Package 'EfficientMaxEigenpair'

General matrix maximal eigenpair

Description

Calculate the maximal eigenpair for the general matrix.

Usage

eff.ini.maxeig.general(A, v0_tilde = NULL, z0 = NULL, z0numeric, xi = 1,
  digit.thresh = 6)

Arguments

A	The input general matrix.
v0_tilde	The unnormalized initial vector $\tilde{v0}$.
z0	The type of initial $z_0$ used to calculate the approximation of $\rho(Q)$. There are three types: 'fixed', 'Auto' and 'numeric' corresponding to three choices of $z_0$ in paper.
z0numeric	The numerical value assigned to initial $z_0$ as an approximation of $\rho(Q)$ when z_0='numeric'.
xi	The coefficient used to form the convex combination of $\delta_1^{-1}$ and $(v_0,-Q*v_0)_\mu$, it should between 0 and 1.
digit.thresh	The precise level of output results.

Value

A list of eigenpair object are returned, with components $z$, $v$ and $iter$.

z	The approximating sequence of the maximal eigenvalue.
v	The approximating eigenfunction of the corresponding eigenvector.
iter	The number of iterations.

See Also

eff.ini.maxeig.tri for the tridiagonal matrix maximal eigenpair by rayleigh quotient iteration algorithm. eff.ini.maxeig.shift.inv.tri for the tridiagonal matrix maximal eigenpair by shifted inverse iteration algorithm.

Examples

A = matrix(c(1, 1, 3, 2, 2, 2, 3, 1, 1), 3, 3)
eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = 'fixed')

A = matrix(c(1, 1, 3, 2, 2, 2, 3, 1, 1), 3, 3)
eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = 'Auto')

##Symmetrizing A converge to second largest eigenvalue
A = matrix(c(1, 3, 9, 5, 2, 14, 10, 6, 0, 11, 11, 7, 0, 0, 1, 8), 4, 4)
S = (t(A) + A)/2
N = dim(S)[1]
a = diag(S[-1, -N])
b = diag(S[-N, -1])
c = rep(NA, N)
c[1] = -diag(S)[1] - b[1]
c[2:(N - 1)] = -diag(S)[2:(N - 1)] - b[2:(N - 1)] - a[1:(N - 2)]
c[N] = -diag(S)[N] - a[N - 1]

z0ini = eff.ini.maxeig.tri(a, b, c, xi = 7/8)$z[1]
eff.ini.maxeig.general(A, v0_tilde = rep(1, dim(A)[1]), z0 = 'numeric',
z0numeric = 28 - z0ini)

---

Tridiagonal matrix maximal eigenpair

Description

Calculate the maximal eigenpair for the tridiagonal matrix by shifted inverse iteration algorithm.

Usage

eff.ini.maxeig.shift.inv.tri(a, b, c, xi = 1, digit.thresh = 6)

Arguments

a	The lower diagonal vector.
b	The upper diagonal vector.
c	The shifted main diagonal vector. The corresponding unshift diagonal vector is -c(b[1] + c[1], a[1:N - 1] + b[2:N] + c[2:N], a[N] + c[N + 1]) where N+1 is the dimension of matrix.
xi	The coefficient used to form the convex combination of $\delta_1^{-1}$ and $(v_0,-Q*v_0)_\mu$, it should between 0 and 1.
digit.thresh	The precise level of output results.

Value

A list of eigenpair object are returned, with components $z$, $v$ and $iter$.

z	The approximating sequence of the maximal eigenvalue.
v	The approximating eigenfunction of the corresponding eigenvector.
iter	The number of iterations.

See Also

eff.ini.maxeig.tri for the tridiagonal matrix maximal eigenpair by rayleigh quotient iteration algorithm. eff.ini.maxeig.general for the general matrix maximal eigenpair.

Examples

a = c(1:7)^2
b = c(1:7)^2
c = rep(0, length(a) + 1)
c[length(a) + 1] = 8^2
eff.ini.maxeig.shift.inv.tri(a, b, c, xi = 1)

---

Tridiagonal matrix maximal eigenpair

Description

Calculate the maximal eigenpair for the tridiagonal matrix by rayleigh quotient iteration algorithm.

Usage

eff.ini.maxeig.tri(a, b, c, xi = 1, digit.thresh = 6)

Arguments

a	The lower diagonal vector.
b	The upper diagonal vector.
c	The shifted main diagonal vector. The corresponding unshift diagonal vector is -c(b[1] + c[1], a[1:N - 1] + b[2:N] + c[2:N], a[N] + c[N + 1]) where N+1 is the dimension of matrix.
xi	The coefficient used to form the convex combination of $\delta_1^{-1}$ and $(v_0,-Q*v_0)_\mu$, it should between 0 and 1.
digit.thresh	The precise level of output results.

Value

A list of eigenpair object are returned, with components $z$, $v$ and $iter$.

z	The approximating sequence of the maximal eigenvalue.
v	The approximating eigenfunction of the corresponding eigenvector.
iter	The number of iterations.

See Also

eff.ini.maxeig.shift.inv.tri for the tridiagonal matrix maximal eigenpair by shifted inverse iteration algorithm. eff.ini.maxeig.general for the general matrix maximal eigenpair.

Examples

a = c(1:7)^2
b = c(1:7)^2
c = rep(0, length(a) + 1)
c[length(a) + 1] = 8^2
eff.ini.maxeig.tri(a, b, c, xi = 1)

---

General conservative matrix maximal eigenpair

Description

Calculate the next to maximal eigenpair for the general conservative matrix.

Usage

eff.ini.seceig.general(Q, z0 = NULL, c1 = 1000, digit.thresh = 6)

Arguments

Q	The input general matrix.
z0	The type of initial $z_0$ used to calculate the approximation of $\rho(Q)$. There are two types: 'fixed' and 'Auto' corresponding to two choices of $z_0$ in paper.
c1	A large constant.
digit.thresh	The precise level of output results.

Value

A list of eigenpair object are returned, with components $z$, $v$ and $iter$.

z	The approximating sequence of the maximal eigenvalue.
v	The approximating eigenfunction of the corresponding eigenvector.
iter	The number of iterations.

Note

The conservativity of matrix $Q=(q_{ij})$ means that the sums of each row of matrix $Q$ are all 0.

See Also

eff.ini.seceig.tri for the tridiagonal matrix next to the maximal eigenpair.

Examples

Q = matrix(c(-30, 1/5, 11/28, 55/3291, 30, -17, 275/42, 330/1097,
0, 84/5, -20, 588/1097, 0, 0, 1097/84, -2809/3291), 4, 4)
eff.ini.seceig.general(Q, z0 = 'Auto', digit.thresh = 5)
eff.ini.seceig.general(Q, z0 = 'fixed', digit.thresh = 5)

---

Tridiagonal matrix next to the maximal eigenpair

Description

Calculate the next to maximal eigenpair for the tridiagonal matrix whose sums of each row should be 0.

Usage

eff.ini.seceig.tri(a, b, xi = 1, digit.thresh = 6)

Arguments

a	The lower diagonal vector.
b	The upper diagonal vector.
xi	The coefficient used in the improved initials to form the convex combination of $\delta_1^{-1}$ and $(v_0,-Q*v_0)_\mu$, it should between 0 and 1.
digit.thresh	The precise level of output results.

Value

A list of eigenpair object are returned, with components $z$, $v$ and $iter$.

z	The approximating sequence of the maximal eigenvalue.
v	The approximating eigenfunction of the corresponding eigenvector.
iter	The number of iterations.

Note

The sums of each row of the input tridiagonal matrix should be 0.

See Also

eff.ini.seceig.general for the general conservative matrix next to the maximal eigenpair.

Examples

a = c(1:7)^2
b = c(1:7)^2

eff.ini.seceig.tri(a, b, xi = 0)
eff.ini.seceig.tri(a, b, xi = 1)
eff.ini.seceig.tri(a, b, xi = 2/5)

---

EfficientMaxEigenpair: A package for computating the maximal eigenpair for a matrix.

Description

The EfficientMaxEigenpair package provides some auxillary functions and five categories of important functions: tridiag, tri.sol, find_deltak, ray.quot.tri, shift.inv.tri, ray.quot.seceig.tri, ray.quot.general, ray.quot.seceig.general, eff.ini.maxeig.tri, eff.ini.maxeig.shift.inv.tri, eff.ini.maxeig.general, eff.ini.seceig.tri and eff.ini.seceig.general.

EfficientMaxEigenpair functions

tridiag: generate tridiagonal matrix Q based on three input vectors.

tri.sol: construct the solution of linear equation (-Q-zI)w=v.

find_deltak: compute $\delta_k$ for given vector $v$ and matrix $Q$.

ray.quot.tri: rayleigh quotient iteration algorithm to computing the maximal eigenpair of tridiagonal matrix $Q$.

shift.inv.tri: shifted inverse iteration algorithm to computing the maximal eigenpair of tridiagonal matrix $Q$.

ray.quot.seceig.tri: rayleigh quotient iteration algorithm to computing the next to maximal eigenpair of tridiagonal matrix $Q$.

ray.quot.general: rayleigh quotient iteration algorithm to computing the maximal eigenpair of general matrix $A$.

ray.quot.seceig.general: rayleigh quotient iteration algorithm to computing the next to maximal eigenpair of general matrix $A$.

eff.ini.maxeig.tri: calculate the maximal eigenpair for the tridiagonal matrix by rayleigh quotient iteration algorithm.

eff.ini.maxeig.shift.inv.tri: calculate the maximal eigenpair for the tridiagonal matrix by shifted inverse iteration algorithm.

eff.ini.maxeig.general: calculate the maximal eigenpair for the general matrix.

eff.ini.seceig.tri: calculate the next to maximal eigenpair for the tridiagonal matrix whose sums of each row should be 0.

eff.ini.seceig.general: calculate the next to maximal eigenpair for the general conservative matrix.

---

Compute $\delta_k$

Description

Compute $\delta_k$ for given vector $v$ and matrix $Q$.

Usage

find_deltak(Q, v)

Arguments

Q	The given tridiagonal matrix.
v	The column vector on the right hand of equation.

Value

A list of $\delta_k$ for given vector $v$ and matrix $Q$.

Examples

a = c(1:7)^2
b = c(1:7)^2
c = rep(0, length(a) + 1)
c[length(a) + 1] = 8^2
N = length(a)
Q = tridiag(b, a, -c(b[1] + c[1], a[1:N - 1] + b[2:N] + c[2:N], a[N] + c[N + 1]))
find_deltak(Q, v=rep(1,dim(Q)[1]))

---

Rayleigh quotient iteration

Description

Rayleigh quotient iteration algorithm to computing the maximal eigenpair of general matrix $A$.

Usage

ray.quot.general(A, mu, v0_tilde, zstart, digit.thresh = 6)

Arguments

A	The input matrix to find the maximal eigenpair.
mu	A vector.
v0_tilde	The unnormalized initial vector $\tilde{v0}$.
zstart	The initial $z_0$ as an approximation of $\rho(Q)$.
digit.thresh	The precise level of output results.

Value

A list of eigenpair object are returned, with components $z$, $v$ and $iter$.

z	The approximating sequence of the maximal eigenvalue.
v	The approximating eigenfunction of the corresponding eigenvector.
iter	The number of iterations.

Examples

A = matrix(c(1, 1, 3, 2, 2, 2, 3, 1, 1), 3, 3)
ray.quot.general(A, mu=rep(1,dim(A)[1]), v0_tilde=rep(1,dim(A)[1]), zstart=6,
 digit.thresh = 6)

---

Rayleigh quotient iteration

Description

Rayleigh quotient iteration algorithm to computing the maximal eigenpair of matrix Q.

Usage

ray.quot.seceig.general(Q, mu, v0_tilde, zstart, digit.thresh = 6)

Arguments

Q	The input matrix to find the maximal eigenpair.
mu	A vector.
v0_tilde	The unnormalized initial vector $\tilde{v0}$.
zstart	The initial $z_0$ as an approximation of $\rho(Q)$.
digit.thresh	The precise level of output results.

Value

A list of eigenpair object are returned, with components $z$, $v$ and $iter$.

z	The approximating sequence of the maximal eigenvalue.
v	The approximating sequence of the corresponding eigenvector.
iter	The number of iterations.

Examples

Q = matrix(c(1, 1, 3, 2, 2, 2, 3, 1, 1), 3, 3)
ray.quot.seceig.general(Q, mu=rep(1,dim(Q)[1]), v0_tilde=rep(1,dim(Q)[1]), zstart=6,
 digit.thresh = 6)

---

Rayleigh quotient iteration for Tridiagonal matrix

Description

Rayleigh quotient iteration algorithm to computing the next to maximal eigenpair of tridiagonal matrix Q.

Usage

ray.quot.seceig.tri(Q, mu, v0_tilde, zstart, digit.thresh = 6)

Arguments

Q	The input matrix to find the maximal eigenpair.
mu	A vector.
v0_tilde	The unnormalized initial vector $\tilde{v0}$.
zstart	The initial $z_0$ as an approximation of $\rho(Q)$.
digit.thresh	The precise level of output results.

Value

A list of eigenpair object are returned, with components $z$, $v$ and $iter$.

z	The approximating sequence of the maximal eigenvalue.
v	The approximating eigenfunction of the corresponding eigenvector.
iter	The number of iterations.

Examples

a = c(1:7)^2
b = c(1:7)^2
c = rep(0, length(a) + 1)
c[length(a) + 1] = 8^2
N = length(a)
Q = tridiag(b, a, -c(b[1] + c[1], a[1:N - 1] + b[2:N] + c[2:N], a[N] + c[N + 1]))
ray.quot.seceig.tri(Q, mu=rep(1,dim(Q)[1]), v0_tilde=rep(1,dim(Q)[1]), zstart=6,
 digit.thresh = 6)

---

Rayleigh quotient iteration for Tridiagonal matrix

Description

Rayleigh quotient iteration algorithm to computing the maximal eigenpair of tridiagonal matrix Q.

Usage

ray.quot.tri(Q, mu, v0_tilde, zstart, digit.thresh = 6)

Arguments

Q	The input matrix to find the maximal eigenpair.
mu	A vector.
v0_tilde	The unnormalized initial vector $\tilde{v0}$.
zstart	The initial $z_0$ as an approximation of $\rho(Q)$.
digit.thresh	The precise level of output results.

Value

A list of eigenpair object are returned, with components $z$, $v$ and $iter$.

z	The approximating sequence of the maximal eigenvalue.
v	The approximating eigenfunction of the corresponding eigenvector.
iter	The number of iterations.

Examples

a = c(1:7)^2
b = c(1:7)^2
c = rep(0, length(a) + 1)
c[length(a) + 1] = 8^2
N = length(a)
Q = tridiag(b, a, -c(b[1] + c[1], a[1:N - 1] + b[2:N] + c[2:N], a[N] + c[N + 1]))
ray.quot.tri(Q, mu=rep(1,dim(Q)[1]), v0_tilde=rep(1,dim(Q)[1]), zstart=6,
 digit.thresh = 6)

---

Shifted inverse iteration algorithm for Tridiagonal matrix

Description

Shifted inverse iteration algorithm algorithm to computing the maximal eigenpair of tridiagonal matrix $Q$.

Usage

shift.inv.tri(Q, mu, v0_tilde, zstart, digit.thresh = 6)

Arguments

Q	The input matrix to find the maximal eigenpair.
mu	A vector.
v0_tilde	The unnormalized initial vector $\tilde{v0}$.
zstart	The initial $z_0$ as an approximation of $\rho(Q)$.
digit.thresh	The precise level of output results.

Value

A list of eigenpair object are returned, with components $z$, $v$ and $iter$.

z	The approximating sequence of the maximal eigenvalue.
v	The approximating eigenfunction of the corresponding eigenvector.
iter	The number of iterations.

Examples

a = c(1:7)^2
b = c(1:7)^2
c = rep(0, length(a) + 1)
c[length(a) + 1] = 8^2
N = length(a)
Q = tridiag(b, a, -c(b[1] + c[1], a[1:N - 1] + b[2:N] + c[2:N], a[N] + c[N + 1]))
shift.inv.tri(Q, mu=rep(1,dim(Q)[1]), v0_tilde=rep(1,dim(Q)[1]), zstart=6,
 digit.thresh = 6)

---

Solve the linear equation (-Q-zI)w=v.

Description

Construct the solution of linear equation (-Q-zI)w=v.

Usage

tri.sol(Q, z, v)

Arguments

Q	The given tridiagonal matrix.
z	The Rayleigh shift.
v	The column vector on the right hand of equation.

Value

A solution sequence $w$ to the equation (-Q-zI)w=v.

Examples

a = c(1:7)^2
b = c(1:7)^2
c = rep(0, length(a) + 1)
c[length(a) + 1] = 8^2
N = length(a)
zstart = 6
Q = tridiag(b, a, -c(b[1] + c[1], a[1:N - 1] + b[2:N] + c[2:N], a[N] + c[N + 1]))
tri.sol(Q, z=zstart, v=rep(1,dim(Q)[1]))

---

Tridiagonal matrix

Description

Generate tridiagonal matrix Q based on three input vectors.

Usage

tridiag(upper, lower, main)

Arguments

upper	The upper diagonal vector.
lower	The lower diagonal vector.
main	The main diagonal vector.

Value

A tridiagonal matrix is returned.

Examples

a = c(1:7)^2
b = c(1:7)^2
c = -c(1:8)^2
tridiag(b, a, c)

---