## stability

stability A multidiscipline term with a variety of (related) meanings. In numerical analysis it is used with what appears to be a bewildering array of possible prefixes. There are, however, two important basic usages.

Given a well-defined numerical procedure it is important that roundoff errors do not seriously influence the accuracy of the results. This is referred to as numerical stability and depends on the error-propagation properties of the procedure.

Discretization methods for the solution of integral and differential equations are based on a subdivision of the region in which the solution is required. Stability here means that perturbations in the data (initial or boundary conditions) have a bounded effect on the solution obtained (ignoring roundoff error) for a given subdivision. The existence of a uniform bound on this effect over all sufficiently fine subdivisions is a necessary condition for the convergence of the method as the subdivision is refined.

In the solution of ordinary differential equations much of the stability theory has been developed in the study of stiff systems of equations. Of great importance to this development was the concept of A-stability introduced by Dahlquist in 1963. A method is A-stable if it produces bounded solutions for the test problem y′ = qy, y(0) = 1 Re(q) < 0

for all stepsizes. The trapezoidal rule (see ordinary differential equations) is an example of an A-stable method. Much of the later theory has investigated similar properties for more general test problems.

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## stability

stability
1. Atmospheric condition in which air that is forced to rise tends to return to its pre-existing position in the absence of the uplifting force. If the adiabatic lapse rate of uplifted air is greater than the environmental lapse rate, then the vertically displaced air will become colder than the surrounding air and as its density increases it will tend to sink back. See also INSTABILITY.

2. In engineering, the resistance of a structure to collapse or sliding, dependent upon the shearing strength of the material.

3. In geochemistry, the state of equilibrium towards which a system will move from any other state under the same conditions.

4. In thermodynamics, the condition when a slight disturbance of temperature, pressure, or composition does not result in the appearance of a new phase.

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## stability

stability An atmospheric condition in which air that is forced to rise tends to return to its pre-existing level in the absence of the uplifting force. If the adiabatic lapse rate of uplifted air is greater than the environmental lapse rate, the vertically displaced air will become colder than the surrounding air and as its density increases it will tend to sink back. Compare instability.

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## stability

sta·bil·i·ty / stəˈbilitē/ • n. the state of being stable: there are fears for the political stability of the area.

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## stability

stability: see equilibrium.

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## stability

stabilitybanditti, bitty, chitty, city, committee, ditty, gritty, intercity, kitty, nitty-gritty, Pitti, pity, pretty, shitty, slitty, smriti, spitty, titty, vittae, witty •fifty, fifty-fifty, nifty, shifty, swiftie, thrifty •guilty, kiltie, silty •flinty, linty, minty, shinty •ballistae, Christie, Corpus Christi, misty, twisty, wristy •sixty •deity, gaiety (US gayety), laity, simultaneity, spontaneity •contemporaneity, corporeity, femineity, heterogeneity, homogeneity •anxiety, contrariety, dubiety, impiety, impropriety, inebriety, notoriety, piety, satiety, sobriety, ubiety, variety •moiety •acuity, ambiguity, annuity, assiduity, congruity, contiguity, continuity, exiguity, fatuity, fortuity, gratuity, ingenuity, perpetuity, perspicuity, promiscuity, suety, superfluity, tenuity, vacuity •rabbity •improbity, probity •acerbity • witchetty • crotchety •heredity •acidity, acridity, aridity, avidity, cupidity, flaccidity, fluidity, frigidity, humidity, hybridity, insipidity, intrepidity, limpidity, liquidity, lividity, lucidity, morbidity, placidity, putridity, quiddity, rabidity, rancidity, rapidity, rigidity, solidity, stolidity, stupidity, tepidity, timidity, torpidity, torridity, turgidity, validity, vapidity •commodity, oddity •immodesty, modesty •crudity, nudity •fecundity, jocundity, moribundity, profundity, rotundity, rubicundity •absurdity • difficulty • gadgety •majesty • fidgety • rackety •pernickety, rickety •biscuity •banality, duality, fatality, finality, ideality, legality, locality, modality, morality, natality, orality, reality, regality, rurality, tonality, totality, venality, vitality, vocality •fidelity •ability, agility, civility, debility, docility, edibility, facility, fertility, flexility, fragility, futility, gentility, hostility, humility, imbecility, infantility, juvenility, liability, mobility, nihility, nobility, nubility, puerility, senility, servility, stability, sterility, tactility, tranquillity (US tranquility), usability, utility, versatility, viability, virility, volatility •ringlety •equality, frivolity, jollity, polity, quality •credulity, garrulity, sedulity •nullity •amity, calamity •extremity • enmity •anonymity, dimity, equanimity, magnanimity, proximity, pseudonymity, pusillanimity, unanimity •comity •conformity, deformity, enormity, multiformity, uniformity •subcommittee • pepperminty •infirmity •Christianity, humanity, inanity, profanity, sanity, urbanity, vanity •amnesty •lenity, obscenity, serenity •indemnity, solemnity •mundanity • amenity •affinity, asininity, clandestinity, divinity, femininity, infinity, masculinity, salinity, trinity, vicinity, virginity •benignity, dignity, malignity •honesty •community, immunity, importunity, impunity, opportunity, unity •confraternity, eternity, fraternity, maternity, modernity, paternity, taciturnity •serendipity, snippety •uppity •angularity, barbarity, bipolarity, charity, circularity, clarity, complementarity, familiarity, granularity, hilarity, insularity, irregularity, jocularity, linearity, parity, particularity, peculiarity, polarity, popularity, regularity, secularity, similarity, singularity, solidarity, subsidiarity, unitarity, vernacularity, vulgarity •alacrity • sacristy •ambidexterity, asperity, austerity, celerity, dexterity, ferrety, posterity, prosperity, severity, sincerity, temerity, verity •celebrity • integrity • rarity •authority, inferiority, juniority, majority, minority, priority, seniority, sonority, sorority, superiority •mediocrity • sovereignty • salubrity •entirety •futurity, immaturity, impurity, maturity, obscurity, purity, security, surety •touristy •audacity, capacity, fugacity, loquacity, mendacity, opacity, perspicacity, pertinacity, pugnacity, rapacity, sagacity, sequacity, tenacity, veracity, vivacity, voracity •laxity •sparsity, varsity •necessity •complexity, perplexity •density, immensity, propensity, tensity •scarcity • obesity •felicity, toxicity •fixity, prolixity •benedicite, nicety •anfractuosity, animosity, atrocity, bellicosity, curiosity, fabulosity, ferocity, generosity, grandiosity, impecuniosity, impetuosity, jocosity, luminosity, monstrosity, nebulosity, pomposity, ponderosity, porosity, preciosity, precocity, reciprocity, religiosity, scrupulosity, sinuosity, sumptuosity, velocity, verbosity, virtuosity, viscosity •paucity • falsity • caducity • russety •adversity, biodiversity, diversity, perversity, university •sacrosanctity, sanctity •chastity •entity, identity •quantity • certainty •cavity, concavity, depravity, gravity •travesty • suavity •brevity, levity, longevity •velvety • naivety •activity, nativity •equity •antiquity, iniquity, obliquity, ubiquity •propinquity