Numerical Methods Contents - SAM

9 ylabel ( ’ { \ b f c o n d i t i o n number } ’ , ’ f o n t s i z e ’ ,14) ;
10 t i t l e ( ’ Condition numbers o f row t r a n s f o r m a t i o n matrices ’ ) ;
11 legend ( ’2−norm ’ , ’maximum norm ’ , ’1−norm ’ , ’ l o c a t i o n ’ , ’ southeast ’ ) ;
12 p r i n t −depsc2 ’ . . / PICTURES/ rowtrfcond . eps ’ ;
all rows/columns are pairwise orthogonal (w.r.t. Euclidean inner product),
| detQ| = 1, and all eigenvalues ∈ {z ∈ C: |z| = 1}.
‖QA‖ 2 = ‖A‖ 2 for any matrix A ∈ K n,m
Observation: cond(T(µ)) large for large µ
As explained in Sect. 2.2, Gaussian (forward) elimination can be viewed as successive multiplication
with elimination matrices. If an elimination matrix has a large condition number, then small relative errors
in the entries of intermediate matrices caused by earlier roundoff errors can experience massive
amplification and, thus, spoil all further steps (➤ loss of numerical stability, Def. 2.5.5).
Drawbacks of LU-factorization:
often pivoting required (➞ destroys structure, Ex. 2.6.21, leads to fill-in)
Possible (theoretical) instability of partial pivoting → Ex. 2.5.2
Therefore, the entries of elimination matrices should be kept small, and this is the main rationale
behind (partial) pivoting (→ Sect. 2.3), which ensures that multipliers have modulus ≤ 1 throughout
forward elimination.
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✸
Stability problems of Gaussian elimination without pivoting are due to the fact that row transformations
can convert well-conditioned matrices to ill-conditioned matrices, cf. Ex. 2.5.2
Which bijective row transformations preserve the Euclidean condition number of a matrix ?
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➣ transformations hat preserve the Euclidean norm of a vector !
Recall from linear algebra:
Investigate algorithms that use orthogonal/unitary row transformations to convert a matrix to
upper triangular form.
Definition 2.8.1 (Unitary and orthogonal matrices).
• Q ∈ K n,n , n ∈ N, is unitary, if Q −1 = Q H .
• Q ∈ R n,n , n ∈ N, is orthogonal, if Q −1 = Q T .
✬
Theorem 2.8.2 (Criteria for Unitarity).
✩
Goal:
find unitary row transformation rendering certain matrix elements zero.
⎛ ⎞ ⎛ ⎞
Q⎜
⎟
⎝ ⎠ = ⎜
⎝ 0
⎟
⎠ with QH = Q −1 .
Q ∈ C n,n unitary ⇔ ‖Qx‖ 2 = ‖x‖ 2 ∀x ∈ K n .
✫
Q unitary ⇒ cond(Q) = 1
If Q ∈ K n,n unitary, then
(2.8.1)
➤
unitary transformations enhance (numerical) stability
all rows/columns (regarded as vectors ∈ K n ) have Euclidean norm = 1,
✪
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This “annihilation of column entries” is the key operation in Gaussian forward elimination, where it
is achieved by means of non-unitary row transformations, see Sect. 2.2. Now we want to find a
counterpart of Gaussian elimination based on unitary row transformations on behalf of numerical
stability.
In 2D: two possible orthogonal transformations make 2nd component of a ∈ R 2 vanish:
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