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)!
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
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)!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F153bfb9c-b05e-4b91-94e3-9e151aaf7f28%2Fd43ec349-3056-4847-ac5f-0fc883523c31%2F2i5j4zr_processed.png&w=3840&q=75)
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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