LAHQR - 5.2 English - 68552

LAHQR computes the eigenvalues and Schur factorization of an upper

Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Purpose:

   LAHQR is an auxiliary routine called by SHSEQR to update the
   eigenvalues and Schur decomposition already computed by SHSEQR, by
   dealing with the Hessenberg submatrix in rows and columns ILO to
   IHI.

Parameters:

• WANTT – [in]

  WANTT is LOGICAL

  = .TRUE. : the full Schur form T is required;

  = .FALSE.: only eigenvalues are required.

• WANTZ – [in]

  WANTZ is LOGICAL

  = .TRUE. : the matrix of Schur vectors Z is required;

  = .FALSE.: Schur vectors are not required.

• N – [in]

  N is INTEGER

  The order of the matrix H. N >= 0.

• ILO – [in] ILO is INTEGER

• IHI – [in]

  IHI is INTEGER

  It is assumed that H is already upper quasi-triangular in rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1). SLAHQR works primarily with the Hessenberg submatrix in rows and columns ILO to IHI, but applies transformations to all of H if WANTT is .TRUE..

  1 <= ILO <= fla_max(1,IHI); IHI <= N.

• H – [inout]

  H is REAL array, dimension (LDH,N)

  On entry, the upper Hessenberg matrix H.

  On exit, if INFO is zero and if WANTT is .TRUE., H is upper quasi-triangular in rows and columns ILO:IHI, with any 2-by-2 diagonal blocks in standard form. If INFO is zero and WANTT is .FALSE., the contents of H are unspecified on exit. The output state of H if INFO is nonzero is given below under the description of INFO.

• LDH – [in]

  LDH is INTEGER

  The leading dimension of the array H. LDH >= fla_max(1,N).

• WR – [out] WR is REAL array, dimension (N)

• WI – [out]

  WI is REAL array, dimension (N)

  The real and imaginary parts, respectively, of the computed eigenvalues ILO to IHI are stored in the corresponding elements of WR and WI. If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with WR(i) = H(i,i), and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block,

  WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).

• ILOZ – [in] ILOZ is INTEGER

• IHIZ – [in]

  IHIZ is INTEGER

  Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE..

  1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

• Z – [inout]

  Z is REAL array, dimension (LDZ,N)

  If WANTZ is .TRUE., on entry Z must contain the current matrix Z of transformations accumulated by SHSEQR, and on exit Z has been updated; transformations are applied only to the submatrix Z(ILOZ:IHIZ,ILO:IHI).

  If WANTZ is .FALSE., Z is not referenced.

• LDZ – [in]

  LDZ is INTEGER

  The leading dimension of the array Z. LDZ >= fla_max(1,N).

• INFO – [out]

  INFO is INTEGER

  = 0: successful exit

  > 0: If INFO = i, SLAHQR failed to compute all the eigenvalues ILO to IHI in a total of 30 iterations per eigenvalue; elements i+1:ihi of WR and WI contain those eigenvalues which have been successfully computed.

  If INFO > 0 and WANTT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigenvalues of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H.

  If INFO > 0 and WANTT is .TRUE., then on exit (*) (initial value of H)*U = U*(final value of H) where U is an orthogonal matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI.

  If INFO > 0 and WANTZ is .TRUE., then on exit (final value of Z) = (initial value of Z)*U where U is the orthogonal matrix in (*) (regardless of the value of WANTT.)