Jacobis Method is More Accurate Than Qr
Jacobis Method is More Accurate Than Qr
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Let H he a, positive definite symmetric matrix, and SH a small perturbation of H in the sense that [SHtj/Hi^] < rj/n for all i and j. Then || 0; for the other eigenvalues consider -H and -SH . The Courant-Fischer minimax theorem [12] expresses A, as x^Hx A, = rmn max - S" xeS' "^^^^ where the minimum is over all i-dimensional subspaces S'. Let the spaces Sq and Sj satisfy •, x^Hx ^ ^, x^iH + SH)x A, - = max -, . And A, = max -^=- x€s;^^Ax • ^^s\^^U< + 6K)x Then x^(H + 6H)x^ x^iH + SH)x x^Kx x^Hx l + m. A = min max -r-— tjt— < max =r— • -^jr—- tttt— • x r. < ^'^t S' xeS" ^ (^ + ^min(A) ■ relgapx, relgapx, Proof. Let vjt(O) be abbreviated by Vk. From [9] we have t;. (e) = v, +€Y^ -\ — i . Vk + 0(€2) Let yk = Dvk, so that v, {e) = v. + e ^ |i^ • Vk + 0{e^) (2. 8) i^ A. - Afc The pair (A, , y, ) is an eigenpair of the pencil A - XD ^ . Thus >^k = XkyjD'^yk = VkAyk > X^niA)\\yk\\l and so \\yk\\2 < ( Afc/A^„( . 4))'/2 Letting z^ = yk/Wykh lets us write (A, -A, )/(AfcAji/ ^M = v, +eZ, ,h::lf~:, ,^.
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To quickly assess the difficulty of the text, read a short excerpt:
Let H he a, positive definite symmetric matrix, and SH a small perturbation of H in the sense that [SHtj/Hi^] < rj/n for all i and j. Then || 0; for the other eigenvalues consider -H and -SH . The Courant-Fischer minimax theorem [12] expresses A, as x^Hx A, = rmn max - S" xeS' "^^^^ where the minimum is over all i-dimensional subspaces S'. Let the spaces Sq and Sj satisfy •, x^Hx ^ ^, x^iH + SH)x A, - = max -, . And A, = max -^=- x€s;^^Ax • ^^s\^^U< + 6K)x Then x^(H + 6H)x^ x^iH + SH)x x^Kx x^Hx l + m. A = min max -r-— tjt— < max =r— • -^jr—- tttt— • x r. < ^'^t S' xeS" ^ (^ + ^min(A) ■ relgapx, relgapx, Proof. Let vjt(O) be abbreviated by Vk. From [9] we have t;. (e) = v, +€Y^ -\ — i . Vk + 0(€2) Let yk = Dvk, so that v, {e) = v. + e ^ |i^ • Vk + 0{e^) (2. 8) i^ A. - Afc The pair (A, , y, ) is an eigenpair of the pencil A - XD ^ . Thus >^k = XkyjD'^yk = VkAyk > X^niA)\\yk\\l and so \\yk\\2 < ( Afc/A^„( . 4))'/2 Letting z^ = yk/Wykh lets us write (A, -A, )/(AfcAji/ ^M = v, +eZ, ,h::lf~:, ,^.
What to read after Jacobis Method is More Accurate Than Qr?
You can find similar books in the "Read Also" column, or choose other free books by James Demmel to read online
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