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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F153bfb9c-b05e-4b91-94e3-9e151aaf7f28%2F07e99d57-7f7c-4929-a015-c9c267bc84e2%2Flz4s2ov_processed.png&w=3840&q=75)
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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