Purpose
To compute two Givens rotations (C1,S1) and (C2,S2) such that the orthogonal matrix [ Q 0 ] [ C1 S1 0 ] [ 1 0 0 ] Z = [ ], Q := [ -S1 C1 0 ] * [ 0 C2 S2 ], [ 0 I ] [ 0 0 1 ] [ 0 -S2 C2 ] makes the first column of the real Wilkinson double shift polynomial of the product of matrices in periodic upper Hessenberg form, stored in the array A, parallel to the first unit vector. Only the rotation defined by C1 and S1 is used for the real Wilkinson single shift polynomial (see SLICOT Library routines MB03BE or MB03BF). All factors whose exponents differ from that of the Hessenberg factor are assumed nonsingular. The matrix product is evaluated.Specification
SUBROUTINE MB03AG( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, C1, $ S1, C2, S2, IWORK, DWORK ) C .. Scalar Arguments .. CHARACTER SHFT INTEGER K, LDA1, LDA2, N, SINV DOUBLE PRECISION C1, C2, S1, S2 C .. Array Arguments .. INTEGER AMAP(*), IWORK(*), S(*) DOUBLE PRECISION A(LDA1,LDA2,*), DWORK(*)Arguments
Mode Parameters
SHFT CHARACTER*1 Specifies the number of shifts employed by the shift polynomial, as follows: = 'D': two shifts (assumes N > 2); = 'S': one real shift.Input/Output Parameters
K (input) INTEGER The number of factors. K >= 1. N (input) INTEGER The order of the factors. N >= 2. AMAP (input) INTEGER array, dimension (K) The map for accessing the factors, i.e., if AMAP(I) = J, then the factor A_I is stored at the J-th position in A. AMAP(1) is the pointer to the Hessenberg matrix. S (input) INTEGER array, dimension (K) The signature array. Each entry of S must be 1 or -1. SINV (input) INTEGER Signature multiplier. Entries of S are virtually multiplied by SINV. A (input) DOUBLE PRECISION array, dimension (LDA1,LDA2,K) The leading N-by-N-by-K part of this array must contain the product (implicitly represented by its K factors) in periodic upper Hessenberg form. LDA1 INTEGER The first leading dimension of the array A. LDA1 >= N. LDA2 INTEGER The second leading dimension of the array A. LDA2 >= N. C1 (output) DOUBLE PRECISION S1 (output) DOUBLE PRECISION On exit, C1 and S1 contain the parameters for the first Givens rotation. C2 (output) DOUBLE PRECISION S2 (output) DOUBLE PRECISION On exit, if SHFT = 'D', C2 and S2 contain the parameters for the second Givens rotation. Otherwise, C2 = 1, S2 = 0.Workspace
IWORK INTEGER array, dimension (2*N) DWORK DOUBLE PRECISION array, dimension (2*N*N) On exit, DWORK(N*N+1:N*N+N) and DWORK(N*N+N+1:N*N+2*N) contain the real and imaginary parts, respectively, of the eigenvalues of the matrix product.Method
The necessary elements of the real Wilkinson double shift polynomial are computed, and suitable Givens rotations are found. For numerical reasons, this routine should be called when convergence difficulties are encountered for small order matrices and small K, e.g., N, K <= 6.Further Comments
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