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M = A B 7 where A € Rkxk is nonsingular. a) Verify that the following "block LU decomposition" formula holds: I M = = [CA-+ i] [6 D-CA¹B]₁ 0 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 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 – CA¯¹B)x2 = b2 − CA¯¹b₁. =

M = A B 7 where A € Rkxk is nonsingular. a) Verify that the following "block LU decomposition" formula holds: I M = = [CA-+ i] [6 D-CA¹B]₁ 0 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 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 – CA¯¹B)x2 = b2 − CA¯¹b₁. =

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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M = A B D where A E Rkxk is nonsingular. a) Verify that the following "block LU decomposition" formula holds: M = I A - [CA-¹ ] [₁ 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 ]] via the formula X₂ b₂ 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 C D
Transcribed Image Text:M = A B D where A E Rkxk is nonsingular. a) Verify that the following "block LU decomposition" formula holds: M = I A - [CA-¹ ] [₁ 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 ]] via the formula X₂ b₂ 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 C D
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