The OrderBasis([f1, ..., fn], x, N, [d1, ..., dn]) command computes an order basis for the functions f1, ..., fn with respect to the variable x, the degrees d1, ..., dn, and the order N. It finds all polynomial coefficients that provide an identity of the form

up to a certain number of terms and with the degree of each vi bounded. This is similar to what the IntegerRelations[PSLQ] algorithm does for finding integer relations for floating-point numbers.

More precisely, given functions fi assumed to have a series expansion about 0, the OrderBasis function returns a matrix whose columns provide a basis for the (mathematical) module defined by

That is, for every vector v of polynomials in L there exist n polynomials $\mathrm{a1},\mathrm{a2},\mathrm{\#mo(\…)},\mathrm{an}$ 

Here the degree of $\mathrm{ai}$ is bounded by $\mathrm{di}-\mathrm{degree}\left(M_{i,i}\right)$.

If there are three arguments with the third argument as a positive integer N, that is, OrderBasis([f1, ..., fn], x, N), then the degree bounds d1, ..., dn are assumed to be N, ..., N.

If there are three arguments with the third argument as a list, that is, OrderBasis([f1, ..., fn], x, [d1, ..., dn]), then the order is determined by ${\mathrm{dn}}_1+\dots +{\mathrm{dn}}_n+n-1$.

$F≔\left[1+x^2-x^7+x^{12}\,\sin \left(x\right)\,\exp \left(x\right)\right]\:$

$M≔\mathrm{OrderBasis}\left(F\,x\,8\,\left[3\,1\,2\right]\right)$

$M≔\left[\begin{array}{ccc}{x}^4+\frac{68238360}{4251353}{x}^3+\frac{511942344}{4251353}{x}^2+\frac{971729136}{4251353}x-\frac{2669976}{4251353}& \frac{2011699}{12754059}{x}^3-\frac{23128}{4251353}{x}^2-\frac{4252654}{4251353}x+\frac{34692}{4251353}& \frac{5533799}{12754059}{x}^3+\frac{17049452}{4251353}{x}^2+\frac{33708898}{4251353}x-\frac{66060}{4251353}\\ \frac{829819404}{4251353}+\frac{1293611160x}{4251353}& x^2-\frac{141659}{8502706}x+\frac{8421469}{8502706}& \frac{68785145}{8502706}+\frac{93799511x}{8502706}\\ \frac{2669976}{4251353}-\frac{1804218516x}{4251353}& -\frac{34692}{4251353}+\frac{153223x}{8502706}& x^2-\frac{136335061}{8502706}x+\frac{66060}{4251353}\end{array}\right]$

$\mathrm{map}\left(\mathrm{series}\,\mathrm{map}\left(\mathrm{expand}\,\mathrm{Matrix}\left(1\,3\,F\right)·M\right)\,x\,8\right)$

$\left[\begin{array}{ccc}O\left(x^8\right)& O\left(x^8\right)& O\left(x^8\right)\end{array}\right]$

$\mathrm{map}\left(\mathrm{degree}\,M\,x\right)$

$\left[\begin{array}{ccc}4& 3& 3\\ 1& 2& 1\\ 1& 1& 2\end{array}\right]$

$M≔\mathrm{OrderBasis}\left(F\,x\,8\right)$

$M≔\left[\begin{array}{ccc}{x}^3-\frac{69384}{2011699}{x}^2-\frac{12757962}{2011699}x+\frac{104076}{2011699}& -\frac{40543788}{2011699}{x}^2-\frac{105266112}{2011699}x+\frac{464712}{2011699}& \frac{8097740}{2011699}{x}^2+\frac{21486216}{2011699}x-\frac{76416}{2011699}\\ \frac{12754059}{2011699}{x}^2-\frac{424977}{4023398}x+\frac{25264407}{4023398}& x^3+\frac{32425320}{2011699}{x}^2-\frac{95498640}{2011699}x-\frac{30079248}{2011699}& -\frac{5533799}{2011699}{x}^2+\frac{22284705}{2011699}x+\frac{10793304}{2011699}\\ -\frac{104076}{2011699}+\frac{459669x}{4023398}& -\frac{464712}{2011699}+\frac{135810072x}{2011699}& x^2-\frac{32355936}{2011699}x+\frac{76416}{2011699}\end{array}\right]$

$\mathrm{map}\left(\mathrm{series}\,\mathrm{map}\left(\mathrm{expand}\,\mathrm{Matrix}\left(1\,3\,F\right)·M\right)\,x\,8\right)$

$\left[\begin{array}{ccc}O\left(x^8\right)& O\left(x^8\right)& O\left(x^8\right)\end{array}\right]$

Beckermann, B., and Labahn, G. "Fraction-Free Computation of Matrix Rational Interpolants and Matrix GCDs." SIAM Journal on Matrix Analysis and Applications. Vol. 22 No. 1. (2000): 114-144.

Beckermann, B. and Labahn, G. "A Uniform Approach for the Fast Computation of Matrix-Type Pade Approximants." SIAM Journal on Matrix Analysis and Applications. Vol. 15 No. 3. (1994): 804-823.