Numerical Methods Contents - SAM

Constitutive relations + (2.0.1)
linear system of equations:
➁ : iωC 1 (U 2 − U 1 ) + R1 −1 (U 2 − U 3 ) − iω −1 L −1 (U 2 − U 4 ) + R2 −1 (U 2 − U 5 ) = 0 ,
➂ :
R1 −1 (U 3 − U 2 ) + iωC 2 (U 3 − U 5 ) = 0 ,
➃ :
R5 −1 (U 4 − U 1 ) − iω −1 L −1 (U 4 − U 2 ) + R4 −1 (U 4 − U 5 ) = 0 ,
➄ : R2 −1 (U 5 − U 2 ) + iωC 2 (U 5 − U 3 ) + R4 −1 (U 5 − U 4 ) + R 3 (U 5 − U 6 ) = 0 ,
U 1 = U , U 6 = 0 .
⎛
iωC 1 + R 1 − 1 ωL i + R 1 − 1 i
2 R 1 ωL − 1 ⎞
⎛ ⎞ ⎛ ⎞
R 2
− R 1 1 U 2 iωC 1 U
1 R1
+ iωC 2 0 −iωC 2
⎜ i
⎝ ωL 0 1
R5
− ωL i + R 1 − 1 ⎜U 3
⎟
⎟⎝
4 R 4 ⎠
U 4 ⎠ = ⎜
0
⎝ 1 ⎟
R U ⎠
5
−R 1 −iωC 2 2 −R 1 1
4 R2
+ iωC 2 + R 1 U 5 0
4
✸
✬
Theorem 2.0.3 (Criteria for invertibility of matrix).
A matrix A ∈ K n,n is invertible/regular if one of the following equivalent conditions is satisfied:
1. ∃B ∈ K n,n : BA = AB = I,
2. x ↦→ Ax defines an endomorphism of K n ,
3. the columns of A are linearly independent (full column rank),
4. the rows of A are linearly independent (full row rank),
5. detA ≠ 0 (non-vanishing determinant),
6. rank(A) = n (full rank).
✫
Formal way to denote solution of LSE:
A ∈ K n,n regular & Ax = b ⇒ x = A −1 b .
✩
✪
Theory
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2.1 Gaussian Elimination
matrix inverse
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Known from linear algebra:
!
Exceptional feature of linear systems of equations (LSE):
☞ “exact” solution computable with finitely many elementary operations
Definition 2.0.1 (Invertible matrix).
A ∈ K n,n invertible /
regular
B ˆ= inverse of A, (✎ notation B = A −1 )
:⇔ ∃ 1 B ∈ K n,n : AB = BA = I .
Algorithm: Gaussian elimination (→ secondary school, linear algebra)
Wikipedia: Although the method is named after mathematician Carl Friedrich Gauss, the earliest
presentation of it can be found in the important Chinese mathematical text Jiuzhang
suanshu or The Nine Chapters on the Mathematical Art, dated approximately 150 B.C.E,
and commented on by Liu Hui in the 3rd century.
Definition 2.0.2 (Rank of a matrix).
The rank of a matrix M ∈ K m,n , denoted by rank(M), is the maximal number of linearly
independent rows/columns of M.
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Idea: transformation to “simpler”, but equivalent LSE by means of successive row
transformations
Ex. 1.2.5: row transformations ↔ left-multiplication with transformation matrix
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