers. And the following is found to be the most simple and straightforward way. The following ingredients are the simplified version of here.

First download and unpacked your desired version of gcc compiler. Here an official gcc download address are provided https://ftp.gnu.org/gnu/gcc . And we will take version 5.2.0 as an example. If other versions are desired, please replace 5.2.0 in all following instructions by the version number, e.g. replace gccver=5.2.0 by gccver=4.9.3.
```
gccver=5.2.0
mkdir ~/gcc-tmp
cd ~/gcc-tmp
wget https://ftp.gnu.org/gnu/gcc/gcc-${gccver}/gcc-${gccver}.tar.gz
tar -xzf gcc-${gccver}.tar.gz 
```
Download the prerequisite sources into the same gcc source directory.
```
cd ~/gcc-tmp/gcc-${gccver}
./contrib/download_prerequisites
```
Create a building directory and configure gcc. One important thing here is that the building directory must be different from the source directory.
```
mkdir ~/gcc-tmp/build
cd ~/gcc-tmp/build
~/gcc-tmp/gcc-${gccver}/configure --prefix=/path/to/install/the/gcc/ --disable-multilib
```
Build gcc.
```
cd ~/gcc-tmp/build
make
```
Install gcc.
```
cd ~/gcc-tmp/build
make install
```
Clean all temporary folders and files.
```
cd ~
rm -rf ~/gcc-tmp 
```

Installation of MPICH

MPICH is a high performance and widely portable implementation of the Message Passing Interface (MPI) standard. The following ingredients are the simplified version of here.

First download and unpacked your desired version of mpich. Here an official mpich download address are provided http://www.mpich.org/static/downloads/. And we will take version 3.2 as an example. If other versions are desired, please replace 3.2 in all following instructions by the version number, e.g. replace mpichver=3.2 by mpichver=3.1.4.
```
mpichver=3.2
mkdir ~/mpich-tmp
cd ~/mpich-tmp
wget http://www.mpich.org/static/downloads/${mpichver}/mpich-${mpichver}.tar.gz
tar -xzf mpich-${mpichver}.tar.gz 
```

Configure mpich.

cd ~/mpich-tmp/mpich-${mpichver}
LD_LIBRARY_PATH=/path/to/installed/gcc/lib64 ./configure CC=/path/to/installed/gcc/bin/gcc CXX=/path/to/installed/gcc/bin/g++ F77=/path/to/installed/gcc/bin/gfortran FC=/path/to/installed/gcc/bin/gfortran --prefix=/path/to/install/the/mpich/

Install mpich.

cd ~/mpich-tmp/mpich-${mpichver}
make install

Clean all temporary folders and files.
```
cd ~
rm -rf ~/mpich-tmp 
```

Updating List:

2015-11-19: Added the first version of the compilers installation instructions.
2015-11-20: Added the installation instructions for mpich.

Please let me know if any of other compilers are useful to be installed under local directory.

Yingzhou Li

ICME Related Seminar Calendar

2015-10-06T00:00:00+00:00

Yesterday, I got the ICME weekly seminar digest from ICME office and found it helpful to have all the speakers, titles and abstracts listed in a single email. However, the date and time listed in the email are not very convenient to follow. Verifying the seminar time slot in your own calendar costs a lot of energy.

Since then, I created my own seminar calendar for many ICME related seminars, i.e., ICME Departmental Seminar, ICME First Year Seminar, Linear Algebra and Optimization Seminar, Applied Math Seminar, Probability Seminar and Statistics Department Seminar. And there is no reason to keep it private. Hence, I published the calendar and embedded it into my homepage in a Seminar Calendar page. Currently, the speaker, title and abstract are listed in description as well.

In the future, I will keep on adding more related seminar calendar and more ICME academic events.

The new seminar calendar for Winter 2016 is available online.

Yingzhou Li

Ubuntu installation list

2015-09-23T00:00:00+00:00

Installation of Ubuntu

Readers are referred to the following links for the creation of a bootable USB stick on different platform.

Ubuntu: bootable usb
Mac: bootable usb
Windows: bootable usb

Installation instruction of Ubuntu is available here. However, the installation is relatively easy even without these instruction.

PPA List

Chrome (double check if it is necessary)

wget -q -O - https://dl-ssl.google.com/linux/linux_signing_key.pub | sudo apt-key add -
sudo sh -c 'echo "deb http://dl.google.com/linux/chrome/deb/ stable main" >> /etc/apt/sources.list.d/chrome.list'

Thunderbird

sudo add-apt-repository ppa:mozillateam/thunderbird-next

RStudio ( This repository currently does not support Ubuntu Vivid. Please ignore this line if you are using Ubuntu 15.04. )
```
sudo add-apt-repository ppa:opencpu/rstudio
```

Julia

sudo add-apt-repository ppa:staticfloat/juliareleases

At the end of adding these repository, we need the following commands to update the package list and upgrade installed packages.

sudo apt-get update
sudo apt-get upgrade
sudo apt-get dist-upgrade

It is also suggested to run sudo apt-get autoremove to remove unnecessary packages.

Installation List

Ubuntu Restricted Extras

sudo apt-get install ubuntu-restricted-extras

Chrome
```
sudo apt-get install google-chrome-beta
```
And following extensions are useful:
- Hangouts
- Unblock Youku
Git
```
sudo apt-get install git
```

LaTeX and Texmaker

sudo apt-get install texlive-full
sudo apt-get install texmaker

Vim

sudo apt-get install vim
git clone https://github.com/VundleVim/Vundle.vim.git ~/.vim/bundle/Vundle.vim

The vimrc file can be found here.

Dropbox
```
sudo apt-get install nautilus-dropbox
```
Thunderbird Thunderbird is installed by default and only upgrading is required. And following add-ons are useful:
- Grammar Checker
- LaTeX It!
- MinimizeToTray revived
Openssh Server
```
sudo apt-get install openssh-server
```

Kerberos

sudo apt-get install krb5-user

Configure the kerberos. Here is an example for corn machines at Stanford.

Create a folder for Control ticket.
```
mkdir ~/.krblog
```

Set up configuration for ssh. Copy the following lines into ~/.ssh/config file.

Host corn corn?? corn.stanford.edu corn??.stanford.edu
GSSAPIKeyExchange yes
GSSAPIAuthentication yes
GSSAPIDelegateCredentials yes
ControlMaster auto
ControlPath ~/.krblog/.%r@%h:%p
ControlPersist yes
ServerAliveInterval 60
ServerAliveCountMax 3

MPICH
```
sudo apt-get install mpich
```

R and RStudio

sudo apt-get install r-base
sudo apt-get install r-base-dev

Python, numpy, scipy, matplotlib, ipython and ipython-notebook

sudo apt-get install python-numpy python-scipy python-matplotlib ipython ipython-notebook

Jekyll

sudo apt-get install ruby-dev
sudo gem install jekyll
sudo gem install jekyll-sitemap
sudo apt-get install nodejs

sudo apt-get install ruby-dev
sudo gem install jekyll
sudo gem install jekyll-sitemap
sudo apt-get install nodejs

Julia
```
sudo apt-get install julia
```
CMake
```
sudo apt-get install cmake
```
Valgrind
```
sudo apt-get install valgrind
```
Pinyin Input

Install ibus-pinyin for Ubuntu 10-14.
```
sudo apt-get install ibus-pinyin
```
Install fcitx for Ubuntu 15.
```
sudo apt-get install fcitx
```
With the ibus-pinyin or fcitx installed, we follow the following steps:
1. Click System Settings/Language Support
2. Click Install/Remove Languages... and add Chinese
3. Click System Settings/Language Support and select IBus (Ubuntu 10-14) or fcitx (Ubuntu 15) as the Keyboard input method system.
4. Logout and Re-Login.
5. Configure pinyin input based on different systems. Here we only show an example for Ubuntu 15,
  - Click keyboard button on the status bar, and select Configure.
  - Add Chinese pinyin ( uncheck Only Show Current Language ). The instructions for Ubuntu 10-14 are detailed here.
For detailed instructions, readers are referred to Pinyin Joe’s.
Dput, Uploading packages to PPA
```
sudo apt-get install dput
```

Exfat related packages

sudo apt-get install exfat-utils
sudo apt-get install exfat-fuse

Updating List:

2015-09-25: Added the first version of the Ubuntu package installation instructions.
2015-09-28: Added instructions for Chinese pinyin input.
2015-09-30: Added instructions for dput to upload packages to PPA. And updated the instruction order to avoid the usage of the not installed package.
2015-10-03: Added instructions for Ubuntu restricted extras. This installation includes many useful plug-ins.
2015-10-24: Added instructions for Exfat format related packages. Such that any Exfat format UBS can be read.

Please let me know if any of these packages are not updated via the repositories.

Yingzhou Li

Vim Setting

2015-07-22T00:00:00+00:00

My Vim setting (.vimrc) is reorganized today. Most of these settings are referred from Vim Wiki Mainly, it supports basic auto read, line number, ruler (specifically for Vim on Mac, this is not necessary for Linux). Also it enables auto spell check with underline highlight for spell bad words and others, and the functionality of returning to the last edit position for opening files is added to it. The last line of basic setting part adds a user defined command Clean for removing spaces at the end of each line.

Another parts of the .vimrc file enables Vundle with plugins julia-vim, tabular and vim-markdown. A few settings of vim-markdown are also presented next to Vundle.

The detailed .vimrc file is attached as follows. And an updating list and a TODO list are added at the end of this post. Hopefully, it will be helpful for others.

" Configuration file for vim
" Edited by Yingzhou Li

"=================================================================
" Basic setting for vim
set history=20
set autoread
set number
set ruler                           " status bar setting for Mac

" Setting for spell check
autocmd BufRead *.txt,*.tex,*.md,*.markdown setlocal spell
set spelllang=en_us

" Setting for indent
set tabstop=4
set shiftwidth=4
set softtabstop=4
set autoindent
set expandtab

set viminfo=%,'500,\"500,n~/.viminfo

" Setting for highlight
syntax on
highlight Comment ctermfg=Yellow
highlight clear SpellBad
highlight SpellBad term=standout ctermfg=1 term=underline cterm=underline
highlight clear SpellCap
highlight SpellCap term=underline cterm=underline
highlight clear SpellRare
highlight SpellRare term=underline cterm=underline
highlight clear SpellLocal
highlight SpellLocal term=underline cterm=underline

" Return to last edit position when opening files
autocmd BufReadPost *
	\ if line("'\"") > 0 && line("'\"") <= line("$") |
	\   exe "normal! g`\"" |
	\ endif

" Remove space at the end of each line
command Clean :%s/\s\+$//e


"=================================================================
" Setting for Vundle, julia-vim
filetype off
set rtp+=~/.vim/bundle/Vundle.vim
call vundle#begin()
Plugin 'gmarik/Vundle.vim'
Plugin 'JuliaLang/julia-vim'
Plugin 'godlygeek/tabular'
Plugin 'plasticboy/vim-markdown'
call vundle#end()
filetype plugin indent on


"=================================================================
" Setting for Markdown+LaTeX
let g:vim_markdown_folding_disabled=1
let g:vim_markdown_math=1
let g:vim_markdown_frontmatter=1

Installation:

Vim installation

The vimrc file here only support self-installed vim. The installation instructions for different platforms are different. Here is an example for Ubuntu:
```
sudo apt-get install vim
```
Vundle

The installation of Vundle is available here.
Download the vimrc file.
Copy .vimrc to your user folder.
```
cp /path/to/vimrc ~/.vimrc
```
Install plugins.
```
vim ~/.vimrc
:PluginInstall
```

Updating List:

2015-07-22: Added new .vimrc file.
2015-08-24: Specified spell check for *.txt, *.tex, *.md, *.markdown files only.
2015-09-23: Updated this page and added vim installation step in the installation instruction.

TODO List:

Add ‘‘SMART” hard wrap for text files, especially for .tex files.

Yingzhou Li

MATLAB Random Seed

2015-07-07T00:00:00+00:00

A shocking thing about random seed in MATLAB is that the random seed is initialized by the same value every time we start MATLAB.

The story is as follows.

We were trying to run numerical results on Sherlock for our new paper “Multidimensional Butterfly Factorization” which is collaborating with Haizhao Yang and Lexing Ying. As been pointed out in previous papers [1,2], an evaluation of true relative error is too expansive to compute. Hence, the relative error was evaluated in a subsampled set of points, i.e.,

\[\epsilon = \sqrt{\frac{\sum_{x\in S}\lvert u^d(x)-u^a(x) \rvert^2} {\sum\_{x\in S}\lvert u^d(x) \rvert^2}}\]

where $u^d$ is the result of direct evaluation, $u^a$ is the result of the fast algorithm, and $S$ is a point set of size 256.

On Sherlock, all jobs have to be submitted via sbatch files. In each sbatch file, we will start MATLAB and run the script. However, we found that the evaluated relative error is constant over several runs. At the beginning, we thought there was a bug in the code, especially, in the relative error evaluation part. After a careful debugging and double checking the relative error, we finally came to this finding.

We start a new MATLAB, and the following lines are always true on my Mac Pro.

>> rand()

ans =

    0.8147

>> randn()

ans =

    1.8339

A nice way to have true random values every run, we could suggest to initialize the random seed with current computer time. And as been pointed out in the help file of rng, we could put the following line at the beginning of the code:

rng('shuffle');

such that it seeds the random number generator based on the current time.

Yingzhou Li

New Homepage

2015-06-24T00:00:00+00:00

My new homepage was posted yesterday. The key reason to post a new homepage is to have Posts section in it. Another reason is that the theme in previous homepage is too dark. Hence I spent two days to code this new homepage and added the post feature to it.

Jack Poulson is the first one suggested me to write post daily or weekly to improve my writing skill and he also recommended me Jekyll which is supper useful for blog-style web designing. It was a long time ago, I downloaded this template called Left, where the post section is naturely embeded. Finally, until this week, I moved my previous homepage to this template and started posting. As metioned earlier, the main purpose of my posting is to improve my academic writing skill. At the same time, I will share my understandings of papers, algorithms, or theorems. The frequency of posts is not guaranteed. But I will try my best to post at least one each week.

Hopefully, you will enjoy my posts. Clicking the red figure below or the email link, an auto prepared email would pop out. You are welcome to send me any discussion and suggestion. Our email might be posted below the related post with your permission.

Yingzhou Li

Left

2015-06-23T00:00:00+00:00

Left is a clean, whitespace-happy layout for Jekyll.

This is designed to be an easy layout to modify for your own blog. It was extracted from zachholman.com, which means it was battle-hardened from years of posting serious blog posts about emoji and swear words.

You’re actually reading an example post right now. See how cool it is? Oh, be quiet, you’re making me blush.

You’ll want to get the code and read the README to learn how to install and set up Left for your own purposes.

If you have any questions about this Jekyll template, follow and ping Holman on Twitter @holman.

I personally like the layout of Left. I modified the blog style template to a personal homepage style. Email me if you happen to share the same flavor as me.

Yingzhou Li

Yingzhou Li

Probability bound in factoring to order-finding

Probability of Random Choosing an Even Order Number

Revisit the Bound

Example 1 ($N=3$)

Example 2 ($N=15$)

Example 3 ($N=63$)

Fudan Library Access Button

Installation Instruction

Mozilla Firefox

Google Chrome, Edge or Safari

Internet Explorer

Mobile Users

Usage Instruction

Known Issue

Von Neumann's Trace Inequalities

Von Neumann’s trace inequality and related results

Lemma 1 for general Hermitian matrices

Setup mex with MSVC on Windows

Reducible Singular Value Decomposition

Preliminary

Excise

Solution

Setup mex with GFORTRAN on Mac

Paper Corrections for A multiscale butterfly algorithm for multidimensional Fourier integral operators

Paper Corrections for Butterfly Factorization

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