\（\ usepackage {array} \ setlength {\ oddsidemargin} {-0.0in} \ setlength {\ evensidemargin} {-0.0in} \ setlength {\ textheight} {8.75in} \ setlength {\ textwidth} {6.5in} \ setlength {\ topmargin} {-0.25in} \ newcommand {\ R} {\ mathbb R} \ newcommand {\ Rm} {\ mathbb R ^ m} \ newcommand {\ Rn} {\ mathbb R ^ n} \ newcommand { \ Rnxn} {\ mathbb R ^ {n \ times n}} \ newcommand {\ Rmxn} {\ mathbb R ^ {m \ times n}} \ newcommand {\ Rmxm} {\ mathbb R ^ {m \ times m} } \ newcommand {\ Rmxk} {\ mathbb R ^ {m \ times k}} \ newcommand {\ Rkxn} {\ mathbb R ^ {k \ times n}} \ newcommand {\ C} {\ mathbb C} \ newcommand {\ Cm} {\ mathbb C ^ m} \ newcommand {\ Cmxm} {\ mathbb C ^ {m \ times m}} \ newcommand {\ Cnxn} {\ mathbb C ^ {n \ times n}} \ newcommand { \ Cmxn} {\ mathbb C ^ {m \ times n}} \ newcommand {\ Ckxk} {\ mathbb C ^ {k \ times k}} \ newcommand {\ Cn} {\ mathbb C ^ n} \ newcommand {\ Ck} {\ mathbb C ^ k} \ newcommand {\ Null} {{\ cal N}} \ newcommand {\ Col} {{\ cal C}} \ newcommand {\ Rowspace} {{\ cal R}} \ newcommand {\ Span} {{\ rm {Span}}} \ newcommand {\ rank} {{\ rm rank}} \ newcommand {\ triu} {{\ rm triu}} \ newcommand {\ tril} {{\ rm tril }} \ newco mmand {\ sign} {{\ rm sign}} \ newcommand {\ FlaTwoByTwo} [4] {\ left（\ begin {array} {c | c}＃1 \ amp＃2 \\ \ hline＃3 \ amp＃4 \ end {array} \ right）} \ newcommand {\ FlaTwoByTwoSingleLine} [4] {\ left（\ begin {array} {cc}＃1 \ amp＃2 \\＃3 \ amp＃4 \ end {array} \ right）} \ newcommand {\ FlaTwoByTwoSingleLineNoPar} [4] {\ begin {array} {cc}＃1 \ amp＃2 \\＃3 \放大器＃4 \ end {array}} \ newcommand {\ FlaOneByTwo} [2] {\ left（\ begin {array} {c | c}＃1 \ amp＃2 \ end {array} \ right）} \ newcommand { \ FlaOneByTwoSingleLine} [2] {\ left（\ begin {array} {cc}＃1 \ amp＃2 \ end {array} \ right）} \ newcommand {\ FlaTwoByOne} [2] {\ left（\ begin {array } {c}＃1 \\ \ hline＃2 \ end {array} \ right）} \ newcommand {\ FlaTwoByOneSingleLine} [2] {\ left（\ begin {array} {c}＃1 \\＃2 \ end {array} \ right）} \ newcommand {\ FlaThreeByOneB} [3] {\ left（\ begin {array} {c}＃1 \\ \ hline＃2 \\＃3 \ end {array} \ right）}} \ newcommand {\ FlaThreeByOneT} [3] {\ left（\ begin {array} {c}＃1 \\＃2 \\ \ hline＃3 \ end {array} \ right）} \ newcommand {\ FlaOneByThreeR} [3] {\ left（\ begin {array} {c | cc}＃1 \ amp＃2 \ amp＃3 \ end {array} \ right）} \ newcommand {\ FlaOneByThreeL} [3] {\ left（\ beg in {array} {c c | c}＃1 \ amp＃2 \ amp＃3 \ end {array} \ right）} \ newcommand {\ FlaThreeByThreeBR} [9] {\ left（\ begin {array} {c | cc}＃1 \ amp＃2 \ amp＃3 \\ \ hline＃4 \ amp＃5 \ amp＃6 \\＃7 \ amp＃8 \ amp＃9 \ end {array} \ right）} \ newcommand {\ FlaThreeByThreeTL} [9] {\左（\ begin {array} {cc | c}＃1 \ amp＃2 \ amp＃3 \\＃4 \ amp＃5 \ amp＃6 \\ \ hline＃7 \ amp＃8 \ amp＃9 \ end {array} \ right）} \ newcommand {\ diag} [1] {{\ rm diag}（＃1）} \ newcommand {\ URt} {{\ sc HQR}} \ newcommand {\ FlaAlgorithm} {\ begin { array} {| l |} \ hline \ routinename \\ \ hline \ partitionings \\ ~~~ \ begin {array} {l} \ partitionsizes \ end {array} \\ {\ bf \ color {blue} {while} 〜} \ guard \\ ~~~ \ begin {array} {l} \ repartitionings \ end {array} \\ ~~~ \ color {red} {\ begin {array} {l} \ hline \ color {black} {\ update} \\ \ hline \ end {array}} \\ ~~~ \ begin {array} {l} \ moveboundaries \ end {array} \\ {\ bf \ color {blue} {endwhile}} \\\ \ hline \ end {array}} \ newcommand {\ FlaAlgorithmWithInit} {\ begin {array} {| l |} \ hline \ routinename \\ \ hline \ initialize \\ \ partitionings \\ ~~~ \ begin {array} { l} \ p artitionsizes \ end {array} \\ {\ bf \ color {blue} {while}〜} \ guard \\ ~~~ \ begin {array} {l} \ repartitionings \ end {array} \\ ~~~ \ color {red} {\ begin {array} {l} \ hline \ color {black} {\ update} \\ \ hline \ end {array}} \\ ~~~ \ begin {array} {l} \ moveboundaries \ end {array} \\ {\ bf \ color {blue} {endwhile}} \\ \ hline \ end {array}} \ newcommand {\ FlaBlkAlgorithm} {\ begin {array} {| l |} \ hline \ routinename \\ \ hline \ partitionings \\ ~~~ \ begin {array} {l} \ partitionsizes \ end {array} \\ {\ bf \ color {blue} {while}〜} \ guard \\ ~~~ {\ bf选择〜block〜size〜} \ blocksize \\ ~~~ \ begin {array} {l} \ repartitionings \ end {array} \\ ~~~ ~~~ \ repartitionsizes \\ ~~~ \ color {red} {\开始{array} {l} \ hline \ color {black} {\ update} \\ \ hline \ end {array}} \\ ~~~ \ begin {array} {l} \ moveboundaries \ end {array} \\ {\ bf \ color {blue} {endwhile}} \\ \ hline \ end {array}} \ newcommand {\ complexone} {\ begin {array} {| c |} \ hline \！\ pm \！ \\ \ hline \ end {array}〜} \ newcommand {\ HQR} {{\ rm HQR}} \ newcommand {\ QR} {{\ rm QR}} \ newcommand {\ st} {{\ rm \ s.t. }} \ newcommand {\ QRQ} {{\ rm {\ normalsize \ bf Q} {\ rm \ tiny R}}}} \ newcommand {\ QRR} {{\ rm {\ rm \ tiny Q} {\ bf \ normalsize R}}} \ newcommand {\ deltaalpha} {\ delta \！\ alpha} \ newcommand {\ deltax} {\ delta \！x} \ newcommand {\ deltay} {\ delta \！y} \ newcommand {\ deltaz} {\ delta !! z} \ newcommand {\ deltaw} {\ delta \！w} \ newcommand {\ DeltaA} {\ delta \！\！A} \ newcommand {\ meps} {\ epsilon _ {\ rm mach}} \ newcommand {\ fl} [1] {{\ rm fl（＃1}}} \ newcommand {\ becomes} {：=} \ newcommand {\ defrowvector} [2] {\ left（＃1_0，＃1_1，\ ldots，＃1 _ {＃2-1} \ right）} \ newcommand {\ tr} [1] {{＃1} ^ T} \ newcommand {\ LUpiv} [1] {{\ rm LU}（＃1） } \ newcommand {\ maxi} {{\ rm maxi}} \ newcommand {\ Chol} [1] {{\ rm Chol}（＃1}} \ newcommand {\ lt} {} \ newcommand {\ amp} {＆} \）