Transcription of Conjugate Gradient Method - Stanford University
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Conjugate Gradient Method - Stanford University
Conjugate Gradient Method direct and indirect methods positive definite linear systems Krylov sequence derivation of the Conjugate Gradient Method spectral analysis of Krylov sequence preconditioningEE364b, Stanford UniversityProf. Mert Pilanciupdated: May 5, 2022 Three classes of methods for linear equationsmethods to solve linear systemAx=b,A Rn n dense direct (factor-solve methods) runtime depends only on size; independent of data, structure, orsparsity work well fornup to a few thousand sparse direct (factor-solve methods) runtime depends on size, sparsity pattern.
• proposed by Hestenes and Stiefel in 1952 (as direct method) • solves SPD system Ax = b – in theory (i.e., exact arithmetic) in n iterations – each iteration requires a few inner products in Rn, and one matrix-vector multiply z → Az • for A dense, matrix-vector multiply z → Az costs n2, so total cost is n3, same as direct methods
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