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 needed for the real Wilkinson single shift polynomial (see the SLICOT Library routines MB03BE or MB03BF). The shifts are defined based on the eigenvalues (computed externally by the SLICOT Library routine MB03BB) of the trailing 2-by-2 submatrix of the matrix product. See the definitions of the arguments W1 and W2.Specification
SUBROUTINE MB03AB( SHFT, K, N, AMAP, S, SINV, A, LDA1, LDA2, W1, $ W2, C1, S1, C2, S2 ) C .. Scalar Arguments .. CHARACTER SHFT INTEGER K, LDA1, LDA2, N, SINV DOUBLE PRECISION C1, C2, S1, S2, W1, W2 C .. Array Arguments .. INTEGER AMAP(*), S(*) DOUBLE PRECISION A(LDA1,LDA2,*)Arguments
Mode Parameters
SHFT CHARACTER*1 Specifies the number and type of shifts employed by the shift polynomial, as follows: = 'C': two complex conjugate shifts; = 'D': two real identical shifts; = 'R': two real shifts; = 'S': one real shift. When the eigenvalues are complex conjugate, this argument must be set to 'C'.Input/Output Parameters
K (input) INTEGER The number of factors. K >= 1. N (input) INTEGER The order of the factors in the array A. N >= 2, for a single shift polynomial; N >= 3, for a double shift polynomial. 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, defined by A(:,:,AMAP(1)). 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 an n-by-n 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. W1 (input) DOUBLE PRECISION The real part of the first eigenvalue. If SHFT = 'S', this argument is not used. W2 (input) DOUBLE PRECISION The second eigenvalue, if both eigenvalues are real, else the imaginary part of the complex conjugate pair. 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, C2 and S2 contain the parameters for the second Givens rotation. If SHFT = 'S', C2 = 1, S2 = 0.Method
Givens rotations are properly computed and applied.Further Comments
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Program Text
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