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A [[ B]. D M = where A E Rkxk is nonsingular. a) Verify that the following "block LU decomposition" formula holds: M = I A - [CA-+ i] [4 [CA-₁ в BA-'B]. D-CA-¹B The matrix D-CA-¹B is called the Schur complement of A and is the matrix we get after B eliminating the first block of unknowns x₁ in the system via the formula C D X₂ x₁ = A-¹(b₁ - Bx2); plugging this formula into the second block row of the equation yields: b₂ = Cx₁ + Dx2 = CA¯¹(b₁ – Bx2) + Dx2 (D- - CA¹B)x2 = b₂ - CA-¹b₁. b) Describe how you can construct the full LU decomposition of M by computing an LU decomposition of A, evaluating and factorizing D - CA-¹B, and finally manipulating the result using efficient operations (and never explicitly constructing A-¹)!

A [[ B]. D M = where A E Rkxk is nonsingular. a) Verify that the following "block LU decomposition" formula holds: M = I A - [CA-+ i] [4 [CA-₁ в BA-'B]. D-CA-¹B The matrix D-CA-¹B is called the Schur complement of A and is the matrix we get after B eliminating the first block of unknowns x₁ in the system via the formula C D X₂ x₁ = A-¹(b₁ - Bx2); plugging this formula into the second block row of the equation yields: b₂ = Cx₁ + Dx2 = CA¯¹(b₁ – Bx2) + Dx2 (D- - CA¹B)x2 = b₂ - CA-¹b₁. b) Describe how you can construct the full LU decomposition of M by computing an LU decomposition of A, evaluating and factorizing D - CA-¹B, and finally manipulating the result using efficient operations (and never explicitly constructing A-¹)!

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter5: Orthogonality
Section5.5: Applications
Problem 74EQ
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we have a matrix M ∈ R ^n×n decomposed into blocks:

A [[ B]. D M = where A E Rkxk is nonsingular. a) Verify that the following "block LU decomposition" formula holds: M = I A - [CA-¹ i 6 в BA-¹B]· D-CA-¹B The matrix D-CA-¹B is called the Schur complement of A and is the matrix we get after eliminating the first block of unknowns x₁ in the system x₁ = A-¹(b₁ - Bx2); plugging this formula into the second block row of the equation yields: b₂ = Cx₁ + Dx2 = CA¯¹(b₁ – Bx2) + Dx2 (D- - CA¹B)x2 = b₂ - CA-¹b₁. Q-NE B CD via the formula b) Describe how you can construct the full LU decomposition of M by computing an LU decomposition of A, evaluating and factorizing D - CA-¹B, and finally manipulating the result using efficient operations (and never explicitly constructing A-¹)!
Transcribed Image Text:A [[ B]. D M = where A E Rkxk is nonsingular. a) Verify that the following "block LU decomposition" formula holds: M = I A - [CA-¹ i 6 в BA-¹B]· D-CA-¹B The matrix D-CA-¹B is called the Schur complement of A and is the matrix we get after eliminating the first block of unknowns x₁ in the system x₁ = A-¹(b₁ - Bx2); plugging this formula into the second block row of the equation yields: b₂ = Cx₁ + Dx2 = CA¯¹(b₁ – Bx2) + Dx2 (D- - CA¹B)x2 = b₂ - CA-¹b₁. Q-NE B CD via the formula b) Describe how you can construct the full LU decomposition of M by computing an LU decomposition of A, evaluating and factorizing D - CA-¹B, and finally manipulating the result using efficient operations (and never explicitly constructing A-¹)!
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