\hyphenation{in-ter-val in-ter-vals}
\hyphenation{com-po-nent}
\hyphenation{sys-tems}
\hyphenation{pol-y-no-mi-als}
\hyphenation{pol-y-no-mi-al}
\hyphenation{pre-con-di-tion-er}
\hyphenation{Lem-ma}

\font\sevenbm = cmmib10 at 7pt
\font\eightbm = cmmib10 at 8pt
\font\ninebm = cmmib10 at 9pt
\font\scripttt = cmtt10 at 7pt
\font\tinytt = cmtt10 at 5pt
\font\eighttt = cmtt10 at 8pt
\font\subsubstt = cmtt10 scaled \magstep2
\font\sevenbf = cmb10 at 7pt
\font\bbs=msbm7
\font\ninebsy= cmbsy10 at 9pt


\newtheorem{algorithm-shell}{Algorithm}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\def\qed{\unskip\kern10pt{\unitlength1pt\linethickness{.4pt}\framebox(6,6){}}}
\newenvironment{proof}{{\it Proof:\/}}{\hfill\qed}

\newcommand{\eqref}[1]{(\ref{#1})}
\newcommand{\bigch}[1]{\makebox{\large\makebox{$#1$}}}
\newcommand{\etal}{{\it et al.\/}}
\newcommand{\bpbx}[2]{\parbox{#1}{\vskip.02in #2 \vskip.02in}}
\newcommand{\stretchline}{\mbox{\large\vphantom{$|$}}}

% Commands for intervals, interval vectors, functions and ranges:

\newcommand{\ivc}[1]{\protect{\mbox{$#1$}}}
\newcommand{\ivl}[1]{\protect{\makebox{\boldmath $#1$}}}
\newcommand{\ivlfn}[1]{\protect{\makebox{\sevenbm #1}}}
\newcommand{\ivlsm}[1]{\protect{\makebox{\ninebm #1}}}
\newcommand{\ivlfg}[1]{\protect{\makebox{\eightbm #1}}}
\newcommand{\pv}[1]{\protect{\makebox{{\tt #1}}}}
\newcommand{\pvs}[1]{\protect{\makebox{{\scripttt #1}}}}
\newcommand{\pve}[1]{\protect{\makebox{{\eighttt #1}}}}
\newcommand{\iX}{{\ivl X}}
\newcommand{\lowerface}[2]{{\ivl #1}_{\underline #2}}
\newcommand{\upperface}[2]{{\ivl #1}_{\overline #2}}
\newcommand{\iXsm}{\ivlsm{X}}
\newcommand{\iXfn}{\makebox{\sevenbm X}}
\newcommand{\iY}{{\ivl Y}}
\newcommand{\iYsm}{{\ivlsm Y}}
\newcommand{\iyl}{{\underline y}}
\newcommand{\iyu}{{\overline y}}
\newcommand{\iXo}{{{\ivl X}_0}}
\newcommand{\ix}{{\ivl x}}
\newcommand{\ixsm}{{\ivlsm x}}
\newcommand{\ixfn}{{\ivlfn x}}
\newcommand{\ixk}{{\ivl x}^{(k)}}
\newcommand{\ixl}{{\underline{x}}}
\newcommand{\ixu}{{\overline{x}}}
\newcommand{\iy}{{\ivl y}}
\newcommand{\iysm}{{\ivlsm y}}
\newcommand{\iz}{{\ivl z}}
\newcommand{\izsm}{{\ivlsm z}}
\newcommand{\is}{{\ivl s}}
\newcommand{\izl}{{\underline{z}}}
\newcommand{\izu}{{\overline{z}}}
\newcommand{\ifn}{{\ivl f}}
\newcommand{\iF}{{\ivl F}}
\newcommand{\ifl}{{\underline{f}}}
\newcommand{\ifu}{{\overline{f}}}
\newcommand{\rangef}{{\ivl f}^u}
\newcommand{\rangeF}{{\ivl F}^u}
\newcommand{\rangeFsm}{{\ivlsm F}^u}
\newcommand{\range}[1]{{\ivl #1}^u}
\newcommand{\std}{\omega}
\newcommand{\istdfn}{\makebox{\sevenbm\char33}}
\newcommand{\istd}{\ivl\omega}
\newcommand{\cancelminus}{\ominus}
\newcommand{\cancelplus}{\oplus}
\newcommand{\canceldivide}{\oslash}
\newcommand{\cancelop}{\hskip2pt\bigcirc\hskip-10.pt\mbox{\scriptsize{\mit op}}\hskip3pt}
\newcommand{\canceltimes}{\otimes}
\newcommand{\domain}{{\bf D}}
\newcommand{\Fnoti}{F_{{\sim}i}}
\newcommand{\Xnotj}{X_{{\sim}j,t}}
\newcommand{\signum}[1]{{\rm sgn}(#1)}
\newcommand{\midpoint}[1]{\mbox{\rm m}{\left({#1}\right)}}
\newcommand{\width}[1]{{\mbox{\rm w}{\left(#1\right)}}}
\newcommand{\iS}{\ivl{S}}

\newcommand{\udiff}{{(u,u')}}
\newcommand{\vdiff}{{(v,v')}}
\newcommand{\dao}[2]{{\left\langle #1,#2 \right\rangle}}
\newcommand{\sao}[3]{{\makebox{\large$\ll$} #1,#2, #3 \makebox{\large$\gg$}}}
\newcommand{\slp}[1]{{#1^{\makebox{\tiny$(s)$}}}}
\newcommand{\pnt}[1]{{\check #1}}
\newcommand{\iu}{{\ivl u}}
\newcommand{\iv}{{\ivl v}}
\newcommand{\ivfn}{{\ivlfn v}}
\newcommand{\ivtil}{\tilde{\ivl v}}
\newcommand{\iV}{\ivl V}

% Commands for interval linear systems and interval Newton methods:

\newcommand{\Xp}{{\pnt X}}
\newcommand{\xp}{{\pnt x}}
\newcommand{\xpk}{{\pnt x}^{(k)}}
\newcommand{\iXp}{{\pnt{\ivl X}}}
\newcommand{\iXpfn}{{\pnt{\ivlfn X}}}
\newcommand{\iXpsm}{{\pnt{\ivlsm X}}}
\newcommand{\ixp}{{\pnt{\ivl x}}}
\newcommand{\ixpfn}{{\pnt{\ivlfn x}}}
\newcommand{\ixpsm}{{\pnt{\ivlsm x}}}
\newcommand{\iXh}{{\widehat{\ivl X}}}
\newcommand{\iXhsm}{{\widehat{\ivlsm X}}}
\newcommand{\ixh}{{\widehat{\ivl x}}}
\newcommand{\ixhsm}{{\widehat{\ivlsm x}}}
\newcommand{\xsol}{x^{*}}
\newcommand{\Xsol}{{X^{*}}}
\newcommand{\iXsol}{{\ivl X}^{*}}
\newcommand{\iXsolsm}{{\ivlsm X}^{*}}
\newcommand{\ixsol}{{\ivl x}^{*}}
\newcommand{\ixsolsm}{{\ivlsm x}^{*}}
\newcommand{\Xhat}{{\widehat X}}
\newcommand{\gradc}{\ivl\nabla\ivl C}
\newcommand{\gradcsm}{\mbox{\ninebsy \char114}\ivlsm C}
\newcommand\newX{{\iXh}_{\makebox{\rm new}}}
\newcommand\newXsm{{\iXhsm}_{\makebox{\small\rm new}}}
\newcommand{\inewt}[3]{{\ivl N}(#1;#2,#3)}
\newcommand{\iXtil}{{{\tilde{\ivl X}}}}
\newcommand{\iXtilsm}{{{\tilde{\ivlsm X}}}}
\newcommand{\iXtilfn}{{{\tilde{\ivlfn X}}}}
\newcommand{\ixtil}{{{\tilde{\ivl x}}}}
\newcommand{\ixtilsm}{{{\tilde{\ivlsm x}}}}
\newcommand{\ixtill}{{{\underline{\tilde{x}}}}}
\newcommand{\ixtilu}{{{\overline{\tilde{x}}}}}
\newcommand{\iA}{{\ivl A}}
\newcommand{\iAsm}{{\ivlsm A}}
\newcommand{\Ap}{{\pnt A}}
\newcommand{\ia}{{\ivl a}}
\newcommand{\ial}{\underline a}
\newcommand{\iau}{\overline a}
\newcommand{\iB}{{\ivl B}}
\newcommand{\iBo}{{{\ivl B}_0}}
\newcommand{\iYA}{{\ivl M}}
\newcommand{\iya}{{\ivl m}}
\newcommand{\iYB}{{\ivl C}}
\newcommand{\iYb}{{\ivl c}}
\newcommand{\unitedset}{\sum_{\exists\exists}}
\newcommand{\solset}{{\Sigma}}
\newcommand{\controlledset}{{\sum_{\exists\forall}}}
\newcommand{\tolerableset}{{\sum_{\forall\exists}}}
\newcommand{\ihull}{{\raise2.25pt\hbox{\framebox[3pt]{\phantom{\tiny|}}}\Sigma}}
\newcommand{\ch}{\hskip2pt\underline\cup\hskip2pt}
\newcommand{\invmid}{Y^{\mbox{\scriptsize\rm mid}}}
\newcommand{\matspread}{\ivl{\Delta}}
\newcommand{\norm}[1]{{\left\|#1\right\|}}

%  Commands for the preconditioner chapter

\newcommand{\precnum}{{\bf n}_i(Y_i)}
\newcommand{\precdenom}{{{\bf d}_i(Y_i)}}
\newcommand{\precnuml}{{\underline{n}_i(Y_i)}}
\newcommand{\precnumu}{{\overline{n}_i(Y_i)}}
\newcommand{\precdenoml}{{\underline{d}_i(Y_i)}}
\newcommand{\precdenomu}{{\overline{d}_i(Y_i)}}
\newcommand{\CLp}{\makebox{$\makebox{C}^{\makebox{\scriptsize L}}$}{\rm-preconditioner}}
\newcommand{\CRp}{\makebox{$\makebox{C}^{\makebox{\scriptsize R}}$}{\rm-preconditioner}}
\newcommand{\ELp}{\makebox{$\makebox{E}^{\makebox{\scriptsize L}}$}{\rm-preconditioner}}
\newcommand{\ERp}{\makebox{$\makebox{E}^{\makebox{\scriptsize R}}$}{\rm-preconditioner}}
\newcommand{\EWp}{\makebox{$\makebox{E}^{\makebox{\scriptsize W}}$}{\rm-preconditioner}}
\newcommand{\ithgap}{\makebox{$\ix_i\setminus\ixtil_i$}}
\newcommand{\yp}{y^{+}}
\newcommand{\ym}{y^{-}}
\newcommand{\vp}{v^{+}}
\newcommand{\vm}{v^{-}}
\newcommand{\Mp}{{\overline s}}
\newcommand{\Mm}{{\underline s}}
\newcommand{\objective}{\mbox{${\cal O}$}}
\newcommand{\CWsc}{{\makebox{$\makebox{\scriptsize C}^{\makebox{\tiny W}}$}}}
\newcommand{\CW}{{\makebox{$\makebox{C}^{\makebox{\scriptsize W}}$}}}
\newcommand{\CMsc}{{\makebox{$\makebox{\scriptsize C}^{\makebox{\tiny M}}$}}}
\newcommand{\CM}{{\makebox{$\makebox{C}^{\makebox{\scriptsize M}}$}}}
\newcommand{\EM}{{\makebox{$\makebox{E}^{\makebox{\scriptsize M}}$}}}
\newcommand{\optimalitycondition}{\mbox{${\cal NLP}$}}
\newcommand{\lpproblem}{\mbox{${\cal LP}$}}
\newcommand{\quasifeasible}{\mbox{${\bf Q}$}}

\newcommand{\lpa}{{\cal A}}
\newcommand{\lpb}{{\cal B}}
\newcommand{\lpc}{{\cal C}}
\newcommand{\lpx}{{\cal X}}
\newcommand{\midminit}{{\pnt A}_{\neg i}^T}
\newcommand{\midmini}{{\pnt A}_{\neg i}}
\newcommand{\wxmit}{\width{X_{\neg i}}^T}
\newcommand{\wxmi}{\width{X_{\neg i}}}
\newcommand{\lmit}{{\underline A}_{\neg i}^T}
\newcommand{\lmi}{{\underline A}_{\neg i}}
\newcommand{\umit}{{\overline A}_{\neg i}^T}
\newcommand{\umi}{{\overline A}_{\neg i}}
\newcommand{\lit}{{\underline A}_i^T}
\newcommand{\uit}{{\overline A}_i^T}

%  Commands for functions, derivatives, and slopes:

\newcommand{\iphi}{\ivl\phi}
\newcommand{\iphifn}{\protect{\makebox{\sevenbm \char30}}}
\newcommand{\iphism}{\protect{\makebox{\ninebm \char30}}}
\newcommand{\igrad}{{\ivl\nabla\ivl\phi}}
\newcommand{\slope}{{{\bf S}^\sharp}}
\newcommand{\slopediff}{{{\bf S}^{(d)}}}
\newcommand{\slopebound}{{\bf S}}
\newcommand{\hansenslope}[1]{{{\bf S}_{#1}^{H,S}}}
\newcommand{\indexset}{{(i_1,i_2,\dots i_n)}}
\newcommand{\grad}{{\nabla\phi}}

\newcommand{\iphimv}{\iphi_2}
\newcommand{\iphimvfn}{\iphifn_2}
\newcommand{\iphimvsm}{\iphism_2}
\newcommand{\iphin}{{{\iphi_{\makebox{n}}}}}
\newcommand{\iphis}{{\iphi_2}}
\newcommand{\rangephi}{{\iphi^u}}
\newcommand{\hansenphi}{{\iphi^H_{(i_1,i_2,\dots i_n)}}}

\newcommand{\phiu}{{\overline{\phi}}}
\newcommand{\phil}{{\underline{\phi}}}

\newcommand{\isin}{\makebox{\bf sin}}
\newcommand{\icos}{\makebox{\bf cos}}

\newcommand{\Krawczyk}[2]{{\bf K}(#1,#2)}
\newcommand{\Krawczykfn}[2]{\makebox{\sevenbf K}(#1,#2)}
\newcommand{\GS}[1]{{\bf GS}(#1)}
\newcommand{\iFp}{\ivl{F'}}
\newcommand{\iFpsm}{\ivlsm{F}{\bf'}}
\newcommand{\iFpfn}{\ivlfn{F}{\bf'}}

\newcommand{\pdo}[2]{\frac{\partial{#1}}{\partial{#2}}}
\newcommand{\ipdo}[2]{\frac{\ivl{\partial}{#1}}{\ivl{\partial}{#2}}}
\newcommand{\pspd}[2]{\frac{\partial^2{#1}}{\partial{#2}^2}}
\newcommand{\spd}[3]{\frac{\partial^2{#1}}{\partial{#2}\partial{#3}}}

\newcommand{\tf}{{\tilde f}}

%  Lists and branch-and-bound algorithms:


\newcommand{\todolist}{{\cal L}}
\newcommand{\unknownlist}{{\cal U}}
\newcommand{\completedlist}{{\cal C}}
\newcommand{\roots}{{\cal R}}
\newcommand{\boxlist}{{\cal L}_{\ivlfn Y}}
\newcommand{\newlist}{{\cal L}_{\mbox{\scriptsize\rm{new}}}}
\newcommand{\donecoords}{{\cal I}}
\newcommand{\donecoordsnew}{{\cal I}_{\makebox{\rm new}}}
\newcommand{\donecoordsnewsm}{{\cal I}_{\makebox{\small\rm new}}}
\newcommand{\Xc}{\iX^{(c)}}
\newcommand{\Xcsm}{\iXsm^{(c)}}
\newcommand{\Xcfn}{\iXfn^{(c)}}
\newcommand{\Xcp}{{\pnt X}^{(c)}}
\newcommand{\xc}{\ix^{(c)}}
\newcommand{\xcsm}{\ixsm^{(c)}}
\newcommand{\reldiam}[1]{\|\mbox{\rm w}({#1})\|_{\mbox{\scriptsize\rm rel}}}
\newcommand{\smear}{\sigma}
\newcommand{\scale}{s}
\newcommand{\Tmpbox}{\ivl{W}}
\newcommand{\Tmpboxfn}{\ivlfn{W}}
\newcommand{\Tmpboxsm}{\ivlsm{W}}
\newcommand{\tmpbox}{\ivl w}
\newcommand{\tmpboxfn}{\ivlfn w}

%  Sets of numbers:

% Comment out the following if AMS fonts can be used
%\newcommand{\Bbb}{\bit}
\newcommand{\R}[1]{{\Bbb R}^{#1}}
\newcommand{\Rs}[1]{\makebox{\bbs R}^{#1}}
\newcommand{\Ro}{{\Bbb R}}
\newcommand{\IR}[1]{{\Bbb{I}\Bbb{R}}^{#1}}
\newcommand{\IRo}{{\Bbb{I}\Bbb{R}}}
\newcommand{\IC}{{\Bbb{I}\Bbb{C}}}

%  Algorithm constructs:

\newcommand {\If} {{\em IF }}
\newcommand {\Then} {{\em THEN }}
\newcommand {\Else} {{\em ELSE }}
\newcommand {\Endif} {{\em END IF}}
\newcommand {\return} {{\em RETURN }}
\newcommand \Input[1] {\noindent{\em INPUT: #1}}
\newcommand \Output[1] {\noindent{\em OUTPUT: #1}}
\newcommand {\Exit} {{\em EXIT}}
\newcommand {\Dowhile} {{\em DO~WHILE }}
\newcommand {\Do} {{\em DO }}
\newcommand {\Enddo} {{\em END DO }}

%  Tolerances --

\newcommand {\epsmach}{{\epsilon_m}}
\newcommand{\epsx}{\epsilon_d}
\newcommand{\maxtoprocess}{\makebox{\tt M}}
\newcommand {\maxerr} {\makebox{{\rm M}}}

%  Miscellaneous --

\newcommand {\interior}[1]{{\makebox{{\rm int}}(#1)}}
\newcommand {\bdy}[1]{\makebox{$\ivl\partial$}#1}
\newcommand {\union}{\mbox{\Large$\cup$}}
\newcommand {\degree}[2]{\makebox{\rm d}(#1,#2,0)}
\newcommand{\mig}[1]{{\makebox{\large$<$} #1 \makebox{\large$>$}}}
\newcommand\opinv{\hskip2pt\makebox{\mit op}^{-1}\hskip2pt}
\newcommand\op{\hskip2pt\makebox{\mit op}\hskip2pt}
\newcommand\nintermediate{\makebox{\tt NI}}
\newcommand\OP{\makebox{OP}}
\newcommand{\bigoh}[1]{{\cal O}\left({#1}\right)}
\newcommand{\nop}{N_{\mbox{\scriptsize\rm OPS}}}
\newcommand{\neqn}{N_{\mbox{\scriptsize\rm EQ}}}
\newcommand{\subsit}{\protect{\inx{sub\-sti\-tu\-tion-ite\-ra\-tion}}}

\def\BibTeX{{\rm B\kern-.05em{\sc i\kern-.025em b}\kern-.08em
    T\kern-.1667em\lower.7ex\hbox{E}\kern-.125emX}}

%  Routine names of own code --

\newcommand{\subsitcode}{\pv{\inx{SUBSIT}}}

