LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

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Functions/Subroutines  
subroutine  slaqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO) 
SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. 
subroutine slaqr4  (  logical  WANTT, 
logical  WANTZ,  
integer  N,  
integer  ILO,  
integer  IHI,  
real, dimension( ldh, * )  H,  
integer  LDH,  
real, dimension( * )  WR,  
real, dimension( * )  WI,  
integer  ILOZ,  
integer  IHIZ,  
real, dimension( ldz, * )  Z,  
integer  LDZ,  
real, dimension( * )  WORK,  
integer  LWORK,  
integer  INFO  
) 
SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
Download SLAQR4 + dependencies [TGZ] [ZIP] [TXT]SLAQR4 implements one level of recursion for SLAQR0. It is a complete implementation of the small bulge multishift QR algorithm. It may be called by SLAQR0 and, for large enough deflation window size, it may be called by SLAQR3. This subroutine is identical to SLAQR0 except that it calls SLAQR2 instead of SLAQR3. SLAQR4 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasitriangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors. Optionally Z may be postmultiplied into an input orthogonal matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
[in]  WANTT  WANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required. 
[in]  WANTZ  WANTZ is LOGICAL = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required. 
[in]  N  N is INTEGER The order of the matrix H. N .GE. 0. 
[in]  ILO  ILO is INTEGER 
[in]  IHI  IHI is INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO1 and IHI+1:N and, if ILO.GT.1, H(ILO,ILO1) is zero. ILO and IHI are normally set by a previous call to SGEBAL, and then passed to SGEHRD when the matrix output by SGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI should be set to 1 and N, respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0. 
[in,out]  H  H is REAL array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and WANTT is .TRUE., then H contains the upper quasitriangular matrix T from the Schur decomposition (the Schur form); 2by2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is .FALSE., then the contents of H are unspecified on exit. (The output value of H when INFO.GT.0 is given under the description of INFO below.) This subroutine may explicitly set H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO1 or j = IHI+1, IHI+2, ... N. 
[in]  LDH  LDH is INTEGER The leading dimension of the array H. LDH .GE. max(1,N). 
[out]  WR  WR is REAL array, dimension (IHI) 
[out]  WI  WI is REAL array, dimension (IHI) The real and imaginary parts, respectively, of the computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) and WI(ILO:IHI). If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the ith and (i+1)th, with WI(i) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then 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 2by2 diagonal block, WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = WI(i). 
[in]  ILOZ  ILOZ is INTEGER 
[in]  IHIZ  IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N. 
[in,out]  Z  Z is REAL array, dimension (LDZ,IHI) If WANTZ is .FALSE., then Z is not referenced. If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value of Z when INFO.GT.0 is given under the description of INFO below.) 
[in]  LDZ  LDZ is INTEGER The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1. 
[out]  WORK  WORK is REAL array, dimension LWORK On exit, if LWORK = 1, WORK(1) returns an estimate of the optimal value for LWORK. 
[in]  LWORK  LWORK is INTEGER The dimension of the array WORK. LWORK .GE. max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended. If LWORK = 1, then SLAQR4 does a workspace query. In this case, SLAQR4 checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed. 
[out]  INFO  INFO is INTEGER \verbatim INFO is INTEGER = 0: successful exit .GT. 0: if INFO = i, SLAQR4 failed to compute all of the eigenvalues. Elements 1:ilo1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO .GT. 0 and WANT is .FALSE., then on exit, the remaining unconverged eigenvalues are the eigen values of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H. If INFO .GT. 0 and WANTT is .TRUE., then on exit (*) (initial value of H)*U = U*(final value of H) where U is a orthogonal matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI. If INFO .GT. 0 and WANTZ is .TRUE., then on exit (final value of Z(ILO:IHI,ILOZ:IHIZ) = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal matrix in (*) (regard less of the value of WANTT.) If INFO .GT. 0 and WANTZ is .FALSE., then Z is not accessed. 
Definition at line 265 of file slaqr4.f.