Numerical Linear Algebra Rayleigh Quotient. Power Iteration. Inverse Iteration. Rayleigh Quotient Iteration. Arnoldi Iteration. Lanczos Iteration. 1/13 It addresses the question, given an x, what scalar, α, "acts most like the eigenvalue" for x in the sense of minimizing jjMx − αxjj? Calculating the gradient: 2 rr(x) = (Mx − r(x)x) x∗x This shows that the eigenvectors of M are the stationary points of the function r(x) and the eigenvalues of M are the values of r(x) at these stationary points. The Rayleigh quotient is a quadratically accurate estimate of the eigenvalue! Rayleigh Quotient For a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient r(x), is defined as: x∗Mx r(x) = : x∗x 2/13 Calculating the gradient: 2 rr(x) = (Mx − r(x)x) x∗x This shows that the eigenvectors of M are the stationary points of the function r(x) and the eigenvalues of M are the values of r(x) at these stationary points. The Rayleigh quotient is a quadratically accurate estimate of the eigenvalue! Rayleigh Quotient For a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient r(x), is defined as: x∗Mx r(x) = : x∗x It addresses the question, given an x, what scalar, α, "acts most like the eigenvalue" for x in the sense of minimizing jjMx − αxjj? 2/13 This shows that the eigenvectors of M are the stationary points of the function r(x) and the eigenvalues of M are the values of r(x) at these stationary points. The Rayleigh quotient is a quadratically accurate estimate of the eigenvalue! Rayleigh Quotient For a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient r(x), is defined as: x∗Mx r(x) = : x∗x It addresses the question, given an x, what scalar, α, "acts most like the eigenvalue" for x in the sense of minimizing jjMx − αxjj? Calculating the gradient: 2 rr(x) = (Mx − r(x)x) x∗x 2/13 The Rayleigh quotient is a quadratically accurate estimate of the eigenvalue! Rayleigh Quotient For a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient r(x), is defined as: x∗Mx r(x) = : x∗x It addresses the question, given an x, what scalar, α, "acts most like the eigenvalue" for x in the sense of minimizing jjMx − αxjj? Calculating the gradient: 2 rr(x) = (Mx − r(x)x) x∗x This shows that the eigenvectors of M are the stationary points of the function r(x) and the eigenvalues of M are the values of r(x) at these stationary points. 2/13 Rayleigh Quotient For a given complex Hermitian matrix M and nonzero vector x, the Rayleigh quotient r(x), is defined as: x∗Mx r(x) = : x∗x It addresses the question, given an x, what scalar, α, "acts most like the eigenvalue" for x in the sense of minimizing jjMx − αxjj? Calculating the gradient: 2 rr(x) = (Mx − r(x)x) x∗x This shows that the eigenvectors of M are the stationary points of the function r(x) and the eigenvalues of M are the values of r(x) at these stationary points. The Rayleigh quotient is a quadratically accurate estimate of the eigenvalue! 2/13 Algorithm 1 Power Iteration 1: v(0) = some vector with jjv(0)jj = 1 2: for k = 1; 2;::: do 3: w = Av(k−1) 4: v(k) = w=jjwjj (k) T (k) 5: λk = (v ) Av 6: end for Power Iteration Suppose v(0) is a normalized vector. The following is expected to produce a sequence v(i) that converges to an eigenvector corresponding to the largest eigenvalue of A. 3/13 Power Iteration Suppose v(0) is a normalized vector. The following is expected to produce a sequence v(i) that converges to an eigenvector corresponding to the largest eigenvalue of A. Algorithm 2 Power Iteration 1: v(0) = some vector with jjv(0)jj = 1 2: for k = 1; 2;::: do 3: w = Av(k−1) 4: v(k) = w=jjwjj (k) T (k) 5: λk = (v ) Av 6: end for 3/13 Then one can write: (k) k (0) v =ckA v k k k =ckλ1(a1q1 + a2(λ2/λ1) q2 + ··· + am(λm/λ1) qk In the limit k ! 1, k 2k (k) λ2 (k) λ2 jjv − (±q1)jj = O jλ − λ1j = O λ1 λ1 Power Iteration We can analyze the power iteration as: (0) v = a1q1 + a2q2 + ::: + amqm where qi are orthonormal eigenvectors with corresponding eigenvalues satisfying jλ1j ≥ jλ2j ≥ · · · ≥ jλjm ≥ 0. 4/13 In the limit k ! 1, k 2k (k) λ2 (k) λ2 jjv − (±q1)jj = O jλ − λ1j = O λ1 λ1 Power Iteration We can analyze the power iteration as: (0) v = a1q1 + a2q2 + ::: + amqm where qi are orthonormal eigenvectors with corresponding eigenvalues satisfying jλ1j ≥ jλ2j ≥ · · · ≥ jλjm ≥ 0. Then one can write: (k) k (0) v =ckA v k k k =ckλ1(a1q1 + a2(λ2/λ1) q2 + ··· + am(λm/λ1) qk 4/13 Power Iteration We can analyze the power iteration as: (0) v = a1q1 + a2q2 + ::: + amqm where qi are orthonormal eigenvectors with corresponding eigenvalues satisfying jλ1j ≥ jλ2j ≥ · · · ≥ jλjm ≥ 0. Then one can write: (k) k (0) v =ckA v k k k =ckλ1(a1q1 + a2(λ2/λ1) q2 + ··· + am(λm/λ1) qk In the limit k ! 1, k 2k (k) λ2 (k) λ2 jjv − (±q1)jj = O jλ − λ1j = O λ1 λ1 4/13 Algorithm 3 Inverse Iteration 1: v(0) = some vector with jjv(0)jj = 1 2: for k = 1; 2;::: do 3: Solve (A − µI)w = v(k−1) for w 4: v(k) = w=jjwjj (k) T (k) 5: λk = (v ) Av 6: end for −1 Then by a judicious choice of µ close to λJ , (λJ − µ) can be made much larger for j 6= J. Applying the power iteration should converge to qJ . Inverse Iteration For any µ that is not an eigenvalue of A, the eigenvectors of (A − µI)−1 has the same eigenvectors as A and the −1 corresponding eigenvalues are (λj − µ) . 5/13 Algorithm 4 Inverse Iteration 1: v(0) = some vector with jjv(0)jj = 1 2: for k = 1; 2;::: do 3: Solve (A − µI)w = v(k−1) for w 4: v(k) = w=jjwjj (k) T (k) 5: λk = (v ) Av 6: end for Applying the power iteration should converge to qJ . Inverse Iteration For any µ that is not an eigenvalue of A, the eigenvectors of (A − µI)−1 has the same eigenvectors as A and the −1 corresponding eigenvalues are (λj − µ) . −1 Then by a judicious choice of µ close to λJ , (λJ − µ) can be made much larger for j 6= J. 5/13 Algorithm 5 Inverse Iteration 1: v(0) = some vector with jjv(0)jj = 1 2: for k = 1; 2;::: do 3: Solve (A − µI)w = v(k−1) for w 4: v(k) = w=jjwjj (k) T (k) 5: λk = (v ) Av 6: end for Inverse Iteration For any µ that is not an eigenvalue of A, the eigenvectors of (A − µI)−1 has the same eigenvectors as A and the −1 corresponding eigenvalues are (λj − µ) . −1 Then by a judicious choice of µ close to λJ , (λJ − µ) can be made much larger for j 6= J. Applying the power iteration should converge to qJ . 5/13 Inverse Iteration For any µ that is not an eigenvalue of A, the eigenvectors of (A − µI)−1 has the same eigenvectors as A and the −1 corresponding eigenvalues are (λj − µ) . −1 Then by a judicious choice of µ close to λJ , (λJ − µ) can be made much larger for j 6= J. Applying the power iteration should converge to qJ . Algorithm 6 Inverse Iteration 1: v(0) = some vector with jjv(0)jj = 1 2: for k = 1; 2;::: do 3: Solve (A − µI)w = v(k−1) for w 4: v(k) = w=jjwjj (k) T (k) 5: λk = (v ) Av 6: end for 5/13 In this algorithm, one combines the two: Algorithm 7 Rayleigh quotient Iteration 1: v(0) = some vector with jjv(0)jj = 1 2: λ(0) = (v(0))T Av(0) 3: for k = 1; 2;::: do 4: Solve (A − λ(k−1)I)w = v(k−1) for w 5: v(k) = w=jjwjj 6: λ(k) = (v(k))T Av(k) 7: end for Rayleigh Quotient Iteration The Rayleigh Quotient is a method for obtaining eigenvalue from and eigenvector estimate while the Inverse iteration obtains an eigenvector from estimate of the eigenvalue. 6/13 Rayleigh Quotient Iteration The Rayleigh Quotient is a method for obtaining eigenvalue from and eigenvector estimate while the Inverse iteration obtains an eigenvector from estimate of the eigenvalue. In this algorithm, one combines the two: Algorithm 8 Rayleigh quotient Iteration 1: v(0) = some vector with jjv(0)jj = 1 2: λ(0) = (v(0))T Av(0) 3: for k = 1; 2;::: do 4: Solve (A − λ(k−1)I)w = v(k−1) for w 5: v(k) = w=jjwjj 6: λ(k) = (v(k))T Av(k) 7: end for 6/13 Given a matrix A and a vector b, the associated sequence of vectors: b; Ab; A2b; A3b : : : is called Krylov sequence or Krylov subspace. Projection into Krylov Subspace Iterative methods are based on projecting an m−dimensional problem into a lower dimensional Krylov subspace. 7/13 Projection into Krylov Subspace Iterative methods are based on projecting an m−dimensional problem into a lower dimensional Krylov subspace.