Given a list (or a Vector) v of $n$ real numbers, the PSLQ(v) command outputs a list (or a Vector) u of $n$ integers such that $\sum _{i=1}^n{u}_i{v}_i$ is minimized.  Thus the PSLQ function finds an integer relation between a vector of linearly dependent real numbers if the input has enough precision.

Given a list (or a Vector) v of $n$ complex numbers, the PSLQ(v) command outputs a list (or a vector) u of $n$ complex integers (Gaussian integers) such that the norm of $\sum _{i=1}^n{u}_i{v}_i$ is minimized.

This is an implementation of Bailey and Ferguson's PSLQ algorithm. You can also use the Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm to find a linear relation.  For more information, see IntegerRelations[LLL]. Generally speaking, PSLQ is faster.

One application of PSLQ is to find the minimal polynomial of an algebraic number given a decimal approximation of the algebraic number.  The examples below illustrate this.  Generally speaking, if the height of the minimal polynomial is $m$ and its degree is $n$, then you need more than $n{\log }_{10}\left(m\right)$ correct decimal digits for the algebraic number.  If the input has $d$ digits and this is insufficient, the output of PSLQ will typically be a list of $n$ integers each with approximately $d$/$n$ digits long.

$\mathrm{with}\left(\mathrm{IntegerRelations}\right)\:$

$v≔\left[1.570796326\,1.414213562\,-1.101048032\right]$

$v≔\left[1.570796326\,1.414213562\,\mathrm{-1.101048032}\right]$

$u≔\mathrm{PSLQ}\left(v\right)$

$u≔\left[\mathrm{-2}\,3\,1\right]$

$u\left[1\right]v\left[1\right]+u\left[2\right]v\left[2\right]+u\left[3\right]v\left[3\right]$

$2.\times {10}^{\mathrm{-9}}$

$\mathrm{Logs}≔\mathrm{Vector}\left(\left[\log \left(2\right)\,\log \left(3\right)\,\log \left(5\right)\,\log \left(6\right)\,\log \left(10\right)\right]\right)$

$\mathrm{Logs}≔\left[\begin{array}{c}\ln \left(2\right)\\ \ln \left(3\right)\\ \ln \left(5\right)\\ \ln \left(6\right)\\ \ln \left(10\right)\end{array}\right]$

$u≔\mathrm{PSLQ}\left(\mathrm{Logs}\right)$

$u≔\left[\begin{array}{c}1\\ 0\\ 1\\ 0\\ \mathrm{-1}\end{array}\right]$

$\mathrm{Relation}≔\mathrm{add}\left(\mathrm{Logs}\left[i\right]u\left[i\right]\,i=1..5\right)=0$

$\mathrm{Relation}≔\ln \left(2\right)+\ln \left(5\right)-\ln \left(10\right)=0$

$r≔\mathrm{sqrt}\left(2\right)+\mathrm{sqrt}\left(3\right)$

$r≔\sqrt{2}+\sqrt{3}$

$v≔\mathrm{expand}\left(\left[\mathrm{seq}\left(r^i\,i=0..4\right)\right]\right)$

$v≔\left[1\,\sqrt{2}+\sqrt{3}\,5+2\sqrt{2}\sqrt{3}\,11\sqrt{2}+9\sqrt{3}\,49+20\sqrt{2}\sqrt{3}\right]$

$v≔\mathrm{evalf}\left(v\,12\right)$

$v≔\left[1.\,3.14626436994\,9.89897948556\,31.1448064542\,97.9897948556\right]$

$v≔\mathrm{evalf}\left(v\right)$

$v≔\left[1.\,3.146264370\,9.898979486\,31.14480645\,97.98979486\right]$

$u≔\mathrm{PSLQ}\left(v\right)$

$u≔\left[1\,0\,\mathrm{-10}\,0\,1\right]$

$\mathrm{add}\left(u\left[i\right]v\left[i\right]\,i=1..5\right)$

$0.$

$m≔\mathrm{add}\left(u\left[i\right]z^{i-1}\,i=1..5\right)$

$m≔z^4-10z^2+1$

$\mathrm{simplify}\left(\mathrm{eval}\left(m\,z=r\right)\right)$

$0$

$r≔\cos \left(\frac{\mathrm{\pi }}{5}\right)+\mathrm{I}\sin \left(\frac{\mathrm{\pi }}{5}\right)$

$r≔\cos \left(\frac{\mathrm{\pi }}{5}\right)+\mathrm{I}\sin \left(\frac{\mathrm{\pi }}{5}\right)$

$a≔\mathrm{evalf}\left(r\right)$

$a≔0.8090169943+0.5877852524\mathrm{I}$

$v≔\left[1\,a\,a^2\,a^3\,a^4\right]\:$

$u≔\mathrm{PSLQ}\left(v\right)$

$u≔\left[1\,\mathrm{-1}\,1\,\mathrm{-1}\,1\right]$

$\mathrm{add}\left(u\left[i\right]v\left[i\right]\,i=1..5\right)$

$\mathrm{-1.}\times {10}^{\mathrm{-10}}-3.\times {10}^{\mathrm{-10}}\mathrm{I}$

$m≔\mathrm{add}\left(u\left[i+1\right]z^i\,i=0..4\right)$

$m≔z^4-z^3+z^2-z+1$

$\mathrm{simplify}\left(\mathrm{eval}\left(m\,z=r\right)\right)$

$0$

$r≔\mathrm{sqrt}\left(1+2\mathrm{I}\right)$

$r≔\sqrt{1+2\mathrm{I}}$

$v≔\mathrm{evalf}\left(\left[1\,r\,r^2\right]\right)$

$v≔\left[1.\,1.272019650+0.7861513778\mathrm{I}\,1.+2.\mathrm{I}\right]$

$u≔\mathrm{PSLQ}\left(v\right)$

$u≔\left[1+2\mathrm{I}\,0\,\mathrm{-1}\right]$

$m≔u\left[1\right]+u\left[2\right]x+u\left[3\right]x^2$

$m≔-x^2+1+2\mathrm{I}$

$\mathrm{eval}\left(m\,x=r\right)$

$0$

$m≔\mathrm{subs}\left(\mathrm{I}=z\,m\right)$

$m≔-x^2+2z+1$

$m≔\mathrm{resultant}\left(m\,z^2+1\,z\right)$

$m≔x^4-2x^2+5$

$\mathrm{eval}\left(m\,x=r\right)$

$0$

$r≔1+2^{\frac{1}{3}}-3^{\frac{1}{3}}$

$r≔1+2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}-3^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$\mathrm{Digits}≔50$

$\mathrm{Digits}≔50$

$\mathrm{interface}\left(\mathrm{rtablesize}=11\right)\:$

$v≔\mathrm{Vector}\left(\left[\mathrm{seq}\left(r^i\,i=0..10\right)\right]\right)\:$

$v≔\mathrm{evalf}\left(v\,55\right)\:$

$v≔\mathrm{evalf}\left(v\right)$

$v≔\left[\begin{array}{c}1.\\ 0.81767147958746478244557229649811876217838221120216\\ 0.66858664853075383639048354397191016622964216439499\\ 0.54668423413656577641146431210930096937662127202181\\ 0.44700810659358576105260725836989622226160789792505\\ 0.36550577990596844124463402408070205786826727820194\\ 0.29886365185348346975488348385448631176655996274722\\ 0.24437228440595079023352327191229754014349264986649\\ 0.19981624736038252994573312957596287544943722672979\\ 0.16338404662477883755114718040485659711884715361349\\ 0.13359447514467024355385019361427542093983423064155\end{array}\right]$

$u≔\mathrm{PSLQ}\left(v\right)$

$u≔\left[\begin{array}{c}\mathrm{-162}\\ 324\\ 0\\ \mathrm{-297}\\ 108\\ 27\\ 27\\ \mathrm{-45}\\ 27\\ \mathrm{-8}\\ 1\end{array}\right]$

$\mathrm{add}\left(u\left[i\right]v\left[i\right]\,i=1..11\right)$

$\mathrm{-1.75}\times {10}^{\mathrm{-48}}$

$m≔\mathrm{add}\left(u\left[i\right]z^{i-1}\,i=1..11\right)$

$m≔z^{10}-8z^9+27z^8-45z^7+27z^6+27z^5+108z^4-297z^3+324z-162$

$\mathrm{simplify}\left(\mathrm{eval}\left(m\,z=r\right)\right)$

$0$

$\mathrm{factor}\left(m\right)$

$\left(z+1\right)\left(z^9-9z^8+36z^7-81z^6+108z^5-81z^4+189z^3-486z^2+486z-162\right)$

$u≔\mathrm{PSLQ}\left(v\left[1..7\right]\right)$

$u≔\left[\begin{array}{c}6336266\\ \mathrm{-4491855}\\ 6130884\\ \mathrm{-12073116}\\ 3103150\\ \mathrm{-6012678}\\ 2169170\end{array}\right]$

$\mathrm{add}\left(u\left[i\right]v\left[i\right]\,i=1..7\right)$

$1.99\times {10}^{\mathrm{-42}}$