The Extended ABM Algorithm

Now present the so-called extended ABM algorithm and discuss its properties.

Algorithm 24: Extended ABM Algorithm

Input: A tuple of ane points X = [p1, ..., ps] with pi∈ Cⁿ, V = [v1, ..., vs] with vi∈ C and kVk > εmachine, small numbers ε ≥ 0, 0 ≤ τ < kVk, D ∈ N such that D < s, and a degree compatible term ordering σ on Tⁿ

Output: An approximate O-border basis G, a list H of polynomials (see Theorem 4.4.3 for details)

1 d := 1, O := [1], G := [ ], H := [ ], M := (1, ..., 1)^tr∈ Mat_s,1(C);

2 L := [t₁, ..., t_`] = all terms of degree 1 ordered decreasingly w.r.t. σ;

3 repeat

4 for i := 1 to ` do

5 m := |O|;

// We assume that O = [om, ..., o1]

6 A := (eval_X(t_i) , M );

7 B := A^∗A;

8 γ := smallest eigenvalue of B ;

9 if √

γ ≤ ε then

10 s := (s_m+1, s_m, ..., s₁) := norm one eigenvector of B w.r.t. to γ ;

11 g := sm+1ti+ smom+ ... + s1o1;

12 G := concat(G, [g]);

13 else

14 s := (s_m+1, s_m, ..., s₁) = solution of the least squares problem mins

As − V^tr

(compare algorithms in Section 2.11);

15 if As − V^tr

≤ τ then

16 h := sm+1ti+ smom+ ... + s1o1;

17 H := concat(H, [h]);

18 end

19 O := concat([ti] , O);

20 M := A;

21 end

22 end

23 d := d + 1;

24 L := [t₁, ..., t_`] := all terms of degree d in ∂O ordered decreasingly w.r.t. σ;

25 until L = [ ] or d > D;

26 return (G, O, H);

Theorem 4.4.3. This algorithm computes three sets G = {g1, ..., g_ν}, O = {t1, ..., t_µ}, and H = {h₁, ..., h_κ} which have the following properties:

1. All the polynomials in G are unitary and generate an ε-approximate vanishing ideal of X.

2. There is no unitary polynomial in hOi_C which vanishes ε-approximately on X.

3. For all polynomials hi in H we have keval_X(h_i) − Vk ≤ τ .

4. The condition number κ of the inhomogeneous least squares problem which is solved in line 14 is bounded by

s · kXk^D_max

ε + s · kXk^D_max ε

!2s

1 −(kVk − τ )² kVk²

 kVk − τ kVk .

If additionally the parameter D is chosen large enough, e.g. D = s − 1, such that the algorithm does not terminate prematurely, then also the following properties hold:

5. If O is an order ideal of terms, then the set ˜G = { (1/ LC_σ(g)) g| g ∈ G} is an O-border prebasis.

6. If O is an order ideal of terms, then the set ˜G is an δ-approximate border basis with

δ = 2 kXk_max/ min

i |γ_i| + ν/



mini |γ_i|

2

.

Here kXk_max denotes the maximal absolute coordinate in X and mini|γ_i| the minimal border coecient of all polynomials in G.

Proof. First of all we note that the ABM and the extended ABM algorithm share the same basic structure. The latter algorithm only contains some additional processing steps in lines 14 to 18 and an additional degree check in line 25.

The individual steps are well-dened because of the same reasons as in the ABM algorithm. An explanation is contained in the rst part of the proof of Theorem 4.3.1.

Next we discuss niteness. In line 25 termination is assured by checking if the current degree d exceeds a user specied upper bound D. If so the execution will stop and a possibly partial result, i.e. no complete approximate border basis, will be returned. However, even if we would drop the additional check in line 25 the algorithm would still terminate. This is shown in the second part of the proof of Theorem 4.3.1.

Claims 1, 2, 5, and 6 in fact refer to properties of the result of the ABM algorithm, which we have already discussed and proven in Theorem 4.3.1. Note, that we have to assume that D is chosen large enough, i.e. D = s − 1, such that a full approximate border basis will be returned in the end.

Claim 3 holds because the polynomials which are added to H are tested to have this property in line 15.

For each polynomial hi∈ H we let Ai = eval_X(supp (hi)). Let us denote the coecient vector of h_i with respect to supp (hi) by ci. This means that evalX(h_i) = A_ic_i holds. To prove claim 4 let us rst cast a result from Theorem 2.12.1 into our setting which states that the general condition number of the inhomogeneous least squares problem is

κ (Ai) +tan (θi) κ (Ai)²

η_i ,

where cos (θi) = ^keval_kVk^X(hi)k = ^kA_kVk^icik for all hi ∈ H and ηi = ^kA_kA^ikkcik

icik . First we observe that 1 ≤ η_i ≤ κ (A). The rst inequality follows from kAk kcik ≥ kAc_ik. For the second inequality, note that ci is not the zero vector as kVk 6= 0 and A has only a trivial kernel which means that kAcik 6= 0. With the help of Proposition 2.6.5 we can now compute κ (Ai) = kA_ik

A⁺_i = kA_ikkA⁺_i k^kAicik

kA_icik ≥ kA_ikkA⁺_iAicik

kA_icik = ^kA_kA^ikkcik

icik . Therefore, we obtain κ (Ai) +tan (θi) κ (Ai)²

ηi

≤ κ (A_i) + tan (θi) κ (Ai)². Next we use the equationtan cos⁻¹(x)
=

√ 1−x²

x in order to obtain tan (θ_i) =

s

1 −kA_ic_ik² kVk²

 kA_ic_ik kVk .

Because of the triangle inequality and because of claim 5 we can establish that kAicik ≥ kVk − kA_ic_i− Vk ≥ kVk − τ > 0. So

tan (θ_i) ≤ s

1 −(kVk − τ )² kVk²

 kVk − τ kVk . Now it remains to bound κ (Ai) = kAik

A⁺_i

. Because we make sure during the computation that the evaluation matrices Ai have no singular values smaller than ε we thus know that A⁺_i

< ¹_ε. Additionally, for any Ai ∈ Mat_m,n(C) the inequality kAik ≤ √

mn kA_ik_max holds (see Proposition 2.3.20). In our case the matrix dimensions can at most be m = n = s. The entries of the evaluation matrix are products of powers of the entries of X. As O can at most contain s elements the maximal degree is naturally bounded by s, but as we are using an articial degree bound in form of the input parameter D (see line 25 of the algorithm) we can be more precise and state that the limit is in fact D. So kAik ≤ s · kXk^D_max where kXk_max is the maximal absolute entry in the input data set X. Now we have established that κ (Ai) ≤ s · kXk^D_max¹_ε. If we collect all the facts we obtain the inequality

κ = κ (Ai) ≤ s · kXk^D_max

ε + s · kXk^D_max ε

!2s

1 −(kVk − τ )² kVk²

 kVk − τ kVk , which concludes the proof.

Remark 4.4.4. In case kVk is very small, e.g. kVk ≈ εmachine, we are essentially considering a homogeneous least squares problem. Thus, the ABM algorithm (22) should be used instead of the extended ABM algorithm (24), as the polynomials in H (if any) have very small coecients.

Now we tend to have a look at an example which demonstrates the operation of the extended ABM algorithm.

Example 4.4.5. Let P = R [x1], X = [(−2) , (−1) , (−0.01) , (0) , (0.01) , (1) , (2)],

V = [(−1) , (0) , (0.99) , (1) , (0.99) , (0) , (−1)] , and let σ be the DegRevLex term ordering. We now apply the extended ABM algorithm (24) with ε = 0.1, τ = 10⁻³, and D = 5:

• d = 1, O = {1}, G = ∅, M = (1, ..., 1)^tr, and L = [x1]

• The algorithm terminates because L is empty. Finally, we obtain the sets G = {g1}, H = {h₁} and O = x⁴₁, x³₁, x²₁, x₁, 1

, where G is an approximate O-border basis.

Remark 4.4.6. The extended ABM algorithm can be used to check if a polynomial function (in general or up to a specied degree D) exists such that V can be expressed τ -approximately

by the input data X. For this purpose we need to set the parameter ε to 0. If H is empty upon termination of the algorithm, then we have the guarantee that no such relations exists.

Next, we look at an example that emphasises that in general no solution needs to exist, which means that H will contain no polynomials.

Example 4.4.7. [Existence of Solutions]

Let P = C [x1, x₂], X = (0, 0), V = (1, 0), ε = 0 and τ = 0.1. When we use the extended ABM algorithm, all least squares problems we have to solve in line 14 are of the form

mins

0 · · · 0 1 0 · · · 0 1

!

s − 1 0

!

with s = (sm+1, ..., s₁)^tr∈ C^m+1. A polynomial is added to H if it satises q

|s₁− 1|²+ |s1|² < 0.1.

This is equivalent to the following conditions:

(s1− 1) (¯s1− 1) + s₁s¯1 < 0.01 2s1¯s1− s₁− ¯s1+ 1 < 0.01 2 s₁s¯₁− 0.5s₁− 0.5¯s₁+ 0.5²+ 0.25

< 0.01 2 (s₁− 0.5) (¯s₁− 0.5) + 0.5 < 0.01

2 |s₁− 0.5|² < −0.49.

As we can see, there exists no number s that would satisfy this inequality.

In document Berechnung und Anwendungen Approximativer Randbasen (Page 145-149)