Verified Error Bounds for Real Eigenvalues of Real Symmetric and Persymmetric Matrices

Algorithm 1

VerifyEig.

	Input: an symmetric or persymmetric matrix; : the tolerance of numerical rank.
	Output { }, , and an index set .
	Step 1 Use eig code to calculate all distinct real eigenvalues {} of .
	Step 2 For each , compute the singular value decomposition of the matrix or .
	Step 3 Solve (25) and (27) or (29) and (31) to obtain Jacobi matrix .
	Step 4 If is full rank, choose the index set such that is nonsingular.
	Step 5 Compute the constants by (36).
	Step 6 Compute the constant by (38).
	Step 7 If , then compute by (5).
	Step 8 If , return , , and .