What Does Rank Two Update Mean

• Journal List
• MethodsX
• v.5; 2018
• PMC6314274

MethodsX. 2018; five: 103–117.

Multiple-rank modification of symmetric eigenvalue problem

Received 2017 Sep 28; Accepted 2018 Jan 7.

Graphical abstruse

Keywords: Eigenvalue decomposition, Modification, Secular equation, Positive semi-definite matrices

Abstract

Rank-1 modifications practical k-times (k > one) often are performed to achieve a rank-k modification. Nosotros propose a rank- 1000 modification for enhancing computational efficiency. As the first step toward a rank- k modification, an algorithm to perform a rank-2 modification is proposed and tested. The computation toll of our proposed algorithm is in O ( n two ) where due north is the cardinality of the matrix of interest. We also advise a general rank-k update algorithm based on the Sturm Theorem, and compare our results to those of the direct eigenvalue decomposition and of a perturbation method.

Method details

Optimal ability flow plays a central function in the operation and planning studies for power systems. Due to its nonlinearity and non-convexity, a numerical solution is pursued using heuristic methods [[1], [2]]. The nearly widely used method is a Lagrange relaxation, which relies on computationally expensive matrix factorizations. A matrix normally involves a low-rank update; therefore, information technology would be a viable option to update the factors instead of the expensive re-factorization process to evaluate them. An LU modification method is first introduced, and many study results show its effectiveness [[8], [9]]. While very relevant to power system studies because it can preserve the sparsity, the lack of numerically stability is an important issue. A stable update is accomplished by updating Cholesky factorization [[9], [x], [xi]]; nonetheless, its applicability is limited to positive semi-definite (PSD) matrices. The matrices associated with power systems are not, in general, PSD. Modifying eigenvalue/eigenvector pairs [12] would be a good candidate as they are numerically stable and can modify a large-scale matrix.

Consider a existent symmetric matrix A ∈ Rn×n with known eigenvalue decomposition A = QΛQT, to which a symmetric perturbation is added. Q is the matrix comprised of eigenvectors and Λ is a diagonal eigenvalue matrix, i.e., i^thursday column vector q_i of Q is the i^th eigenvector, and the i^th diagonal element of Λ (Λ_i) is the corresponding eigenvalue. If the perturbation is a rank-1 matrix, σvvT where σ and v are a scalar and a vector in Rn×1, the eigenvalue of new matrix A + σvv^T could exist given by a "secular equation" [4]:

where λi is the ith eigenvalue of original matrix A, and ζ_i is the ith element in vector z = QTv. If ζi are all nonzero and λi are distinct, so this equation has due north solutions. On interval [λ_i, λ_i + one], function f is monotonic. These properties lead to several efficient and stable methods to solve the secular equation [[4], [v]].

Notwithstanding, it is hard to find the rank-1 modification matrix during the heuristic algorithm. In well-nigh situations, the perturbation is not a rank-1 matrix. A reduction of the rank in the perturbation matrix is possible past utilizing eigenvalue decomposition later selecting a subset of eigen-pairs. The disadvantages of this approach are: 1) computationally expensive eigenvalue decomposition, and 2) poor accurateness when a subset of eigenvalue pairs are included. It will exist benign to modify high-rank eigenvalues straight. Eigenvalue decomposition is performed for a symmetric matrix, and the perturbation is made in preserving the symmetry. In an iterative method, bounden constraint sets change, which involves an update such every bit e_j ^Ta+a^Te_j where a is a vector in Rn×1 and eastward_j is the j^th column vector of the identity matrix in Rn×n. Such a alter is easily recognized without any farther analysis to place the perturbation vector as a rank-1 update process. This is the primary motivation for a rank-two update.

In this paper, we advise a new, direct rank-chiliad modification where n ≤ chiliad ≪ n, i.east., k is small but non negligible such equally k = one or two. In Department Theory of eigenvalue updates, we nowadays the theoretical background for the multiple-rank modification for rank-k updates. Section Numerical result lists the numerical results, and Department Decision outlines conclusions and future works.

Theory of eigenvalue updates

Secular equation of multiple-rank modification

The multiple-rank modification theory was carried out in [3]. Because the eigenvalue decomposition of a real symmetric matrix A + KK^T , where Thousand ∈ R^n×k , north ≤ k ≪ n. The eigenvalue decomposition of matrix A is known. Information technology is pointed out that the solution of a multiple-rank modification problem is equivalent to roots of:

f ( λ ) = det [ I k − K T ( λ I northward − A ) − ane Yard ] = det [ I g − U T ( λ I due north − Λ ) − i U ] = one + ∑ r = 1 k ( − ane ) r ∑ one ≤ k 1 ≤ ⋯ ≤ m r ≤ m 1 ≤ j ane ≤ ⋯ ≤ j r ≤ north [ det ( U j ane ⋯ j r ; one thousand one ⋯ k r ) ] 2 ( λ − λ j 1 ) ⋯ ( λ − λ j r )

(1)

where U = Q T K , U j 1 ⋯ j r ; k 1 ⋯ k r = ( e j ane ⋯ e j r ) T U ( e j one ⋯ e j r ) , and e_j is the j ^th column vector from an identity matrix [3]. Arbenz et al. point out that solving f ( λ ) = 0 is numerically hard [3]. While calculating the value of determinants is expensive, rearranging it is possible in the post-obit way:

∑ r = 1 k ( − 1 ) r ∑ 1 ≤ k 1 ≤ ⋯ ≤ k r ≤ k i ≤ j ane ≤ ⋯ ≤ j r ≤ n [ det ( U j 1 ⋯ j r ; g i ⋯ k r ) ] two ( λ − λ j 1 ) ⋯ ( λ − λ j r ) = ∑ r = 1 k ( − ane ) r ∑ one ≤ thousand i ≤ ⋯ ≤ k r ≤ k 1 ≤ j 1 ≤ ⋯ ≤ j r ≤ due north [ det ( U j 1 ⋯ j r ; k 1 ⋯ k r ) ] 2 ∑ i ∈ { j 1 ⋯ j r } [ ane ∏ southward ≠ i due south ∈ { j 1 ⋯ j r } ( λ s − λ i ) ] 1 λ i − λ = ∑ i = 1 n 1 λ i − λ { ∑ r = ane grand ∑ 1 ≤ 1000 1 ≤ ⋯ ≤ thou r ≤ k 1 ≤ j i ≤ ⋯ ≤ j r ≤ northward [ det ( U j i ⋯ j r ; k 1 ⋯ thou r ) ] ii ∏ southward ≠ i due south ∈ { j 1 ⋯ j r } ( λ s − λ i ) } = ∑ i = ane n α i { 0 } λ i − λ

(2)

where α i { 0 } = det [ 0 due east i T U U T due east i I r + U T D i U ] and D i = d i a g ( i λ 1 − λ i , ⋯ , 1 λ i − 1 − λ i , 0 , one λ i + one − λ i , ⋯ , 1 λ n − λ i )

Therefore, similar to a rank-1 modification, f also can take a secular equation:

This secular equation is called "multiple-rank secular equation". The matrix determinant is (m +one) × (k +one) matrix. When thousand ≪ n, formulating (three) will non increase the computation price significantly; all the same, with increased k, the determinants calculation involves a heavy ciphering, making this approach inefficient.

Another advantage is that this method does not require an orthogonal perturbation matrix 5. Therefore, an additional eigenvalue decomposition for the perturbation matrix is not necessary. Still, solving multiple-rank secular equations is more computationally demanding than that of the rank-1 update process. In a rank-i modification, f is monotonically increasing on each subinterval (λ_i, λ_i+i). Therefore, simply one root exists in the range of [λ_i, λ_i+1]. The multiple-rank secular role is not monotonic on an interval of (λ_i, λ_i+1). At that place would exist at least 0 and at near r solutions. Although multiple-rank modification problems take a similar secular equation to a rank-one, most of the methods used to solve rank-1 problems in [iii] cannot be directly used for multiple-rank bug. To solve the multiple-rank secular equations, we codify two sub-problems: the first sub-trouble is intended to locate eigenvalues in checking how many roots are on each subinterval; the second finds the value for these roots on each subinterval. The details are listed in Section Numerical Effect. We may not need to update all of the eigenvalues—if only the eigenvalues in some specific interval are of interest, then the start sub-problem is plenty.

Location of eigenvalues

In the procedure for the rank-2 modification, eigenvalues are located according to the signs of the coefficients in a polynomial long division, μi. For the case of rank-k, they become more than complicated to locate. In general, the one-column-i-row update approximates the modification while preserving the sparsity and the symmetry [5]:

Step 1. A = A 0 + a b T + b a T = Q ( Λ + x y T + y x T ) Q T

Step 2. Λ + x y T + y x T = Λ + ( 10 y ) [ 0 one 1 0 ] ( x y ) T = Λ + ( x y ) S [ 1 0 0 − ane ] South T ( x y ) T

where S is the Schur complement. Let ( u 1 u 2 ) = ( ten y ) Southward, which yields Λ + x y T + y 10 T = Λ + u i u 1 T − u 2 u 2 T. Therefore, if we can find the dominant column-row pair, it is acceptable to update eigenvalues using several rank-2 modifications.

Considering the secular equation of a rank-2 modification,A + σ 1 five 1 v ane T + σ 2 v 2 v 2 T with σ1 > 0 and σ2 > 0. In this case, the values of all the eigenvalues increase, which is termed a double-correct shift. R is divided into n+1 subintervals past original eigenvalues { λ ane , … , λ n }. When the rank of the perturbation matrix is 2, there are 0, 1, or 2 roots within each subinterval. Nosotros define a location vector L ∈ Z ( n + 1 ) × 1, where each element represents the number of roots within the respective subinterval. We can estimate locations of new eigenvalues using the Courant-Weyl principle [6]:

λ thousand ≤ λ k max ; one ≤ one thousand ≤ n a n d λ thousand max ≤ λ k + 2 ; ane ≤ k ≤ due north − 2

Also, from the signs of μi, the parity of 50 could exist determined.

If μ_i  ·μ _i + ane > 0, there is one eigenvalue in (λ_i, λ_i + i ); otherwise, in that location are either zero or two eigenvalues in this interval (λ_i, λ_i + ane ). These 3 conditions are already sufficient to locate new eigenvalues. Fig. i illustrates a pseudo-lawmaking of Algorithm 1a for the location vector.

Pseudo-code of formulating locating vector when the matrix is under double-right shift update.

Note that the status of Algorithm 1a is σ₁ > 0 and σ_ii > 0. The condition can be generalized as either σ₁ < 0 and σ₂ < 0 (Algorithm 1b), or as σ_i < 0 and σ₂ > 0 (Algorithm 1c). For the first case, instead of searching for a root from the start subinterval, we can beginning from the last subinterval. And for situation (c), the sign of μ1 tin can determine the number of eigenvalues on the first interval. Fig. 2 shows the pseudo-code for these 2 conditions.

Pseudo-code of generating location vector when matrix is under double-left shift update and one-left-one-right shift update.

For a general rank-k modification, the sign of μi is bereft to calculate the location vector. We advise a modified Sturm Chain method as follows.

The original Sturm chain or Sturm sequence [7] is a finite sequence of polynomials p 0 ( λ ) , p ane ( λ ) , ⋯ , p m ( λ ) of a decreasing degree with the following properties:

• •

  p 0 ( λ )

  is foursquare free (no square factors, i.e., no repeated roots)

• •

  p 0 ( λ ) = 0

  , so

  sign ( p ′ ( λ ) ) = sign ( p one ( λ ) )

• •

  If

  p i ( λ ) = 0 , 0 < i < chiliad

  ,

  sign ( p i − 1 ( λ ) ) = − sign ( p i + 1 ( λ ) )

• •

  p m ( λ )

  does non alter its sign.

Then past observing the signs of p 0 ( a ) , p 1 ( a ) … p m ( a ) , and p 0 ( b ) , p 1 ( b ) … p m ( b ), the number of roots of p 0 ( λ ) inside (a, b) could exist determined. Although the original Strum Theorem is for a polynomial office, we found that it can too exist implemented on a secular function. For our secular equation, we define that:

f 1000 ( λ ) = p one thousand ( λ ) π m ( λ ) = c thou − ∑ j = 1 due north − m α j { yard } λ − λ j w h eastward r eastward π yard ( λ ) = ∏ j = ane n − m ( λ − λ j )

(4)

The c_m is the caliber and α j { thousand } is the scalar coefficients of the residual associated with ( λ j − λ ), which tin can be evaluated from p m ( λ ) and π m ( λ ). Here we describe how nosotros compute c_g and α j { m } directly without evaluating p 1000 ( λ ) and π 1000 ( λ ). To expand the Sturm sequence of a non-polynomial secular equation, f 0 ( λ ), we multiply the production of all n denominators in the summation in (4). Then we expand the m^th Sturm sequence and carve up a proper product to make a similar class as in (4). Note that no eigenvalues are repeated. For f 1 ( λ ), to satisfy the second condition, nosotros need to take the derivative of f 0 ( λ ). Direct derivation of f 0 ( λ ) volition lead to second-order terms. Instead, take the derivative of the post-obit part:

p 0 ( λ ) = π 0 ( λ ) ( 1 − ∑ j = 1 north α j { 0 } λ − λ j )

(5)

Then we have:

p 1 ( λ ) = d p 0 ( λ ) d λ = d f 0 ( λ ) d λ π 0 ( λ ) + f 0 ( λ ) ( ∑ j = i n 1 λ − λ j ) π 0 ( λ ) = { ∑ j = 1 n α j { 0 } λ − λ j one λ − λ j + ∑ j = one north ane λ − λ j − ∑ j = 1 n ∑ i = 1 due north 1 λ − λ i α j { 0 } λ − λ j } π 0 ( λ ) = { ∑ j = 1 n α j { 0 } λ − λ j 1 λ − λ j + ∑ j = 1 n one λ − λ j − ∑ j = 1 n ∑ i = 1 n 1 λ − λ j α i { 0 } + α j { 0 } λ j − λ i } π 0 ( λ ) = [ ∑ j = 1 n μ j { ane } λ − λ j ] π 0 ( λ ) [ c ane − ∑ j = ane n − 1 α j { 1 } λ − λ j ] π 1 ( λ ) where μ j { one } = 1 − ∑ j = i i ≠ j n α i { 0 } + α j { 0 } λ j − λ i , c 1 = ∑ j = one north μ j { 1 } , and α j { i } = ( λ n − λ j ) μ j { 1 }

(half-dozen)

Here α i { 1 } is non related to λ. The Sturm concatenation theorem finds the number of solutions inside a given range to satisfy p 0 ( λ ) = 0. Even so, the directly application of the theorem to p 0 ( λ ) is not applied because the ciphering of the coefficients is numerically unstable. Inspired by the fact that only the change in the signs of p i ( λ ) is important, which is the production between π i ( λ ) and f i ( λ ), nosotros propose a modified Sturm series to locate the updated eigenvalues. For this purpose, p 2 ( λ ) … p n ( λ )are generated by long division used for polynomial functions that can be also expanded to a secular function.

In the Sturm chain theorem, the number of changes in the sign of the polynomial functions p 0 ( λ ) … p northward ( λ ) equals the number of solutions inside a region. In [−∞, ∞], the signs of f 0 ( λ ) … f n ( λ ) are same equally those of c₀…c_n, the signs of π 0 ( ∞ ) … π northward ( ∞ ) are all positive, and those of π 0 ( − ∞ ) … π northward ( − ∞ ) are all negative. As a result, the solutions that satisfy p 0 ( λ ) = 0 are the number of positives in c0…cn where c₀  = one. In the computation process of c₀…c_{due north}, the floating number tin can result in a alter in the sign of c₀…c_n. If 1 terminates the Secular long division when it hits the non-positive c for the beginning time, the partial Sturm series yields the lower bound of the solutions between [−∞, ∞]. Furthermore, one can interruption down the regions where at least one solution exists, yielding a solution as in Department Algorithm to find solutions for the secular equation. Suppose ( N s 0 + ane ) c is positive at the first Sturm series, and accordingly Northward s 0 solutions ξs are identified. So a new polynomial function p 0 i ( λ ) is defined every bit follows:

p 0 ( λ ) = [ i − ∑ j = 1 n α j { 0 } λ − λ j ] ∏ j = 1 n ( λ − λ j ) = p 0 1 ( λ ) ∏ s = one N due south 0 ( λ − ξ s ) → p 0 1 ( λ ) = [ 1 − ∑ j = 1 n α j { 0 } λ − λ j ] ∏ j = ane n ( λ − λ j ) ∏ due south = 1 N s 0 ( λ − ξ s )

(7)

By multiplying the first term after the square parenthesis, (7) becomes:

p 0 1 ( λ ) = [ 1 − ∑ j = ane n α j { 0 } λ − λ j ] λ − λ 1 λ − ξ 1 ∏ j = ii due north ( λ − λ j ) ∏ s = 2 North s 0 ( λ − ξ s ) = [ ane − ∑ j = i due north ( λ j − λ 1 λ j − ξ 1 ) α j { 0 } x − λ j − λ 1 − ξ 1 10 − ξ one ( 1 − ∑ j = 1 n α j { 0 } ξ 1 − λ j ) ] ∏ j = ii north ( λ − λ j ) ∏ south = 2 N s 0 ( λ − ξ s )

(8)

Since ξ1 is a solution to p 0 ( λ ) = 0 (i.e., p 0 ( ξ 1 ) = 0), then π 0 ( ξ i ) ≠ 0, and 1 − ∑ j = 1 n α j { 0 } ξ 1 − d j vanishes. Afterwards multiplying all N s 0 terms, p 0 1 ( λ ) becomes:

p 0 1 ( λ ) = [ 1 − ∑ j = 1 n β j { 0 } λ − λ j ] ∏ j = N s 0 + 1 n ( λ − λ j ) where β j { 0 } = [ ∏ p = i N due south 0 d j − d p d j − ξ p ] α j { 0 }

(9)

The formula for p 0 1 ( λ ) resembles that for p 0 ( λ ), which allows the Sturm series to expand with fewer terms by N s 0. Since the first c in p 0 1 ( λ ) is positive, it is guaranteed to accept at to the lowest degree ane solution using the new Sturm serial. This process will continue until all northward solutions are identified.

Algorithm to discover solutions for the secular equation

Unlike from the rank-1 modification, the secular equation of rank-2 is not monotonically increasing at each interval. Therefore, about of the algorithms for a rank-i update may not converge. The Divide-and-Conquer (D&C) method efficiently addresses this problem [13]. Equally long as the location vector is formulated, the D&C is fast and parallelizable. The strategy to find the roots to the secular equation is listed in the D&C zero finder. Fig. 3 is an example of a rank-2 secular equation. In the trouble, the signs of the coefficients are mixed, i.east. positive and negative. This is commonly observed when both updating and downdating are performed in a rank-two update. Nosotros would like to find where the roots exist to apply for the D&C nix finder. Algorithm 1c provides an efficient way to locate the root-termed location vector. Since it is a rank-ii update between two eigenvalues, 0, i, or 2 roots can be. Algorithm 1c utilizes the fact that, every bit shown in Fig. 3, (−∞, 0), (0, 1), (ane, 1.5), (1.5, three), (three, 4), and (4, ∞), at well-nigh ii roots exist.

Graph of rank-two secular role where f ( x ) = one + three − x + three 1 − x + xxx 1.5 − 10 + − 3 3 − 10 + 3 iv − 10.

With a location vector, the D&C method can be utilized.

Algorithm to update eigenvectors

Once an eigenvalue is computed, the corresponding eigenvector 5 tin be updated using the human relationship betwixt an eigenvalue and an eigenvector, i.e., ( Λ + Fifty L T ) u = λ u where L = Q^TYard and u = Q^Tv. Solving the relationship for u yields u = ( Λ − λ I ) − 1 L α where α = − 50^Tu, α ∈ R^k , which implies the eigenvector is in the column space spanned by the rank-chiliad update Grand. α is in the simple-article space of { I + L T ( Λ − λ I ) − 1 Fifty } k × k, and let α₀ denote the simple-article infinite vector. The row-rank update k where n ≤ k ≪ north makes finding the elementary-article infinite computationally inexpensive. Since the eigenvector is a unit vector, v becomes:

v = Q ( Λ − λ I ) − 1 Q T K α 0 ∥ Q ( Λ − λ I ) − i Q T K α 0 ∥ 2

(ten)

Numerical effect

In the D&C method, the tolerance of the error in a root is set to 10⁻⁹ and so that the precision of the solution is within a numerical error range. Fig. four illustrates the comparing of the computation time of a rank-i update, a rank-2 update, and the original eigenvalue decomposition. For the numerical computation to perform straight eigenvalue decomposition, we used the MATLAB eig function [14]. The testing results shows the order p of the computation are in O ( n p ):

• •

  2 times rank-one update: p = 1.81

• •

  Rank-2 update: p = 1.45

• •

  Direct eigenvalue decomposition with MATLAB: p = two.88

Computation fourth dimension of two times rank-ane update, rank-2 update, and original eigenvalue decomposition.

The computation cost of a rank-ii modification is significantly better than that of a rank-1 modification. Parallel ciphering will further enhance the computational efficiency.

In Fig. 5, the red line is the computation fourth dimension for a randomly generated rank-3 update, and the blue line is that of original eigenvalue decomposition. Updating the eigenvalues by the secular Sturm Chain method is in O(due north^1.95), which is more expensive than a rank-two update; nonetheless, it is even so much more efficient than the original eigenvalue decomposition.

Comparison of the computation time for Rank-3 update of the proposed method and the original MATLAB eig office.

When the perturbed matrix K is small, i.due east., norm(1000) << 1, the eigenvalue and eigenvector pairs can be estimated from:

λ j = λ j 0 + ( K T v j 0 ) T ( Yard T v j 0 ) and v j = v j 0 + ∑ i = 1 i ≠ j n ( Thousand T 5 j 0 ) T ( K T v i 0 ) λ i 0 − λ j 0 v j 0

(11)

However, every bit the norm increases, the perturbation method starts to fail as its supposition to ignore higher order terms is invalid. Fig. 7 shows the comparison where the norm(K) is 0.3 and n = 100. While the performance of the proposed method does non change with the norm of M, the eigenvalues, the orthogonality, and the production betwixt eigenvalue and eigenvector pairs all deviate from the desired quantities.

Comparison of the results in terms of the rank of 1000 from direct eigenvalue decomposition, the perturbation method in Ref. [15], and the proposed method outlined in (eleven) when norm(K) = 0.3. (A) 2-norm mistake in the eigenvalues relative to the direct decomposition; (B) ii-norm error of the matrices with various decomposition techniques; (C) 2-norm mistake of the products of eigenvector matrix and its transpose to check the orthogonality; and (D) computation time of the direct eigenvalue decomposition, of the eigenvalue update and of the eigenvector update of the proposed method, and of the perturbation method [15].

Fig. half-dozen illustrates the comparisons in the results amid direct eigenvalue decomposition, the proposed method, and the perturbation method when the norm(K) is 0.01 and n = 100. When the matrix K is very small, the perturbation method [15] provides a reasonably close estimate of the eigenvalue and eigenvector pair. Fig. vii shows that the mistake of the perturbation method [xv] can be large in estimating the eigenvalue and the eigenvector when the norm(K) is non insignificant. The proposed method always provides a precise estimation for both eigen-pairs. While the proposed method is efficient when the rank of the matrix K is modest (approximately 10% of n) but greater than n, the computation time increases as the rank of G increases. The proposed method is more efficient than whatsoever other to compute a subset of eigenvectors when n ≤ k ≪ n. For case, a simple-commodity space is useful in many applied science fields, and the identification of an unabridged eigenvector infinite may not be necessary. In such a circumstance, the eigenvalues are all zeros and the computation fourth dimension to find eigenvectors is however much smaller than that of directly eigenvalue decomposition (compare the dotted blue line and the black solid line in (D) from Fig. 6, Fig. 7). For the consideration of ciphering time, the ciphering of the Sturm series increases in ϑ(n² ) because the major computation task of the proposed method is to compute the eigenvalue of I k − U T ( λ I northward − Λ ) − 1 U. In comparison, the direct eigenvalue decomposition of the matrix takes in ϑ(nk² ). For a given value of k, the ciphering time grows in quadratic (See Fig. 8 for the results where k = forty–60 for various values of northward). The values for p in ϑ(n^p ) decrease with increasing k (2.45, 2.19, and 2.16 for thousand = 40, l, and threescore, respectively).

Comparison of the results in terms of the rank of K from direct eigenvalue decomposition, the perturbation method in Ref. [15], and the proposed method outlined in (11) when norm(K) = 0.01. (A) two-norm error in the eigenvalues relative to the direct decomposition; (B) 2-norm error of the matrices with various decomposition techniques; (C) ii-norm mistake of the products of eigenvector matrix and its transpose to check the orthogonality; and (D) ciphering time of the directly eigenvalue decomposition, of the eigenvalue update and of the eigenvector update of the proposed method, and of the perturbation method [15].

Ciphering times for the proposed method with gear up values for k and with diverse due north. The order p is approximately 2, i,eastward, ϑ(n² ) when k ≪n.

Determination

In this paper, our new algorithm for a rank-thousand modification of eigenvalue decomposition is presented. Computation performance is significantly improved in comparing to rank-i modification methods. We propose a method relying on the location vector, which is tested on several systems and results show our proposed method is efficient. When there is a topology change in a network, marketplace power is greatly affected and is reflected in the symmetric dispatch sensitivity to cost. Nosotros plan to apply this algorithm to update marketplace power computation associated with topological control [16]. Considering a topological change affects many just not all locations, the status (north ≤ k ≪ northward) is satisfied where yard and n are the numbers of the affected substations and of all the substations, respectively.

Acquittance

This work is supported by the U.S. Depart of Energy through the CERTS (Consortium for Electric Reliability Technology Solutions) program, PO# 7069673.

Appendix A

Suppose there exists a secular long sectionalisation series equally follows:

p chiliad − 2 ( x ) − A yard ( x − B m ) p thousand − 1 ( x ) + p k ( x ) = 0 west h due east r e { p one thousand ( x ) = ( c m − ∑ j = 1 n − m α j { one thousand } x − d j ) ∏ k = one n − m ( x − d k ) p thousand − i ( ten ) = ( c thou − one − ∑ j = ane n − 1000 + 1 α j { yard − 1 } x − d j ) ∏ m = 1 due north − m + one ( x − d k ) p m − 2 ( x ) = ( c m − ii − ∑ j = 1 north − m + 2 α j { m − 2 } x − d j ) ∏ 1000 = 1 due north − m + 2 ( x − d grand )

(A1)

By dividing (A1) by c m − 2 ∏ k = 1 n − thousand + 2 ( 10 − d thousand ), one finds

one − ∑ j = ane n − thousand + two β j { k − 2 } ten − d j + h k ( − one + ∑ j = i n − chiliad + 1 β j { m − 1 } 10 − d j + B m − d n − grand + two ten − d n − m + two − B grand − d due north − yard + two x − d northward − m + 2 ∑ j = 1 north − m + 1 β j { m − ane } x − d j ) + g m x − d n − m + 2 one x − d north − k + one − ∑ j = ane n − g γ j { m } x − d j one x − d n − thousand + 2 1 x − d n − thou + ane = 0

(A2)

where 1000 k = c chiliad c m − 2 , h m = c m − i c one thousand − ii A m , β j { m − 2 } = α j { m − two } c thou − ii , β j { m − 1 } = α j { m − 1 } c yard − 1, and γ j { thousand } = α j { m } c m − 2.

Setting the constant term zero leads to hm = 1, i.e., A yard = c m − 2 c yard − 1. Rearranging (A2) yields,

− ∑ j = i n − m + 2 β j { thou − 2 } x − d j + ∑ j = 1 n − m + 1 β j { m − 1 } x − d j + B one thousand − d north − thousand + 2 x − d n − m + 2 − ∑ j = i n − yard + 1 β j { k − one } 10 − d j B thou − d northward − grand + two x − d due north − m + 2 + g m x − d N − m + ii ane x − d n − m + 1 − ∑ j = i n − m γ j { yard } x − d j ane x − d n − m + 2 1 x − d due north − yard + 1 = 0

(A3)

The product of fractions can exist expanded as follows:

{ 1 x − d j one x − d n − k + 2 = one d north − m + ii − d j i x − d n − yard + 2 − 1 d north − g + 2 − d j 1 ten − d j 1 x − d n − m + 2 1 x − d n − thousand + 1 = 1 d n − thousand + 2 − d north − yard + one 1 x − d n − m + two − 1 d n − chiliad + 2 − d north − m + 1 1 x − d north − g + 1 1 ten − d n − m + 2 1 x − d northward − yard + one one x − d j = 1 d n − m + 2 − d n − m + 1 1 d n − yard + two − d j ane 10 − d northward − thousand + two + 1 d northward − m + two − d north − grand + one − 1 d north − m + i − d j i x − d n − 1000 + 1 + i d n − m + 2 − d j one d n − thousand + ane − d j 1 x − d j

(A4)

From (A3) and (A4), one finds:

∑ j = 1 n − one thousand [ β j { m − one } − β j { g − ii } ] 1 ten − d j + [ B 1000 − d n − 1000 + ii − β n − m + ii { g − two } ] 1 x − d n − g + 2 + [ β n − k + ane { thousand − 1 } − β n − m + 1 { one thousand − 2 } ] 1 ten − d n − m + 1 ∑ j = 1 n − m [ β j { one thousand − 1 } − β j { chiliad − two } + β j { yard − 1 } ( B m − d N − m + 2 ) d northward − m + 2 − d j − 1 d n − m + 2 − d j γ j { m } d n − one thousand + 1 − d j ] 1 ten − d j + [ ( β n − grand + 1 { m − one } − β due north − m + ane { m − two } ) ( d n − m + 2 − d due north − m + 1 ) + β northward − m + ane { m − one } ( B m − d northward − m + two ) − chiliad yard − ∑ j = ane n − m γ j { grand } d north − thousand + ane − d j ] × 1 d north − m + ii − d due north − chiliad + 1 i x − d north − m + one + [ ( B 1000 − d due north − m + ii − β n − m + two { m − ii } ) ( d due north − k + 2 − d northward − k + 1 ) + k k + ( B g − d n − chiliad + 2 ) ( d n − thou + 2 − d due north − m + 1 ) ∑ j = one n − m + 1 − β j { m − i } d n − m + two − d j + ∑ j = ane n − m − γ j { grand } d n − m + 2 − d j ] × 1 d n − m + ii − d n − m + 1 1 ten − d n − m + 2 = 0

(A5)

Setting all the constants of x-d_j in (A5) zero yields:

{ B m = d n − m + 2 + ∑ j = 1 northward − g + 2 β j { one thousand − 2 } − ∑ j = 1 due north − thousand + one β j { m − 1 } γ j { m } = ( d north − m + 1 − d j ) [ − ( d n − m + 2 − d j ) β j { chiliad − 2 } + β j { chiliad − 1 } ( B m − d j ) ] g m = B m ∑ j = i n − m + 1 β j { m − ane } − ∑ j = 1 north − g + 2 β j { yard − 2 } ( d n − m + two − d j ) − ∑ j = 1 n − g + 1 β j { m − 1 } d j

(A6)

With (A6), information technology is possible to formulate:

f thou ( 10 ) = p grand ( x ) ∏ k = ane north − 1000 ( x − d k ) = c grand − ∑ j = 1 n − 1000 α j { m } 10 − d j

(A7)

where cm = cm−2gm and α j { 1000 } = c chiliad − two γ j { m }.

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Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6314274/

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