LinearDependency - Maple Help

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.
 • The internal working precision of the LinearDependency command corresponds to the value of Digits. For best results, the same value of Digits should be used with which the input approximation was obtained.

Examples

 > $\mathrm{with}\left(\mathrm{IntegerRelations}\right):$
 > $r≔\sqrt{2}+\sqrt{3}$
 ${r}{≔}\sqrt{{2}}{+}\sqrt{{3}}$ (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)
 > $\mathrm{add}\left({u}_{i}{v}_{i},i=1..5\right)$
 ${0.}$ (6)
 > $m≔\mathrm{add}\left({u}_{i}{z}^{i-1},i=1..5\right)$
 ${m}{≔}{{z}}^{{4}}{-}{10}{}{{z}}^{{2}}{+}{1}$ (7)
 > $\mathrm{simplify}\left(\genfrac{}{}{0}{}{m}{\phantom{z=r}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{m}}{z=r}\right)$
 ${0}$ (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)
 > $\mathrm{add}\left({u}_{i}{v}_{i},i=1..5\right)$
 ${0.}{-}{1.}{×}{{10}}^{{-8}}{}{I}$ (13)
 > $m≔\mathrm{add}\left({u}_{i}{z}^{i-1},i=1..5\right)$
 ${m}{≔}{{z}}^{{4}}{-}{4}{}{{z}}^{{3}}{+}{6}{}{{z}}^{{2}}{-}{2}{}{z}{-}{1}$ (14)
 > $\mathrm{simplify}\left(\genfrac{}{}{0}{}{m}{\phantom{z=r}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}|\phantom{\rule[-0.0ex]{0.1em}{0.0ex}}\genfrac{}{}{0}{}{\phantom{m}}{z=r}\right)$
 ${0}$ (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)