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Lemma 2.3. Let xº be an arbitrary point in R" and let L be any linear subspace of R. Let x be the minimizer of the function f(p) = p Ap-bp restricted to xº + L. Let s € R" be any vector that is A-conjugate to L. Let & be the minimizer of the one-dimensional function obtained by restricting ƒ to the line {x + as: a € R}. Then x is the minimizer of f restricted to xº + span(LU {s}).

Lemma 2.3. Let xº be an arbitrary point in R" and let L be any linear subspace of R. Let x be the minimizer of the function f(p) = p Ap-bp restricted to xº + L. Let s € R" be any vector that is A-conjugate to L. Let & be the minimizer of the one-dimensional function obtained by restricting ƒ to the line {x + as: a € R}. Then x is the minimizer of f restricted to xº + span(LU {s}).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Lemma 2.3. Let xº be an arbitrary point in R" and let L be any linear subspace of R. Let x be the minimizer of the function f(p) = p Ap-bp restricted to xº + L. Let s € R" be any vector that is A-conjugate to L. Let & be the minimizer of the one-dimensional function obtained by restricting f to the line {x + as: a € R}. Then x is the minimizer of f restricted to xº + span(LU {s}).
Transcribed Image Text:Lemma 2.3. Let xº be an arbitrary point in R" and let L be any linear subspace of R. Let x be the minimizer of the function f(p) = p Ap-bp restricted to xº + L. Let s € R" be any vector that is A-conjugate to L. Let & be the minimizer of the one-dimensional function obtained by restricting f to the line {x + as: a € R}. Then x is the minimizer of f restricted to xº + span(LU {s}).
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