In 1969, Krawczyk [Kra69] developed an interval arithmetic version of Newton’s method.
Later in 1977 Moore [Moo77] recognized that Krawczyk’s method can be used to certify the existence and uniqueness of a solution to a system of nonlinear equations. Interval arithmetic and interval Newton’s method are a prominent tool in many areas of applied mathematics;
e.g., in chemical engineering [GS05], thermodynamics [GD05] and robotics [KSS15].
The results in this section are stated for square polynomial systems but they hold equally for square systems of rational functions. Krawczyk’s method is even valid for general square systems of analytic functions. Nevertheless, all statements here are only formulated for polynomial systems. We think that this simplifies the exposition.
Krawczyk’s method
In this section, we recall Krawczyk’s method for zeros of polynomial systems. First, we need three definitions.
Definition 5.2 (Interval enclosure). Let F :Cn→Cn be a system of polynomials. A map
□F : ICn → ICn is an interval enclosure of the system F if for every I ∈ ICn we have {F(x)|x∈I} ⊆□F(I).
In the rest of this chapter, we use the notation □F to denote the interval enclosure of F. Also, we do not distinguish between a point x ∈ Cn and the complex interval [Re(x),Re(x)] +i[Im(x),Im(x)] defined by x. We simply use the symbol “x” for both terms so that □F(x) is well-defined.
Definition 5.3 (Interval matrix norm). Let A∈ICn×n. We define the operator norm of A as ∥A∥∞ := max
B∈Amax
v∈Cn
∥Bv∥∞
∥v∥∞ ,where ∥(v1, . . . , vn)∥∞ = max1≤i≤n|vi| is the infinity norm in Cn.
Next, we introduce an interval version of the Newton operator, theKrawczyk operator[Kra69].
Definition 5.4. Let F : Cn → Cn be a system of polynomials, and JF be its Jacobian matrix seen as a functionCn→ Cn×n. Let □F be an interval enclosure of F and□JF be an interval enclosure of JF. Furthermore, let I ∈ICn andx∈Cn and let Y ∈Cn×n be an invertible matrix. We define the Krawczyk operator
Kx,Y(I) :=x−Y ·□F(x) + (1n−Y ·□JF(I))(I−x).
Here,1n is then×n-identity matrix.
Remark 5.5. In the literature, Kx,Y(I) is often defined using F(x) and not □F(x). Here, we use this definition, because in practice it is usually not feasible to evaluateF(x)exactly.
Instead,F(x) is replaced by an interval enclosure.
Remark 5.6. The second part of Theorem 5.7 motivates to find a matrix Y ∈ Cn×n such that ||1n−Y ·□JF(I)||∞ is minimized. A good choice is an approximation of the inverse ofJF(x).
We are now ready to state the theorem behind Krawczyk’s method. The first proof for real interval arithmetic is due to Moore [Moo77]. One of the few sources that states the theorem in the complex setting is [BLL19]. For completeness, we recall their proof in this section. Note that all the data in the theorem can be computed using interval arithmetic.
Theorem 5.7. Let F :Cn →Cn be a system of polynomials and I ∈ICn. Letx ∈I and Y ∈Cn×n be an invertible complexn×nmatrix. The following holds.
1. If Kx,Y(I)⊂I, there is a zero ofF in I. 2. If additionally √
2∥1n−Y□JF(I)∥∞<1, then F has exactly one zero in I.
To simplify our language when talking about intervals I ∈ ICn satisfying Theorem 5.7, we introduce the following definitions.
Definition 5.8. Let F : Cn → Cn be a square system of polynomials and I ∈ ICn. Let Kx,Y(I)be the associated Krawczyk operator (see Definition 5.4). If there exists an invertible matrixY ∈Cn×n such that Kx,Y(I)⊂I, we say that I is an interval approximate zero F. We callI astrong interval approximate zero ofF if in addition√
2∥1n−Y□JF(I)∥∞<1. Definition 5.9. If I is an interval approximate zero, then, by Theorem 5.7, I contains a zero of F. We call such a zero anassociated zero ofI. IfI is a strong interval approximate zero, then there is a unique associated zero and we refer to is as the associated zero ofI.
The notion of strong interval approximate zero is stronger than the definition suggests at first sight. We do not only certify that a unique zero of F exists inside I but we even certify that we can approximate this zero with arbitrary precision. This is shown in the next proposition. We prove the proposition at the end of this section.
Proposition 5.10. Let I be a strong interval approximate zero of F and let x∗ ∈I be the unique zero of F inside I. Let x ∈ I be any point in I. We define x0 := x and for all i≥ 1 we define the iterates xi := xi−1−Y F(xi−1), where Y ∈ Cn×n is the matrix from Definition 5.8. Then, the sequence (xi)i≥0 converges tox∗.
The idea for the proof of both Theorem 5.7 and Proposition 5.10 is to verify that for strong interval approximate zeros I the map GY(x) = x−Y ·F(x) defines a contraction on I. If this is true, by Banach’s Fixed Point Theorem there is exactly one fixed-point of this map inI. SinceY is invertible, this implies that there is exactly one zero toF(x) inI. Before we give the proof of Theorem 5.7, we need a lemma. It is a direct sequence of a complex version of the mean-value theorem which is shown implicitly in the proof of [BLL19, Lemma 2].
Lemma 5.11. Fix a matrixY ∈Cn×n and defineGY(x) =x−Y F(x). LetI ∈ICn be an interval vector and x, z ∈I. Then, we have
1. GY(z)−GY(x)∈(1n−Y ·□JF(I)) Re(z−x) + (1n−Y ·□JF(I))iIm(z−x).
2. GY(I)⊂Kx,Y(I).
The following proof is adapted from [BLL19, Lemma 2].
Proof of Lemma 5.11. In the proof, we abbreviateG:=GY. We first show the second part assuming the first part of the lemma. Then, we prove the first part. We fix an interval I ∈ICn andx, z∈Cn.
For the second part, we have to show that for all I ∈ICn we haveG(I)⊂Kx,Y(I). To show this, we define the interval matrix M := (1n−Y□JF(I))∈ICn×n. By definition of Kx,Y we have G(x) +M(I −x)⊂Kx,Y(I). Thus, we have to show that G(z)−G(x)∈ M(I −x), since z ∈ I is arbitrary. The first part of the lemma implies that we can find
matricesM1, M2 ∈M such thatG(z)−G(x) =M1Re(z−x)+iM2Im(z−x). Decomposing the matrices into real and imaginary part we find
G(z)−G(x) =Re(M1)Re(z−x)−Im(M2)Im(z−x)+
i(Im(M1)Re(z−x) + Re(M2)Im(z−x)).
Since z−x ∈ I and by definition of the complex interval multiplication from (5.2) and the interval matrix-vector-multiplication (5.3), we see that G(z)−G(x)∈M(I −x). This finishes the proof for the second part.
The first part of the lemma may be shown entry-wise. We will show this by combining a complex version of the mean value theorem with the following observation: JG(x) = 1n−Y ·JF(x), so we have the inclusion
JG(I) =1n−Y ·JF(I)⊆1n−Y ·□JF(I). (5.4) We relate G(z)−G(x) to (5.4) using the mean value theorem. First, we define w :=
Re(z)+iIm(x). Let1≤j≤nand letGj denote thej-th entry ofG. We define the function h(t) :=Gj(tz+(1−t)w). The real and imaginary part ofh(t)are real differentiable functions of the real variable t. The mean value theorem can be applied, and we find 0 < t1, t2 <1 such that Re(h(1))−Re(h(0)) = dtdRe(h(t1)) and Im(h(1))−Im(h(0)) = dtdIm(h(t2)). Settingc1 =t1z+ (1−t1)w andc2 =t2z+ (1−t2)w this implies
Gj(w)−Gj(z) = (∇ReRe(Gj(c1)))T(z−w) +i∇ReIm(G(c2)))T(z−w),
where∇ReGdenotes the vector of partial derivatives with respect to the real variable. Let us denote by G′j the complex derivative of Gj; that is, G′j :Cn→Cn as a function. From the Cauchy Riemann equations it follows that ∇ReRe(Gj(c1)) = Re(G′j(c1)) and likewise
∇ReIm(G(c2)) = Im(G′(c2)). This yieldsGj(z)−Gj(w) = (Re(G′j(c1)+iIm(G′j(c2)))T(z−
w). Putting these equations ranging over jtogether we findG(z)−G(w) = (Re(JG(c1)) + iIm(JG(c2)))(z−w). By construction, c1 and c2 are contained in I, because w andz are contained in I, andI is a product of rectangles and thus convex. Combined with (5.4) this yields
G(z)−G(w)∈(1n−Y ·□JF(I))(z−w).
Using essentially the same arguments for the path fromx to w, we also find G(w)−G(x)∈(1n−Y ·□JF(I))(w−x).
By construction,z−w=iIm(z−x) andw−x= Re(z−x). This implies
G(z)−G(x)∈(1n−Y ·□JF(I)) Re(z−x) + (1n−Y ·□JF(I))iIm(z−x).
This finishes the proof.
Proof of Theorem 5.7 and Proposition 5.10. We fixY ∈Cn×n. The second part of Lemma 5.11 implies that, if we haveKx,Y(I)⊆I, thenGY(I)⊆I. Brouwer’s fixed point Theorem
shows thatGY has a fixed point inI. SinceY is assumed to be invertible, the fixed point is a zero of F. This finishes the proof for the first part of Theorem 5.7. For the second part, let z1, z2 ∈I. The first part of Lemma 5.11 implies
GY(z1)−GY(z2)∈(1n−Y ·□JF(I))Re(z1−z2) + (1n−Y ·□JF(I))iIm(z1−z2).
(Note that we can’t apply the distributivity law because of Theorem 5.1 3.). Applying norms and using submultiplicativity yields
∥GY(z1)−GY(z2)∥∞≤ ∥(1n−Y ·□JF(I))∥∞(∥Re(z1−z2)∥∞+∥Im(z1−z2)∥∞).
Since ∥Re(z1−z2)∥∞+∥Im(z1−z2)∥∞≤√
2∥z1−z2∥∞, it holds
∥GY(z1)−GY(z2)∥∞≤√
2∥1n−Y ·□JF(I)∥∞∥z1−z2∥∞. By assumption √
2∥1n−Y ·□JF(I)∥∞is smaller than 1 so GY is a contraction. Banach’s Fixed Point Theorem implies that GY has a unique zero in I. This shows the second part of Theorem 5.7. The fact that GY is a contraction on I also proves Proposition 5.10.
Certifying Reality
For many applications, only the real zeros of a polynomial system are of interest. Since numerical homotopy continuation computes inCn, it is important to have a rigorous method to determine whether a zero is real.
Recall from Definition 5.8 the notion of strong interval approximate zero.
Lemma 5.12. Let F : Cn → Cn be a real square system of polynomials and I ∈ ICn a strong interval approximate zero of F. Then there exists x ∈ I and Y ∈ Cn×n satisfying Kx,Y(I) ⊂ I and √
2∥1n−Y□JF(I)∥∞ < 1. If additionally {z¯|z ∈Kx,Y(I)} ⊂ I, the associated zero of I is real.
Proof. Theorem 5.7 implies that F has a unique zero s∈Kx,Y(I)⊂ I. Since F is a real polynomial system, it follows that also the element-wise complex conjugate s¯is a zero of F. If we have that s¯∈ {z¯|z ∈Kx,Y(I)} ⊂I, then s¯ = s, since otherwise s¯ and s would be two distinct zeros of F in I, contradicting the uniqueness result from Theorem 5.7.
For a wide range of applications, positive real zeros are of particular interest.
Corollary 5.13. Let F :Cn → Cn be a real square system of polynomials and I ∈ ICn a strong interval approximate zero ofF satisfying the conditions of Lemma 5.12. If Re(I)>0, then the associated zero of I is real and positive.
If the reality test in Lemma 5.12 fails for a strong interval approximate zero I ∈Cn, then this does not necessarily mean that the associated zero ofI is not real. A sufficient condition that I is not real is that there is a coordinate such that the imaginary part of it does not contain zero.
Lemma 5.14. Let F(x) be a square system of polynomials or rational functions and let I ∈ICn be a strong interval approximate zero ofF. If there existsk∈ {1, . . . , n}such that 0∈/ Im(Ik), then the associated zero of I is not real.
Proof. The associated zero x of I is contained in I. Since 0 ∈/ Im(Ik), it follows xk ∈/ R andx /∈Rn.
Now assume that the certification routine produced a list I ofm distinct strong interval approximate zeros for a given system F and that m also agrees with the theoretical upper bound on the number of isolated zeros of F. If we apply Lemma 5.12 to Ik ∈ I, then we obtain only a lower bound, say r, on the number of real zeros of F. However, combined with Lemma 5.14, we can also obtain anupper bound of the number of real zeros. If these two bounds agree, we obtain a certificate thatF hasexactly r real zeros. An application of this is, e.g., the study of the distribution of the number of real solutions of the power flow equations [LZBL20].