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<title>Stability radius</title>

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Stability radius

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In <a href="Mathematics" title="Mathematics">mathematics</a>, the stability radius of an <a href="Mathematical_object" title="Mathematical object">object</a> (system, <a href="Function_(mathematics)" title="Function (mathematics)">function</a>, <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>, <a href="Parameter" title="Parameter">parameter</a>) at a given nominal point is the radius of the largest <a href="Ball_(mathematics)" title="Ball (mathematics)">ball</a>, centered at the nominal point, all of whose elements satisfy pre-determined stability conditions. The picture of this intuitive notion is this:

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {p}}}</annotation>

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</math><img src="./8bd4c026f1b3413adc58b9b65e89e62bce92c85a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}}" loading="lazy"> denotes the nominal point, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> denotes the space of all possible values of the object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">, and the shaded area, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(s)}">

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<annotation encoding="application/x-tex">{\displaystyle P(s)}</annotation>

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</math><img src="./7927ac2ad8a94b105f536045494e75f3193930ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.645ex; height:2.843ex;" alt="{\displaystyle P(s)}" loading="lazy">, represents the set of points that satisfy the stability conditions. The radius of the blue circle, shown in red, is the stability radius.

<div class="mw-heading mw-heading2"><h2 id="Abstract_definition">Abstract definition</h2></div>

The formal definition of this concept varies, depending on the application area. The following abstract definition is quite useful<a href="#cite_note-zlobec09-1">[1]</a><a href="#cite_note-MS10-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}({\hat {p}}):=\max \ \{\rho \geq 0:p\in P(s),\forall p\in B(\rho ,{\hat {p}})\}}</annotation>

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</math><img src="./3c86b16df7bd32332934a18815cfec3937224c5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.469ex; height:2.843ex;" alt="{\displaystyle {\hat {\rho }}({\hat {p}}):=\max \ \{\rho \geq 0:p\in P(s),\forall p\in B(\rho ,{\hat {p}})\}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(\rho ,{\hat {p}})}">

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<annotation encoding="application/x-tex">{\displaystyle B(\rho ,{\hat {p}})}</annotation>

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</math><img src="./871c5baebef0b8b67fc8d94afd54106d2a041da1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.169ex; height:2.843ex;" alt="{\displaystyle B(\rho ,{\hat {p}})}" loading="lazy"> denotes a closed <a href="Ball_(mathematics)" title="Ball (mathematics)">ball</a> of radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> centered at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {p}}}</annotation>

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</math><img src="./8bd4c026f1b3413adc58b9b65e89e62bce92c85a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

It looks like the concept was invented in the early 1960s.<a href="#cite_note-wilf-3">[3]</a><a href="#cite_note-milne-4">[4]</a> In the 1980s it became popular in control theory<a href="#cite_note-Hindrichsen86-5">[5]</a> and optimization.<a href="#cite_note-zlobec88-6">[6]</a> It is widely used as a model of local robustness against small perturbations in a given nominal value of the object of interest.

<div class="mw-heading mw-heading2"><h2 id="Relation_to_Wald's_maximin_model">Relation to Wald's maximin model</h2></div>

It was shown<a href="#cite_note-MS10-2">[2]</a> that the stability radius model is an instance of <a href="Wald's_maximin_model" title="Wald's maximin model">Wald's maximin model</a>. That is,

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<annotation encoding="application/x-tex">{\displaystyle \max \ \{\rho \geq 0:p\in P(s),\forall p\in B(\rho ,{\hat {p}})\}\equiv \max _{\rho \geq 0}\min _{p\in B(\rho ,{\hat {p}})}f(\rho ,p)}</annotation>

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</math><img src="./d794db7709d4cd9f40b3ec4a4ca894494cc4f2fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:58.646ex; height:4.843ex;" alt="{\displaystyle \max \ \{\rho \geq 0:p\in P(s),\forall p\in B(\rho ,{\hat {p}})\}\equiv \max _{\rho \geq 0}\min _{p\in B(\rho ,{\hat {p}})}f(\rho ,p)}" loading="lazy"></dd></dl>

where

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<annotation encoding="application/x-tex">{\displaystyle f(\rho ,p)=\left\{{\begin{array}{cc}\rho &,\ p\in P(s)\\-\infty &,\ p\notin P(s)\end{array}}\right.}</annotation>

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</math><img src="./8f17b20be64271399f7cbe600d649fe2f02c3241.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.811ex; height:6.176ex;" alt="{\displaystyle f(\rho ,p)=\left\{{\begin{array}{cc}\rho &,\ p\in P(s)\\-\infty &,\ p\notin P(s)\end{array}}\right.}" loading="lazy"></dd></dl>

The large penalty (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

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<annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation>

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</math><img src="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy">) is a device to force the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max }">

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<annotation encoding="application/x-tex">{\displaystyle \max }</annotation>

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</math><img src="./b8e49fca3e322708b32d21eaa8b095dc05f09538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.326ex; height:1.676ex;" alt="{\displaystyle \max }" loading="lazy"> player not to perturb the nominal value beyond the stability radius of the system. It is an indication that the stability model is a model of local stability/robustness, rather than a global one.

<div class="mw-heading mw-heading2"><h2 id="Info-gap_decision_theory">Info-gap decision theory</h2></div>

<a href="Info-gap_decision_theory" title="Info-gap decision theory">Info-gap decision theory</a> is a recent non-probabilistic decision theory. It is claimed to be radically different from all current theories of decision under uncertainty. But it has been shown<a href="#cite_note-MS10-2">[2]</a> that its robustness model, namely

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\alpha }}(q,{\tilde {u}}):=\max \ \{\alpha \geq 0:r_{c}\leq R(q,u),\forall u\in U(\alpha ,{\tilde {u}})\}}</annotation>

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</math><img src="./1dfd239a829b2899d6a70709e207e047b055e9ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.817ex; height:2.843ex;" alt="{\displaystyle {\hat {\alpha }}(q,{\tilde {u}}):=\max \ \{\alpha \geq 0:r_{c}\leq R(q,u),\forall u\in U(\alpha ,{\tilde {u}})\}}" loading="lazy"></dd></dl>

is actually a stability radius model characterized by a simple stability requirement of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{c}\leq R(q,u)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle r_{c}\leq R(q,u)}</annotation>

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</math><img src="./ee3af5b7d9e284a40e8d85c6ecb0d33daa4643f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.098ex; height:2.843ex;" alt="{\displaystyle r_{c}\leq R(q,u)}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"> denotes the decision under consideration, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> denotes the parameter of interest, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {u}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {u}}}</annotation>

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</math><img src="./3220dce003e04258dfbb3ade91566109e081e0e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\tilde {u}}}" loading="lazy"> denotes the estimate of the true value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(\alpha ,{\tilde {u}})}">

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<annotation encoding="application/x-tex">{\displaystyle U(\alpha ,{\tilde {u}})}</annotation>

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</math><img src="./220367de81c563c88778c3b83b10a7020515567a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.443ex; height:2.843ex;" alt="{\displaystyle U(\alpha ,{\tilde {u}})}" loading="lazy"> denotes a ball of radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

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</math><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"> centered at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {u}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {u}}}</annotation>

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</math><img src="./3220dce003e04258dfbb3ade91566109e081e0e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\tilde {u}}}" loading="lazy">.

Since stability radius models are designed to deal with small perturbations in the nominal value of a parameter, info-gap's robustness model measures the local robustness of decisions in the neighborhood of the estimate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {u}}}">

<mover>

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {u}}}</annotation>

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</math><img src="./3220dce003e04258dfbb3ade91566109e081e0e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:2.176ex;" alt="{\displaystyle {\tilde {u}}}" loading="lazy">.

Sniedovich<a href="#cite_note-MS10-2">[2]</a> argues that for this reason the theory is unsuitable for the treatment of severe uncertainty characterized by a poor estimate and a vast uncertainty space.

<div class="mw-heading mw-heading2"><h2 id="Alternate_definition">Alternate definition</h2></div>

There are cases where it is more convenient to define the stability radius slightly different. For example, in many applications in control theory the radius of stability is defined as the size of the smallest destabilizing perturbation in the nominal value of the parameter of interest.<a href="#cite_note-paice98-7">[7]</a> The picture is this:

More formally,

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<annotation encoding="application/x-tex">{\displaystyle {\hat {\rho }}(q):=\min _{p\notin P(s)}dist(p,{\hat {p}})}</annotation>

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</math><img src="./8606e97a6497873805ba8e26281c981d279e6812.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.915ex; height:4.509ex;" alt="{\displaystyle {\hat {\rho }}(q):=\min _{p\notin P(s)}dist(p,{\hat {p}})}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dist(p,{\hat {p}})}">

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<annotation encoding="application/x-tex">{\displaystyle dist(p,{\hat {p}})}</annotation>

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</math><img src="./7222f2ec075d5a843b60d8ee105e27783b468ddc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.32ex; height:2.843ex;" alt="{\displaystyle dist(p,{\hat {p}})}" loading="lazy"> denotes the distance of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in P}">

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<annotation encoding="application/x-tex">{\displaystyle p\in P}</annotation>

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</math><img src="./f4dc1da33d78a2bdae06c9edff726feb126dae34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.845ex; height:2.509ex;" alt="{\displaystyle p\in P}" loading="lazy"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {p}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {p}}}</annotation>

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</math><img src="./8bd4c026f1b3413adc58b9b65e89e62bce92c85a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle {\hat {p}}}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Stability_radius_of_functions">Stability radius of functions</h2></div>

The stability radius of a <a href="Continuous_function" title="Continuous function">continuous function</a> f (in a <a href="Functional_space" class="mw-redirect" title="Functional space">functional space</a> F) with respect to an <a href="Open_set" title="Open set">open</a> stability domain D is the <a href="Distance" title="Distance">distance</a> between f and the <a href="Set_(mathematics)" title="Set (mathematics)">set</a> of unstable functions (with respect to D). We say that a function is stable with respect to D if its spectrum is in D. Here, the notion of spectrum is defined on a case-by-case basis, as explained below.

<div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3></div>

Formally, if we denote the set of stable functions by S(D) and the stability radius by r(f,D), then:

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<annotation encoding="application/x-tex">{\displaystyle r(f,D)=\inf _{g\in C}\{\|g\|:f+g\notin S(D)\},}</annotation>

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</math><img src="./b5eca863365134243fea1eca870bc83e793d845c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.986ex; height:4.343ex;" alt="{\displaystyle r(f,D)=\inf _{g\in C}\{\|g\|:f+g\notin S(D)\},}" loading="lazy"></dd></dl>

where C is a subset of F.

Note that if f is already unstable (with respect to D), then r(f,D)=0 (as long as C contains zero).

<div class="mw-heading mw-heading3"><h3 id="Applications">Applications</h3></div>

The notion of stability radius is generally applied to <a href="Special_function" class="mw-redirect" title="Special function">special functions</a> as <a href="Polynomial" title="Polynomial">polynomials</a> (the spectrum is then the roots) and <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> (the spectrum is the <a href="Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a>). The case where C is a proper subset of F permits us to consider structured <a href="Perturbation_theory" title="Perturbation theory">perturbations</a> (e.g. for a matrix, we could only need perturbations on the last row). It is an interesting measure of robustness, for example in <a href="Control_theory" title="Control theory">control theory</a>.

<div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3></div>

Let f be a (<a href="Complex_number" title="Complex number">complex</a>) polynomial of degree n, C=F be the set of polynomials of degree less than (or equal to) n (which we identify here with the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n+1}}</annotation>

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</math><img src="./a6690b0bbf64200f073d88bdbc1bb4bc01de9b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.997ex; height:2.676ex;" alt="{\displaystyle \mathbb {C} ^{n+1}}" loading="lazy"> of coefficients). We take for D the open <a href="Unit_disk" title="Unit disk">unit disk</a>, which means we are looking for the distance between a polynomial and the set of Schur <a href="Stable_polynomial" title="Stable polynomial">stable polynomials</a>. Then:

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<annotation encoding="application/x-tex">{\displaystyle r(f,D)=\inf _{z\in \partial D}{\frac {|f(z)|}{\|q(z)\|}},}</annotation>

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</math><img src="./bc9b6f0c5a70f4e5d43d8de8401be9e5c94679ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.514ex; height:6.509ex;" alt="{\displaystyle r(f,D)=\inf _{z\in \partial D}{\frac {|f(z)|}{\|q(z)\|}},}" loading="lazy"></dd></dl>

where q contains each basis vector (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(z)=(1,z,\ldots ,z^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle q(z)=(1,z,\ldots ,z^{n})}</annotation>

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</math><img src="./9fb992f6f4f405c9a8e5c4302b17cfa0de10aed1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.646ex; height:2.843ex;" alt="{\displaystyle q(z)=(1,z,\ldots ,z^{n})}" loading="lazy"> when q is the usual power basis). This result means that the stability radius is bound with the minimal value that f reaches on the unit circle.

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

<ul><li>The polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(z)=z^{8}-9/10}">

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<annotation encoding="application/x-tex">{\displaystyle f(z)=z^{8}-9/10}</annotation>

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</math><img src="./a8e9f2f846958da78c347fc3f7d80b2f32ccc9b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.909ex; height:3.176ex;" alt="{\displaystyle f(z)=z^{8}-9/10}" loading="lazy"> (whose zeros are the 8th-roots of 0.9) has a stability radius of 1/80 if q is the power basis and the norm is the infinity norm. So there must exist a polynomial g with (infinity) norm 1/90 such that f+g has (at least) a root on the unit circle. Such a g is for example <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(z)=-1/90\sum _{i=0}^{8}z^{i}}">

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<annotation encoding="application/x-tex">{\displaystyle g(z)=-1/90\sum _{i=0}^{8}z^{i}}</annotation>

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</math><img src="./65d37915c71766fbdfba925d07bd8bb1c62db898.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.589ex; height:7.343ex;" alt="{\displaystyle g(z)=-1/90\sum _{i=0}^{8}z^{i}}" loading="lazy">. Indeed, (f+g)(1)=0 and 1 is on the unit circle, which means that f+g is unstable.</li></ul>

<ul><li><a href="Stable_polynomial" title="Stable polynomial">stable polynomial</a></li>

<li><a href="Wald's_maximin_model" title="Wald's maximin model">Wald's maximin model</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-zlobec09-1"><a href="#cite_ref-zlobec09_1-0">^</a> Zlobec S. (2009). Nondifferentiable optimization: Parametric programming. Pp. 2607-2615, in Encyclopedia of Optimization, Floudas C.A and Pardalos, P.M. editors, Springer.

</li>

<li id="cite_note-MS10-2">^ <a href="#cite_ref-MS10_2-0">a</a> <a href="#cite_ref-MS10_2-1">b</a> <a href="#cite_ref-MS10_2-2">c</a> <a href="#cite_ref-MS10_2-3">d</a> Sniedovich, M. (2010). A bird's view of info-gap decision theory. Journal of Risk Finance, 11(3), 268-283.

</li>

<li id="cite_note-wilf-3"><a href="#cite_ref-wilf_3-0">^</a> Wilf, H.S. (1960). Maximally stable numerical integration. Journal of the Society for Industrial and Applied Mathematics, 8(3),537-540.

</li>

<li id="cite_note-milne-4"><a href="#cite_ref-milne_4-0">^</a> Milne, W.E., and Reynolds, R.R. (1962). Fifth-order methods for the numerical solution of ordinary differential equations. Journal of the ACM, 9(1), 64-70.

</li>

<li id="cite_note-Hindrichsen86-5"><a href="#cite_ref-Hindrichsen86_5-0">^</a> Hindrichsen, D. and Pritchard, A.J. (1986). Stability radii of linear systems, Systems and Control Letters, 7, 1-10.

</li>

<li id="cite_note-zlobec88-6"><a href="#cite_ref-zlobec88_6-0">^</a> Zlobec S. (1988). Characterizing Optimality in Mathematical Programming Models. Acta Applicandae Mathematicae, 12, 113-180.

</li>

<li id="cite_note-paice98-7"><a href="#cite_ref-paice98_7-0">^</a> Paice A.D.B. and Wirth, F.R. (1998). Analysis of the Local Robustness of Stability for Flows. <a href="Mathematics_of_Control%2C_Signals%2C_and_Systems" title="Mathematics of Control, Signals, and Systems">Mathematics of Control, Signals, and Systems</a>, 11, 289-302.

</li>

</ol></div></div><div><div class="zim-footer">

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