"[AD0410169AE7]" . "2"^^ . . "861;875" . "Pro libovoln\u00FD algebraick\u00FD syst\u00E9m dan\u00FD matic\u00ED soustavy A a vektorem prav\u00E9 strany b definujeme mno\u017Einu tzv. j\u00E1dra probl\u00E9mu a ukazujeme, \u017Ee ortogon\u00E1ln\u00ED horn\u00ED bidiagonalizace roz\u0161\u00ED\u0159en\u00E9 matice soustavy definuje jeden z prvk\u016F dan\u00E9 mno\u017Einy. Je dok\u00E1z\u00E1no, \u017Ee prvky dan\u00E9 mno\u017Einy maj\u00ED odpov\u00EDdaj\u00EDc\u00ED vlastnosti, zejm\u00E9na minim\u00E1ln\u00ED dimenzi. \u0158e\u0161\u00EDme-li \u00FApln\u00FD probl\u00E9m nejmen\u0161\u00EDch \u010Dtverc\u016F nejprve s nalezen\u00EDm j\u00E1dra probl\u00E9mu, pak vytvo\u0159en\u00E1 teorie je konzistentn\u00ED s existuj\u00EDc\u00EDmi zobecn\u011Bn\u00EDmi z\u00E1kladn\u00EDho probl\u00E9mu \u00FApln\u00FDch nejmen\u0161\u00EDch \u010Dtverc\u016F, je v\u0161ak mnohem jednodu\u0161\u0161\u00ED a matematicky elegantn\u011Bj\u0161\u00ED. Navr\u017Een\u00FD postup je d\u016Fle\u017Eit\u00FD tak\u00E9 pro hled\u00E1n\u00ED \u0159e\u0161en\u00ED v jin\u00E9m smyslu; vede nap\u0159\u00EDklad k jednoduch\u00E9mu \u0159e\u0161en\u00ED data least squares probl\u00E9mu. My\u0161lenky mohou b\u00FDt rovn\u011B\u017E pou\u017Eity p\u0159i \u0159e\u0161en\u00ED ill-posed probl\u00E9m\u016F."@cs . "1"^^ . . "P(1ET400300415), Z(AV0Z10300504)" . . "Strako\u0161, Zden\u011Bk" . "Core Problems in Linear Algebraic Systems" . "Core Problems in Linear Algebraic Systems" . "Paige, C. C." . "469853" . . . . . . . . . . "SIAM Journal on Matrix Analysis and Applications" . . . . "J\u00E1dro probl\u00E9mu v linearn\u00EDch algebraick\u00FDch syst\u00E9mech"@cs . . "27" . . "3" . . . "Core Problems in Linear Algebraic Systems"@en . "US - Spojen\u00E9 st\u00E1ty americk\u00E9" . "scaled total least squares; ill-posed problems; least squares; data least squares; orthogonal regression; core problem; orthogonal reduction; minimum 2-norm solutions; bidiagonalization; singular value decomposition"@en . . . "0895-4798" . "RIV/67985807:_____/06:00031914!RIV07-AV0-67985807" . "For any linear system Ax approximates b we define a set of core problems and show that the orthogonal upper bidiagonalization of [b,A] gives such a core problem. In particular we show that these core problems have desirable properties such as minimal dimensions. When a total least squares problem is solved by first finding a core problem, we show the resulting theory is consistent with earlier generalizations, but much simpler and clearer. The approach is important for other related solutions and leads, for example, to an elegant solution to the data least squares problem. The ideas could be useful for solving ill-posed problems." . . "RIV/67985807:_____/06:00031914" . "For any linear system Ax approximates b we define a set of core problems and show that the orthogonal upper bidiagonalization of [b,A] gives such a core problem. In particular we show that these core problems have desirable properties such as minimal dimensions. When a total least squares problem is solved by first finding a core problem, we show the resulting theory is consistent with earlier generalizations, but much simpler and clearer. The approach is important for other related solutions and leads, for example, to an elegant solution to the data least squares problem. The ideas could be useful for solving ill-posed problems."@en . . "J\u00E1dro probl\u00E9mu v linearn\u00EDch algebraick\u00FDch syst\u00E9mech"@cs . "Core Problems in Linear Algebraic Systems"@en . "15"^^ . .