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IntegerRelations

 LinearDependency
 find an integer dependence (relation)

 Calling Sequence LinearDependency(v,opts)

Parameters

 v - list or Vector of (complex) floating-point numbers opts - (optional); equation of the form method=LLL or method=PSLQ specifying the algorithm used

Description

 • The LinearDependency(v,opts) command finds an integer relation between the numbers in v - if they are linearly dependent. Given a list (or a Vector) of $n$ real or complex numbers, LinearDependency outputs a list (or a Vector) $u$ of $n$ integers such that $\sum _{i=1}^{n}{u}_{i}{v}_{i}$ is close to zero.
 • By default, Bailey and Ferguson's PSLQ (Partial Sum of Least Squares) algorithm is used if the numbers in v are real.
 • The optional argument method=LLL specifies that the LLL (Lenstra-Lenstra-Lovasz) lattice basis reduction algorithm be used, which is the default if v contains non-real values.

Examples

 > $\mathrm{with}\left(\mathrm{IntegerRelations}\right):$
 > $r≔\mathrm{sqrt}\left(2\right)+\mathrm{sqrt}\left(3\right)$
 ${r}{≔}\sqrt{{2}}{+}\sqrt{{3}}$ (1)
 > $v≔\mathrm{expand}\left(\left[\mathrm{seq}\left({r}^{i},i=0..4\right)\right]\right)$
 ${v}{≔}\left[{1}{,}\sqrt{{2}}{+}\sqrt{{3}}{,}{5}{+}{2}{}\sqrt{{2}}{}\sqrt{{3}}{,}{11}{}\sqrt{{2}}{+}{9}{}\sqrt{{3}}{,}{49}{+}{20}{}\sqrt{{2}}{}\sqrt{{3}}\right]$ (2)
 > $v≔\mathrm{evalf}\left(v,12\right)$
 ${v}{≔}\left[{1.}{,}{3.14626436994}{,}{9.89897948556}{,}{31.1448064542}{,}{97.9897948556}\right]$ (3)
 > $v≔\mathrm{evalf}\left(v\right)$
 ${v}{≔}\left[{1.}{,}{3.146264370}{,}{9.898979486}{,}{31.14480645}{,}{97.98979486}\right]$ (4)
 > $u≔\mathrm{LinearDependency}\left(v\right)$
 ${u}{≔}\left[{1}{,}{0}{,}{-10}{,}{0}{,}{1}\right]$ (5)
 > $\mathrm{add}\left(u\left[i\right]v\left[i\right],i=1..5\right)$
 ${0.}$ (6)
 > $m≔\mathrm{add}\left(u\left[i\right]{z}^{i-1},i=1..5\right)$
 ${m}{≔}{{z}}^{{4}}{-}{10}{}{{z}}^{{2}}{+}{1}$ (7)
 > $\mathrm{simplify}\left(\mathrm{eval}\left(m,z=r\right)\right)$
 ${0}$ (8)
 > $r≔1+{\left(-2\right)}^{\frac{1}{3}}$
 ${r}{≔}{1}{+}{\left({-2}\right)}^{{1}}{{3}}}$ (9)
 > $v≔\mathrm{Vector}\left(\mathrm{expand}\left(\left[\mathrm{seq}\left({r}^{i},i=0..4\right)\right],12\right)\right)$
 ${v}{≔}\left[\begin{array}{c}{1}\\ {1}{+}{\left({-2}\right)}^{{1}}{{3}}}\\ {1}{+}{2}{}{\left({-2}\right)}^{{1}}{{3}}}{+}{\left({-2}\right)}^{{2}}{{3}}}\\ {-}{1}{+}{3}{}{\left({-2}\right)}^{{1}}{{3}}}{+}{3}{}{\left({-2}\right)}^{{2}}{{3}}}\\ {-}{7}{+}{2}{}{\left({-2}\right)}^{{1}}{{3}}}{+}{6}{}{\left({-2}\right)}^{{2}}{{3}}}\end{array}\right]$ (10)
 > $v≔\mathrm{evalf}\left(v,12\right):$$v≔\mathrm{evalf}\left(v\right)$
 ${v}{≔}\left[\begin{array}{c}{1.}\\ {1.629960525}{+}{1.091123636}{}{I}\\ {1.466220524}{+}{3.556976909}{}{I}\\ {-1.491220003}{+}{7.397559819}{}{I}\\ {-10.50228211}{+}{10.43062509}{}{I}\end{array}\right]$ (11)
 > $u≔\mathrm{LinearDependency}\left(v,\mathrm{method}=\mathrm{LLL}\right)$
 ${u}{≔}\left[\begin{array}{c}{-1}\\ {-2}\\ {6}\\ {-4}\\ {1}\end{array}\right]$ (12)
 > $\mathrm{add}\left(u\left[i\right]v\left[i\right],i=1..5\right)$
 ${-}{1.}{}{{10}}^{{-8}}{}{I}$ (13)
 > $m≔\mathrm{add}\left(u\left[i\right]{z}^{i-1},i=1..5\right)$
 ${m}{≔}{{z}}^{{4}}{-}{4}{}{{z}}^{{3}}{+}{6}{}{{z}}^{{2}}{-}{2}{}{z}{-}{1}$ (14)
 > $\mathrm{simplify}\left(\mathrm{eval}\left(m,z=r\right)\right)$
 ${0}$ (15)
 > $\mathrm{solve}\left(m=0,z\right)$
 ${1}{,}{-}{{2}}^{{1}}{{3}}}{+}{1}{,}\frac{{{2}}^{{1}}{{3}}}}{{2}}{-}\frac{{I}{}\sqrt{{3}}{}{{2}}^{{1}}{{3}}}}{{2}}{+}{1}{,}\frac{{{2}}^{{1}}{{3}}}}{{2}}{+}\frac{{I}{}\sqrt{{3}}{}{{2}}^{{1}}{{3}}}}{{2}}{+}{1}$ (16)
 > $\mathrm{evalc}\left(r\right)$
 $\frac{{{2}}^{{1}}{{3}}}}{{2}}{+}\frac{{I}{}\sqrt{{3}}{}{{2}}^{{1}}{{3}}}}{{2}}{+}{1}$ (17)