school of mathematics | the university of …higham/talks/fun11.pdfresearch matters february 25,...
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Research Matters
February 25, 2009
Nick HighamDirector of Research
School of Mathematics
1 / 6
Functions of a Matrix: Theory,Applications and Computation
Nick HighamSchool of Mathematics
The University of Manchester
[email protected]://www.ma.man.ac.uk/~higham/
Examples History & Properties Applications Methods
Outline
1 Examples
2 History & Properties
3 Applications
4 Methods for Matrix Square Root
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Examples History & Properties Applications Methods
Matrix Square Root Example
Find a matrix X such that
X 2 = A =
1 1 00 1 00 0 1
.
A solution is
X =
1 1/2 00 1 00 0 1
.All square roots are given by ±X and
Y = ±U
1 1/2 00 1 00 0 −1
U−1, U =
a b d0 a 00 e c
.
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Examples History & Properties Applications Methods
Matrix Square Root Example
Find a matrix X such that
X 2 = A =
1 1 00 1 00 0 1
.A solution is
X =
1 1/2 00 1 00 0 1
.
All square roots are given by ±X and
Y = ±U
1 1/2 00 1 00 0 −1
U−1, U =
a b d0 a 00 e c
.
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Examples History & Properties Applications Methods
Matrix Square Root Example
Find a matrix X such that
X 2 = A =
1 1 00 1 00 0 1
.A solution is
X =
1 1/2 00 1 00 0 1
.All square roots are given by ±X and
Y = ±U
1 1/2 00 1 00 0 −1
U−1, U =
a b d0 a 00 e c
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Examples History & Properties Applications Methods
Root Oddities (1)
B2n = In, where
B4 =
1 1 1 10 −1 −2 −30 0 1 30 0 0 −1
.Arises in BDF solvers for ODEs.
Turnbull (1927): A3n = In, where
A4 =
−1 1 −1 1−3 2 −1 0−3 1 0 0−1 0 0 0
.
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Examples History & Properties Applications Methods
Root Oddities (1)
B2n = In, where
B4 =
1 1 1 10 −1 −2 −30 0 1 30 0 0 −1
.Arises in BDF solvers for ODEs.Turnbull (1927): A3
n = In, where
A4 =
−1 1 −1 1−3 2 −1 0−3 1 0 0−1 0 0 0
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Examples History & Properties Applications Methods
Root Oddities (2)
C2n = I, where
C4 = 2−3/2
1 3 3 11 1 −1 −11 −1 −1 11 −3 3 −1
.
Hill (1932): US patent for involutory matrices incryptography.
Bauer (2002): “since then the value of mathematicalmethods in cryptology has been unchallenged.”
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Examples History & Properties Applications Methods
Root Oddities (2)
C2n = I, where
C4 = 2−3/2
1 3 3 11 1 −1 −11 −1 −1 11 −3 3 −1
.
Hill (1932): US patent for involutory matrices incryptography.
Bauer (2002): “since then the value of mathematicalmethods in cryptology has been unchallenged.”
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Examples History & Properties Applications Methods
Logarithm Example
Find a real log of A = −I2n, i.e., real solution of eX = A.
For real log, map eigenvalues in pairs{−1,−1} → {(2k + 1)πi ,−(2k + 1)πi}.
Let H =[
0 1−1 0
]. All solutions are
X = πU diag((2k1 + 1)H, (2k2 + 1)H, . . . , (2kn + 1)H
)U−1,
for real nonsingular U. Thus, e.g., k1 = 0, k2 = 1, integer U,
eX = −I4 for X = π
39 20 12 6−55 −28 −15 −10
7 3 0 4−71 −36 −24 −11
.
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Examples History & Properties Applications Methods
Logarithm Example
Find a real log of A = −I2n, i.e., real solution of eX = A.
For real log, map eigenvalues in pairs{−1,−1} → {(2k + 1)πi ,−(2k + 1)πi}.
Let H =[
0 1−1 0
]. All solutions are
X = πU diag((2k1 + 1)H, (2k2 + 1)H, . . . , (2kn + 1)H
)U−1,
for real nonsingular U. Thus, e.g., k1 = 0, k2 = 1, integer U,
eX = −I4 for X = π
39 20 12 6−55 −28 −15 −10
7 3 0 4−71 −36 −24 −11
.
MIMS Nick Higham Matrix Functions 6 / 37
Examples History & Properties Applications Methods
Logarithm Example
Find a real log of A = −I2n, i.e., real solution of eX = A.
For real log, map eigenvalues in pairs{−1,−1} → {(2k + 1)πi ,−(2k + 1)πi}.
Let H =[
0 1−1 0
]. All solutions are
X = πU diag((2k1 + 1)H, (2k2 + 1)H, . . . , (2kn + 1)H
)U−1,
for real nonsingular U. Thus, e.g., k1 = 0, k2 = 1, integer U,
eX = −I4 for X = π
39 20 12 6−55 −28 −15 −10
7 3 0 4−71 −36 −24 −11
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Examples History & Properties Applications Methods
Outline
1 Examples
2 History & Properties
3 Applications
4 Methods for Matrix Square Root
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Examples History & Properties Applications Methods
Cayley and Sylvester
Term “matrix” coined in 1850by James Joseph Sylvester,FRS (1814–1897).
Matrix algebra developed byArthur Cayley, FRS (1821–1895).Memoir on the Theory of Ma-trices (1858).
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Examples History & Properties Applications Methods
Cayley and Sylvester on Matrix Functions
Cayley considered matrix squareroots in his 1858 memoir.
Tony Crilly, Arthur Cayley: Mathemati-cian Laureate of the Victorian Age,2006.
Sylvester (1883) gave first defini-tion of f (A) for general f .
Karen Hunger Parshall, James JosephSylvester. Jewish Mathematician in aVictorian World, 2006.
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Examples History & Properties Applications Methods
Two Definitions
Definition (Cauchy integral formula)
f (A) =1
2πi
∫Γ
f (z)(zI − A)−1 dz,
where f analytic on and inside closed contour Γ enclosingλ(A).
Definition (Schwerdtfeger, 1938)For A with distinct e’vals λ1, . . . , λs with indices ni ,
f (A) =s∑
i=1
Ai
ni−1∑j=0
f (j)(λi)
j!(A− λi I)j =
s∑i=1
ni−1∑j=0
f (j)(λi)Zij ,
Ai are Frobenius covariants, Zij depend on A but not f .
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Examples History & Properties Applications Methods
Matrices in Applied Mathematics
Frazer, Duncan & Collar, Aerodynamics Division ofNPL: aircraft flutter, matrix structural analysis.
Elementary Matrices & Some Applications toDynamics and Differential Equations, 1938.Emphasizes importance of eA.
Arthur Roderick Collar, FRS(1908–1986): “First book to treatmatrices as a branch of appliedmathematics”.
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Examples History & Properties Applications Methods
Function of 2× 2 Triangular Matrix
f([
λ1 t12
0 λ2
])=
f (λ1) t12f (λ2)− f (λ1)
λ2 − λ1
0 f (λ2)
, λ1 6= λ2,
[f (λ) t12f ′(λ)
0 f (λ)
], λ1 = λ2 = λ.
(1,2) elements given by t12f [λ2, λ1] always.
Inaccurate if λ1 ≈ λ2.
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Examples History & Properties Applications Methods
Function of 2× 2 Triangular Matrix
f([
λ1 t12
0 λ2
])=
f (λ1) t12f (λ2)− f (λ1)
λ2 − λ1
0 f (λ2)
, λ1 6= λ2,
[f (λ) t12f ′(λ)
0 f (λ)
], λ1 = λ2 = λ.
(1,2) elements given by t12f [λ2, λ1] always.
Inaccurate if λ1 ≈ λ2.
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Examples History & Properties Applications Methods
Log of 2× 2 Triangular Matrix
logλ2 − logλ1 = log(λ2
λ1
)+ 2π i U(logλ2 − logλ1)
= log(
1 + z1− z
)+ 2π i U(logλ2 − logλ1),
where U(z) =⌈
Im z − π2π
⌉, z = (λ2 − λ1)/(λ2 + λ1).
atanh(z) :=12
log(
1 + z1− z
),
f12 = t122 atanh(z) + 2πiU(logλ2 − logλ1)
λ2 − λ1.
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Examples History & Properties Applications Methods
Log of 2× 2 Triangular Matrix
logλ2 − logλ1 = log(λ2
λ1
)+ 2π i U(logλ2 − logλ1)
= log(
1 + z1− z
)+ 2π i U(logλ2 − logλ1),
where U(z) =⌈
Im z − π2π
⌉, z = (λ2 − λ1)/(λ2 + λ1).
atanh(z) :=12
log(
1 + z1− z
),
f12 = t122 atanh(z) + 2πiU(logλ2 − logλ1)
λ2 − λ1.
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Examples History & Properties Applications Methods
Function of Block Triangular MatrixRecall, Fréchet derivative L:
f (X + E)− f (X )− L(X ,E) = o(‖E‖).
Theorem
f([
X E0 X
])=
[f (X ) L(X ,E)
0 f (X )
].
Application: the iteration
Xk+1 = 12(Xk + X−1
k A), X0 = A
converges to A1/2. Apply it to [ A E0 A ] and read off (1,2) block
to get iteration for Fréchet derivative.
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Examples History & Properties Applications Methods
Function of Block Triangular MatrixRecall, Fréchet derivative L:
f (X + E)− f (X )− L(X ,E) = o(‖E‖).
Theorem
f([
X E0 X
])=
[f (X ) L(X ,E)
0 f (X )
].
Application: the iteration
Xk+1 = 12(Xk + X−1
k A), X0 = A
converges to A1/2. Apply it to [ A E0 A ] and read off (1,2) block
to get iteration for Fréchet derivative.
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Examples History & Properties Applications Methods
Outline
1 Examples
2 History & Properties
3 Applications
4 Methods for Matrix Square Root
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Examples History & Properties Applications Methods
Toolbox of Matrix Functions
d2ydt2 + Ay = 0, y(0) = y0, y ′(0) = y ′0
has solution
y(t) = cos(√
At)y0 +(√
A)−1 sin(
√At)y ′0.
But [y ′
y
]= exp
([0 −tA
t In 0
])[y ′0y0
].
In software want to be able evaluate interesting f atmatrix args as well as scalar args.MATLAB has expm, logm, sqrtm, funm.
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Examples History & Properties Applications Methods
Toolbox of Matrix Functions
d2ydt2 + Ay = 0, y(0) = y0, y ′(0) = y ′0
has solution
y(t) = cos(√
At)y0 +(√
A)−1 sin(
√At)y ′0.
But [y ′
y
]= exp
([0 −tA
t In 0
])[y ′0y0
].
In software want to be able evaluate interesting f atmatrix args as well as scalar args.MATLAB has expm, logm, sqrtm, funm.
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Examples History & Properties Applications Methods
Toolbox of Matrix Functions
d2ydt2 + Ay = 0, y(0) = y0, y ′(0) = y ′0
has solution
y(t) = cos(√
At)y0 +(√
A)−1 sin(
√At)y ′0.
But [y ′
y
]= exp
([0 −tA
t In 0
])[y ′0y0
].
In software want to be able evaluate interesting f atmatrix args as well as scalar args.MATLAB has expm, logm, sqrtm, funm.
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Examples History & Properties Applications Methods
The Average Eye
First order character of optical system characterized bytransference matrix
T =
[S δ0 1
]∈ R5×5,
where S ∈ R4×4 is symplectic:
ST JS = J =
[0 I2−I2 0
].
Average m−1∑mi=1 Ti is not a transference matrix.
Harris (2005) proposes the average exp(m−1∑mi=1 log(Ti)).
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Examples History & Properties Applications Methods
Markov Models
Let P be transition probability matrix for discrete-timeMarkov process.If P is transition matrix for 1 year,P(1/12) = P1/12 = e
112 log P is matrix for 1 month.
Problem: log P, P1/k may have wrong sign patterns⇒“regularize”.In credit risk, P is strictly diagonally dominant.
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Examples History & Properties Applications Methods
Email from a Power Company
The problem has arisen through proposedmethodology on which the company will incurcharges for use of an electricity network....
I have the use of a computer and Microsoft Excel....
I have an Excel spreadsheet containing thetransition matrix of how a company’s [Standard &Poor’s] credit rating changes from one year to thenext. I’d like to be working in eighths of a year, sothe aim is to find the eighth root of the matrix.
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Examples History & Properties Applications Methods
HIV to Aids Transition
Estimated 6-month transition matrix.Four AIDS-free states and 1 AIDS state.2077 observations (Charitos et al., 2008).
P =
0.8149 0.0738 0.0586 0.0407 0.01200.5622 0.1752 0.1314 0.1169 0.01430.3606 0.1860 0.1521 0.2198 0.08150.1676 0.0636 0.1444 0.4652 0.1592
0 0 0 0 1
.Want to estimate the 1-month transition matrix.
Λ(P) = {1,0.9644,0.4980,0.1493,−0.0043}.
N. J. Higham and L. Lin.On pth roots of stochastic matrices, LAA, 2011.
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Examples History & Properties Applications Methods
Phi Functions: Definition
ϕ0(z) = ez , ϕ1(z) =ez − 1
z, ϕ2(z) =
ez − 1− zz2 , . . .
ϕk+1(z) =ϕk(z)− 1/k !
z.
ϕk(z) =∞∑
j=0
z j
(j + k)!.
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Examples History & Properties Applications Methods
Phi Functions: Solving DEs
y ∈ Cn, A ∈ Cn×n.
dydt
= Ay , y(0) = y0 ⇒ y(t) = eAty0.
dydt
= Ay + b, y(0) = 0 ⇒ y(t) = t ϕ1(tA)b.
dydt
= Ay + ct , y(0) = 0 ⇒ y(t) = t2ϕ2(tA)c.
...
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Examples History & Properties Applications Methods
Phi Functions: Solving DEs
y ∈ Cn, A ∈ Cn×n.
dydt
= Ay , y(0) = y0 ⇒ y(t) = eAty0.
dydt
= Ay + b, y(0) = 0 ⇒ y(t) = t ϕ1(tA)b.
dydt
= Ay + ct , y(0) = 0 ⇒ y(t) = t2ϕ2(tA)c.
...
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Examples History & Properties Applications Methods
Phi Functions: Solving DEs
y ∈ Cn, A ∈ Cn×n.
dydt
= Ay , y(0) = y0 ⇒ y(t) = eAty0.
dydt
= Ay + b, y(0) = 0 ⇒ y(t) = t ϕ1(tA)b.
dydt
= Ay + ct , y(0) = 0 ⇒ y(t) = t2ϕ2(tA)c.
...
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Examples History & Properties Applications Methods
Exponential Integrators
Considery ′ = Ly + N(y).
N(y(t)) ≈ N(y(0)) implies
y(t) ≈ etLy0 + tϕ1(tL)N(y(0)).
Exponential Euler method:
yn+1 = ehLyn + hϕ1(hL)N(yn).
Lawson (1967); recent resurgence.
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Examples History & Properties Applications Methods
Implementation of Exponential Integrators
u ′(t) = Au(t) + g(t ,u(t)), u(0) = u0, t ≥ 0.
Let uk = g(k−1)(t ,u(t)) |t=0 and ϕ`(z) =∑∞
k=0 zk/(k + `)!.We need to compute
u(t) = etAu0 +∑p
k=1 ϕk(tA)tk uk .
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Examples History & Properties Applications Methods
Evaluating Sum of Phi Functions
Theorem (Al-Mohy & H, 2010)
Let A ∈ Cn×n, U = [u1,u2, . . . ,up] ∈ Cn×p, τ ∈ C, and define
B =
[A U0 J
]∈ C(n+p)×(n+p), J =
[0 Ip−1
0 0
]∈ Cp×p.
Then for X = eτB we have
X (1 : n,n + j) =∑j
k=1 τk ϕk(τA)uj−k+1, j = 1 : p.
u(t) =[
In 0]
exp(
t[
A U0 J
])[u0
ep
].
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Examples History & Properties Applications Methods
Evaluating Sum of Phi Functions
Theorem (Al-Mohy & H, 2010)
Let A ∈ Cn×n, U = [u1,u2, . . . ,up] ∈ Cn×p, τ ∈ C, and define
B =
[A U0 J
]∈ C(n+p)×(n+p), J =
[0 Ip−1
0 0
]∈ Cp×p.
Then for X = eτB we have
X (1 : n,n + j) =∑j
k=1 τk ϕk(τA)uj−k+1, j = 1 : p.
u(t) =[
In 0]
exp(
t[
A U0 J
])[u0
ep
].
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Examples History & Properties Applications Methods
Outline
1 Examples
2 History & Properties
3 Applications
4 Methods for Matrix Square Root
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Examples History & Properties Applications Methods
Matrix Square Root
X is a square root of A ∈ Cn×n ⇐⇒ X 2 = A .Number of square roots may be zero, finite or infinite.
DefinitionFor A with no eigenvalues on R− = {x ∈ R : x ≤ 0} theprincipal square root A1/2 is unique square root X withspectrum in open right half-plane.
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Examples History & Properties Applications Methods
Newton’s Method for Square Root
Apply Newton to F (X ) = X 2 − A = 0: X0 given,
Solve XkEk + EkXk = A− X 2k
Xk+1 = Xk + Ek
}k = 0,1,2, . . .
Modified Newton iteration: freeze Fréchet derivative at X0:
Solve X0Ek + EkX0 = A− X 2k
Xk+1 = Xk + Ek
}k = 0,1,2, . . . ,
X0 diagonal⇒ cheap to solve for Ek .
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Examples History & Properties Applications Methods
Newton’s Method for Square Root
Apply Newton to F (X ) = X 2 − A = 0: X0 given,
Solve XkEk + EkXk = A− X 2k
Xk+1 = Xk + Ek
}k = 0,1,2, . . .
Modified Newton iteration: freeze Fréchet derivative at X0:
Solve X0Ek + EkX0 = A− X 2k
Xk+1 = Xk + Ek
}k = 0,1,2, . . . ,
X0 diagonal⇒ cheap to solve for Ek .
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Examples History & Properties Applications Methods
Pulay Iteration
Let A1/2 = D1/2 + B, D = diag(di) > 0. Squaring gives
D1/2B + BD1/2 = A− D − B2.
Functional iteration gives
Pulay iteration (1966)
D1/2Bk+1 + Bk+1D1/2 = A− D − B2k , B0 = 0.
Can show Pulay ≡ modified Newton with X0 = D1/2:Xk ≡ D1/2 + Bk .
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Examples History & Properties Applications Methods
Convergence of Pulay
“Although no proof of convergence will be given, theprocedure converged rapidly in all cases examined by us”.
Theorem (H, 2008)
Let A ∈ Cn×n with Λ(A) ∩ R− = ∅ and let D = diag(di) > 0and B = A1/2 − D1/2. If
θ =‖B‖
mini d1/2i
<23
then in the Pulay iteration Bk → A1/2 − D1/2 linearly.
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Examples History & Properties Applications Methods
Visser Iteration
Set X0 = (2α)−1I in modified Newton:
Visser iteration (1937)
Xk+1 = Xk + α(A− X 2k ), X0 = (2α)−1I.
Stationary iteration.Richardson iteration.Linear convergence.Choice of α?
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Examples History & Properties Applications Methods
Visser History
Xk+1 = Xk + α(A− X 2k ), X0 = (2α)−1I.
Visser (1937), α = 1/2: show positive operator onHilbert space has a positive square root.Likewise in functional analysis texts, e.g. Riesz &Sz.-Nagy (1956).Enables proof of existence of A1/2 without usingspectral theorem.Used computationally by Liebl (1965), Babuška,Práger & Vitásek (1966), Späth (1966), Duke (1969),Elsner (1970).Elsner proves cgce for A ∈ Cn×n with real, positiveei’vals if 0 < α ≤ ρ(A)−1/2.
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Examples History & Properties Applications Methods
Visser Transformations
Xk+1 = Xk + α(A− X 2k ), X0 = (2α)−1I.
Let θ = 1/(2α), Xk = θYk , and A = θ−2A. Then
Yk+1 = Yk +12(A− Y 2
k ), Y0 = I.
With A ≡ I − C and Yk = I − Pk :
Pk+1 =12(C + P2
k ), P0 = 0.
Qk = Pk/2:
Qk+1 = Q2k +
C4, Q0 = 0.
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Examples History & Properties Applications Methods
Visser ConvergenceXk+1 = Xk + α(A− X 2
k ), X0 = (2α)−1I.
Theorem (H, 2008)
Let A ∈ Cn×n and α > 0. If Λ(I − 4α2A) lies in the cardioid
D = {2z − z2 : z ∈ C, |z| < 1 }
then A1/2 exists and Xk → A1/2 linearly.
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Examples History & Properties Applications Methods
Example
A ∈ R16×16 spd with aii = i2, aij = 0.1, i 6= j .Aim for rel residual < nu in IEEE DP arithmetic.
Pulay iteration D = diag(A): θ = 0.191, 9 iters.Visser iteration α = 0.058 (hand optimized), 245 iters.
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Examples History & Properties Applications Methods
Future Directions
Many applications of f (A), e.g. control theory, computergraphics, theoretical physics.Better understanding of conditioning of f (A).Understanding non-primary functions.Exploit structure, e.g. A ∈ matrix automorphism groupor Jordan or Lie algebra.f (A)b problem.
Al-Mohy & H: Computing the Action of theMatrix Exponential, with an Application toExponential Integrators, SISC, 2011.
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Examples History & Properties Applications Methods
References I
A. H. Al-Mohy and N. J. Higham.Computing the action of the matrix exponential, with anapplication to exponential integrators.SIAM J. Sci. Comput., 33(2):488–511, 2011.
F. L. Bauer.Decrypted Secrets: Methods and Maxims ofCryptology.Springer-Verlag, Berlin, third edition, 2002.ISBN 3-540-42674-4.xii+474 pp.
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Examples History & Properties Applications Methods
References II
G. Boyd, C. A. Micchelli, G. Strang, and D.-X. Zhou.Binomial matrices.Adv. in Comput. Math., 14:379–391, 2001.
T. Charitos, P. R. de Waal, and L. C. van der Gaag.Computing short-interval transition matrices of adiscrete-time Markov chain from partially observeddata.Statistics in Medicine, 27:905–921, 2008.
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Examples History & Properties Applications Methods
References III
T. Crilly.Arthur Cayley: Mathematician Laureate of the VictorianAge.Johns Hopkins University Press, Baltimore, MD, USA,2006.ISBN 0-8018-8011-4.xxi+610 pp.
R. A. Frazer, W. J. Duncan, and A. R. Collar.Elementary Matrices and Some Applications toDynamics and Differential Equations.Cambridge University Press, Cambridge, UK, 1938.xviii+416 pp.1963 printing.
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Examples History & Properties Applications Methods
References IV
W. F. Harris.The average eye.Opthal. Physiol. Opt., 24:580–585, 2005.
N. J. Higham.The Matrix Function Toolbox.http://www.ma.man.ac.uk/~higham/mftoolbox.
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Examples History & Properties Applications Methods
References V
N. J. Higham.Functions of Matrices: Theory and Computation.Society for Industrial and Applied Mathematics,Philadelphia, PA, USA, 2008.ISBN 978-0-898716-46-7.xx+425 pp.
N. J. Higham and L. Lin.On pth roots of stochastic matrices.Linear Algebra Appl., 435(3):448–463, 2011.
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References VI
J. D. Lawson.Generalized Runge-Kutta processes for stable systemswith large Lipschitz constants.SIAM J. Numer. Anal., 4(3):372–380, Sept. 1967.
K. H. Parshall.James Joseph Sylvester. Jewish Mathematician in aVictorian World.Johns Hopkins University Press, Baltimore, MD, USA,2006.ISBN 0-8018-8291-5.xiii+461 pp.
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Examples History & Properties Applications Methods
References VII
P. Pulay.An iterative method for the determination of the squareroot of a positive definite matrix.Z. Angew. Math. Mech., 46:151, 1966.
H. W. Turnbull.The matrix square and cube roots of unity.J. London Math. Soc., 2(8):242–244, 1927.
C. Visser.Note on linear operators.Proc. Kon. Akad. Wet. Amsterdam, 40(3):270–272,1937.
MIMS Nick Higham Matrix Functions 37 / 37