Annotation of rpl/lapack/lapack/dlaqz0.f, revision 1.1
1.1 ! bertrand 1: *> \brief \b DLAQZ0
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DLAQZ0 + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlaqz0.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlaqz0.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlaqz0.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * RECURSIVE SUBROUTINE DLAQZ0( WANTS, WANTQ, WANTZ, N, ILO, IHI, A,
! 22: * $ LDA, B, LDB, ALPHAR, ALPHAI, BETA,
! 23: * $ Q, LDQ, Z, LDZ, WORK, LWORK, REC,
! 24: * $ INFO )
! 25: * IMPLICIT NONE
! 26: *
! 27: * Arguments
! 28: * CHARACTER, INTENT( IN ) :: WANTS, WANTQ, WANTZ
! 29: * INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
! 30: * $ REC
! 31: *
! 32: * INTEGER, INTENT( OUT ) :: INFO
! 33: *
! 34: * DOUBLE PRECISION, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ),
! 35: * $ Q( LDQ, * ), Z( LDZ, * ), ALPHAR( * ),
! 36: * $ ALPHAI( * ), BETA( * ), WORK( * )
! 37: * ..
! 38: *
! 39: *
! 40: *> \par Purpose:
! 41: * =============
! 42: *>
! 43: *> \verbatim
! 44: *>
! 45: *> DLAQZ0 computes the eigenvalues of a real matrix pair (H,T),
! 46: *> where H is an upper Hessenberg matrix and T is upper triangular,
! 47: *> using the double-shift QZ method.
! 48: *> Matrix pairs of this type are produced by the reduction to
! 49: *> generalized upper Hessenberg form of a real matrix pair (A,B):
! 50: *>
! 51: *> A = Q1*H*Z1**T, B = Q1*T*Z1**T,
! 52: *>
! 53: *> as computed by DGGHRD.
! 54: *>
! 55: *> If JOB='S', then the Hessenberg-triangular pair (H,T) is
! 56: *> also reduced to generalized Schur form,
! 57: *>
! 58: *> H = Q*S*Z**T, T = Q*P*Z**T,
! 59: *>
! 60: *> where Q and Z are orthogonal matrices, P is an upper triangular
! 61: *> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
! 62: *> diagonal blocks.
! 63: *>
! 64: *> The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
! 65: *> (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
! 66: *> eigenvalues.
! 67: *>
! 68: *> Additionally, the 2-by-2 upper triangular diagonal blocks of P
! 69: *> corresponding to 2-by-2 blocks of S are reduced to positive diagonal
! 70: *> form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
! 71: *> P(j,j) > 0, and P(j+1,j+1) > 0.
! 72: *>
! 73: *> Optionally, the orthogonal matrix Q from the generalized Schur
! 74: *> factorization may be postmultiplied into an input matrix Q1, and the
! 75: *> orthogonal matrix Z may be postmultiplied into an input matrix Z1.
! 76: *> If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
! 77: *> the matrix pair (A,B) to generalized upper Hessenberg form, then the
! 78: *> output matrices Q1*Q and Z1*Z are the orthogonal factors from the
! 79: *> generalized Schur factorization of (A,B):
! 80: *>
! 81: *> A = (Q1*Q)*S*(Z1*Z)**T, B = (Q1*Q)*P*(Z1*Z)**T.
! 82: *>
! 83: *> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
! 84: *> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
! 85: *> complex and beta real.
! 86: *> If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
! 87: *> generalized nonsymmetric eigenvalue problem (GNEP)
! 88: *> A*x = lambda*B*x
! 89: *> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
! 90: *> alternate form of the GNEP
! 91: *> mu*A*y = B*y.
! 92: *> Real eigenvalues can be read directly from the generalized Schur
! 93: *> form:
! 94: *> alpha = S(i,i), beta = P(i,i).
! 95: *>
! 96: *> Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
! 97: *> Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
! 98: *> pp. 241--256.
! 99: *>
! 100: *> Ref: B. Kagstrom, D. Kressner, "Multishift Variants of the QZ
! 101: *> Algorithm with Aggressive Early Deflation", SIAM J. Numer.
! 102: *> Anal., 29(2006), pp. 199--227.
! 103: *>
! 104: *> Ref: T. Steel, D. Camps, K. Meerbergen, R. Vandebril "A multishift,
! 105: *> multipole rational QZ method with agressive early deflation"
! 106: *> \endverbatim
! 107: *
! 108: * Arguments:
! 109: * ==========
! 110: *
! 111: *> \param[in] WANTS
! 112: *> \verbatim
! 113: *> WANTS is CHARACTER*1
! 114: *> = 'E': Compute eigenvalues only;
! 115: *> = 'S': Compute eigenvalues and the Schur form.
! 116: *> \endverbatim
! 117: *>
! 118: *> \param[in] WANTQ
! 119: *> \verbatim
! 120: *> WANTQ is CHARACTER*1
! 121: *> = 'N': Left Schur vectors (Q) are not computed;
! 122: *> = 'I': Q is initialized to the unit matrix and the matrix Q
! 123: *> of left Schur vectors of (A,B) is returned;
! 124: *> = 'V': Q must contain an orthogonal matrix Q1 on entry and
! 125: *> the product Q1*Q is returned.
! 126: *> \endverbatim
! 127: *>
! 128: *> \param[in] WANTZ
! 129: *> \verbatim
! 130: *> WANTZ is CHARACTER*1
! 131: *> = 'N': Right Schur vectors (Z) are not computed;
! 132: *> = 'I': Z is initialized to the unit matrix and the matrix Z
! 133: *> of right Schur vectors of (A,B) is returned;
! 134: *> = 'V': Z must contain an orthogonal matrix Z1 on entry and
! 135: *> the product Z1*Z is returned.
! 136: *> \endverbatim
! 137: *>
! 138: *> \param[in] N
! 139: *> \verbatim
! 140: *> N is INTEGER
! 141: *> The order of the matrices A, B, Q, and Z. N >= 0.
! 142: *> \endverbatim
! 143: *>
! 144: *> \param[in] ILO
! 145: *> \verbatim
! 146: *> ILO is INTEGER
! 147: *> \endverbatim
! 148: *>
! 149: *> \param[in] IHI
! 150: *> \verbatim
! 151: *> IHI is INTEGER
! 152: *> ILO and IHI mark the rows and columns of A which are in
! 153: *> Hessenberg form. It is assumed that A is already upper
! 154: *> triangular in rows and columns 1:ILO-1 and IHI+1:N.
! 155: *> If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
! 156: *> \endverbatim
! 157: *>
! 158: *> \param[in,out] A
! 159: *> \verbatim
! 160: *> A is DOUBLE PRECISION array, dimension (LDA, N)
! 161: *> On entry, the N-by-N upper Hessenberg matrix A.
! 162: *> On exit, if JOB = 'S', A contains the upper quasi-triangular
! 163: *> matrix S from the generalized Schur factorization.
! 164: *> If JOB = 'E', the diagonal blocks of A match those of S, but
! 165: *> the rest of A is unspecified.
! 166: *> \endverbatim
! 167: *>
! 168: *> \param[in] LDA
! 169: *> \verbatim
! 170: *> LDA is INTEGER
! 171: *> The leading dimension of the array A. LDA >= max( 1, N ).
! 172: *> \endverbatim
! 173: *>
! 174: *> \param[in,out] B
! 175: *> \verbatim
! 176: *> B is DOUBLE PRECISION array, dimension (LDB, N)
! 177: *> On entry, the N-by-N upper triangular matrix B.
! 178: *> On exit, if JOB = 'S', B contains the upper triangular
! 179: *> matrix P from the generalized Schur factorization;
! 180: *> 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S
! 181: *> are reduced to positive diagonal form, i.e., if A(j+1,j) is
! 182: *> non-zero, then B(j+1,j) = B(j,j+1) = 0, B(j,j) > 0, and
! 183: *> B(j+1,j+1) > 0.
! 184: *> If JOB = 'E', the diagonal blocks of B match those of P, but
! 185: *> the rest of B is unspecified.
! 186: *> \endverbatim
! 187: *>
! 188: *> \param[in] LDB
! 189: *> \verbatim
! 190: *> LDB is INTEGER
! 191: *> The leading dimension of the array B. LDB >= max( 1, N ).
! 192: *> \endverbatim
! 193: *>
! 194: *> \param[out] ALPHAR
! 195: *> \verbatim
! 196: *> ALPHAR is DOUBLE PRECISION array, dimension (N)
! 197: *> The real parts of each scalar alpha defining an eigenvalue
! 198: *> of GNEP.
! 199: *> \endverbatim
! 200: *>
! 201: *> \param[out] ALPHAI
! 202: *> \verbatim
! 203: *> ALPHAI is DOUBLE PRECISION array, dimension (N)
! 204: *> The imaginary parts of each scalar alpha defining an
! 205: *> eigenvalue of GNEP.
! 206: *> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
! 207: *> positive, then the j-th and (j+1)-st eigenvalues are a
! 208: *> complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
! 209: *> \endverbatim
! 210: *>
! 211: *> \param[out] BETA
! 212: *> \verbatim
! 213: *> BETA is DOUBLE PRECISION array, dimension (N)
! 214: *> The scalars beta that define the eigenvalues of GNEP.
! 215: *> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
! 216: *> beta = BETA(j) represent the j-th eigenvalue of the matrix
! 217: *> pair (A,B), in one of the forms lambda = alpha/beta or
! 218: *> mu = beta/alpha. Since either lambda or mu may overflow,
! 219: *> they should not, in general, be computed.
! 220: *> \endverbatim
! 221: *>
! 222: *> \param[in,out] Q
! 223: *> \verbatim
! 224: *> Q is DOUBLE PRECISION array, dimension (LDQ, N)
! 225: *> On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in
! 226: *> the reduction of (A,B) to generalized Hessenberg form.
! 227: *> On exit, if COMPQ = 'I', the orthogonal matrix of left Schur
! 228: *> vectors of (A,B), and if COMPQ = 'V', the orthogonal matrix
! 229: *> of left Schur vectors of (A,B).
! 230: *> Not referenced if COMPQ = 'N'.
! 231: *> \endverbatim
! 232: *>
! 233: *> \param[in] LDQ
! 234: *> \verbatim
! 235: *> LDQ is INTEGER
! 236: *> The leading dimension of the array Q. LDQ >= 1.
! 237: *> If COMPQ='V' or 'I', then LDQ >= N.
! 238: *> \endverbatim
! 239: *>
! 240: *> \param[in,out] Z
! 241: *> \verbatim
! 242: *> Z is DOUBLE PRECISION array, dimension (LDZ, N)
! 243: *> On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in
! 244: *> the reduction of (A,B) to generalized Hessenberg form.
! 245: *> On exit, if COMPZ = 'I', the orthogonal matrix of
! 246: *> right Schur vectors of (H,T), and if COMPZ = 'V', the
! 247: *> orthogonal matrix of right Schur vectors of (A,B).
! 248: *> Not referenced if COMPZ = 'N'.
! 249: *> \endverbatim
! 250: *>
! 251: *> \param[in] LDZ
! 252: *> \verbatim
! 253: *> LDZ is INTEGER
! 254: *> The leading dimension of the array Z. LDZ >= 1.
! 255: *> If COMPZ='V' or 'I', then LDZ >= N.
! 256: *> \endverbatim
! 257: *>
! 258: *> \param[out] WORK
! 259: *> \verbatim
! 260: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 261: *> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
! 262: *> \endverbatim
! 263: *>
! 264: *> \param[in] LWORK
! 265: *> \verbatim
! 266: *> LWORK is INTEGER
! 267: *> The dimension of the array WORK. LWORK >= max(1,N).
! 268: *>
! 269: *> If LWORK = -1, then a workspace query is assumed; the routine
! 270: *> only calculates the optimal size of the WORK array, returns
! 271: *> this value as the first entry of the WORK array, and no error
! 272: *> message related to LWORK is issued by XERBLA.
! 273: *> \endverbatim
! 274: *>
! 275: *> \param[in] REC
! 276: *> \verbatim
! 277: *> REC is INTEGER
! 278: *> REC indicates the current recursion level. Should be set
! 279: *> to 0 on first call.
! 280: *> \endverbatim
! 281: *>
! 282: *> \param[out] INFO
! 283: *> \verbatim
! 284: *> INFO is INTEGER
! 285: *> = 0: successful exit
! 286: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 287: *> = 1,...,N: the QZ iteration did not converge. (A,B) is not
! 288: *> in Schur form, but ALPHAR(i), ALPHAI(i), and
! 289: *> BETA(i), i=INFO+1,...,N should be correct.
! 290: *> \endverbatim
! 291: *
! 292: * Authors:
! 293: * ========
! 294: *
! 295: *> \author Thijs Steel, KU Leuven
! 296: *
! 297: *> \date May 2020
! 298: *
! 299: *> \ingroup doubleGEcomputational
! 300: *>
! 301: * =====================================================================
! 302: RECURSIVE SUBROUTINE DLAQZ0( WANTS, WANTQ, WANTZ, N, ILO, IHI, A,
! 303: $ LDA, B, LDB, ALPHAR, ALPHAI, BETA,
! 304: $ Q, LDQ, Z, LDZ, WORK, LWORK, REC,
! 305: $ INFO )
! 306: IMPLICIT NONE
! 307:
! 308: * Arguments
! 309: CHARACTER, INTENT( IN ) :: WANTS, WANTQ, WANTZ
! 310: INTEGER, INTENT( IN ) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK,
! 311: $ REC
! 312:
! 313: INTEGER, INTENT( OUT ) :: INFO
! 314:
! 315: DOUBLE PRECISION, INTENT( INOUT ) :: A( LDA, * ), B( LDB, * ),
! 316: $ Q( LDQ, * ), Z( LDZ, * ), ALPHAR( * ),
! 317: $ ALPHAI( * ), BETA( * ), WORK( * )
! 318:
! 319: * Parameters
! 320: DOUBLE PRECISION :: ZERO, ONE, HALF
! 321: PARAMETER( ZERO = 0.0D0, ONE = 1.0D0, HALF = 0.5D0 )
! 322:
! 323: * Local scalars
! 324: DOUBLE PRECISION :: SMLNUM, ULP, ESHIFT, SAFMIN, SAFMAX, C1, S1,
! 325: $ TEMP, SWAP, BNORM, BTOL
! 326: INTEGER :: ISTART, ISTOP, IITER, MAXIT, ISTART2, K, LD, NSHIFTS,
! 327: $ NBLOCK, NW, NMIN, NIBBLE, N_UNDEFLATED, N_DEFLATED,
! 328: $ NS, SWEEP_INFO, SHIFTPOS, LWORKREQ, K2, ISTARTM,
! 329: $ ISTOPM, IWANTS, IWANTQ, IWANTZ, NORM_INFO, AED_INFO,
! 330: $ NWR, NBR, NSR, ITEMP1, ITEMP2, RCOST, I
! 331: LOGICAL :: ILSCHUR, ILQ, ILZ
! 332: CHARACTER :: JBCMPZ*3
! 333:
! 334: * External Functions
! 335: EXTERNAL :: XERBLA, DHGEQZ, DLASET, DLAQZ3, DLAQZ4, DLABAD,
! 336: $ DLARTG, DROT
! 337: DOUBLE PRECISION, EXTERNAL :: DLAMCH, DLANHS
! 338: LOGICAL, EXTERNAL :: LSAME
! 339: INTEGER, EXTERNAL :: ILAENV
! 340:
! 341: *
! 342: * Decode wantS,wantQ,wantZ
! 343: *
! 344: IF( LSAME( WANTS, 'E' ) ) THEN
! 345: ILSCHUR = .FALSE.
! 346: IWANTS = 1
! 347: ELSE IF( LSAME( WANTS, 'S' ) ) THEN
! 348: ILSCHUR = .TRUE.
! 349: IWANTS = 2
! 350: ELSE
! 351: IWANTS = 0
! 352: END IF
! 353:
! 354: IF( LSAME( WANTQ, 'N' ) ) THEN
! 355: ILQ = .FALSE.
! 356: IWANTQ = 1
! 357: ELSE IF( LSAME( WANTQ, 'V' ) ) THEN
! 358: ILQ = .TRUE.
! 359: IWANTQ = 2
! 360: ELSE IF( LSAME( WANTQ, 'I' ) ) THEN
! 361: ILQ = .TRUE.
! 362: IWANTQ = 3
! 363: ELSE
! 364: IWANTQ = 0
! 365: END IF
! 366:
! 367: IF( LSAME( WANTZ, 'N' ) ) THEN
! 368: ILZ = .FALSE.
! 369: IWANTZ = 1
! 370: ELSE IF( LSAME( WANTZ, 'V' ) ) THEN
! 371: ILZ = .TRUE.
! 372: IWANTZ = 2
! 373: ELSE IF( LSAME( WANTZ, 'I' ) ) THEN
! 374: ILZ = .TRUE.
! 375: IWANTZ = 3
! 376: ELSE
! 377: IWANTZ = 0
! 378: END IF
! 379: *
! 380: * Check Argument Values
! 381: *
! 382: INFO = 0
! 383: IF( IWANTS.EQ.0 ) THEN
! 384: INFO = -1
! 385: ELSE IF( IWANTQ.EQ.0 ) THEN
! 386: INFO = -2
! 387: ELSE IF( IWANTZ.EQ.0 ) THEN
! 388: INFO = -3
! 389: ELSE IF( N.LT.0 ) THEN
! 390: INFO = -4
! 391: ELSE IF( ILO.LT.1 ) THEN
! 392: INFO = -5
! 393: ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
! 394: INFO = -6
! 395: ELSE IF( LDA.LT.N ) THEN
! 396: INFO = -8
! 397: ELSE IF( LDB.LT.N ) THEN
! 398: INFO = -10
! 399: ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
! 400: INFO = -15
! 401: ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
! 402: INFO = -17
! 403: END IF
! 404: IF( INFO.NE.0 ) THEN
! 405: CALL XERBLA( 'DLAQZ0', -INFO )
! 406: RETURN
! 407: END IF
! 408:
! 409: *
! 410: * Quick return if possible
! 411: *
! 412: IF( N.LE.0 ) THEN
! 413: WORK( 1 ) = DBLE( 1 )
! 414: RETURN
! 415: END IF
! 416:
! 417: *
! 418: * Get the parameters
! 419: *
! 420: JBCMPZ( 1:1 ) = WANTS
! 421: JBCMPZ( 2:2 ) = WANTQ
! 422: JBCMPZ( 3:3 ) = WANTZ
! 423:
! 424: NMIN = ILAENV( 12, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
! 425:
! 426: NWR = ILAENV( 13, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
! 427: NWR = MAX( 2, NWR )
! 428: NWR = MIN( IHI-ILO+1, ( N-1 ) / 3, NWR )
! 429:
! 430: NIBBLE = ILAENV( 14, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
! 431:
! 432: NSR = ILAENV( 15, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
! 433: NSR = MIN( NSR, ( N+6 ) / 9, IHI-ILO )
! 434: NSR = MAX( 2, NSR-MOD( NSR, 2 ) )
! 435:
! 436: RCOST = ILAENV( 17, 'DLAQZ0', JBCMPZ, N, ILO, IHI, LWORK )
! 437: ITEMP1 = INT( NSR/SQRT( 1+2*NSR/( DBLE( RCOST )/100*N ) ) )
! 438: ITEMP1 = ( ( ITEMP1-1 )/4 )*4+4
! 439: NBR = NSR+ITEMP1
! 440:
! 441: IF( N .LT. NMIN .OR. REC .GE. 2 ) THEN
! 442: CALL DHGEQZ( WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB,
! 443: $ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK,
! 444: $ LWORK, INFO )
! 445: RETURN
! 446: END IF
! 447:
! 448: *
! 449: * Find out required workspace
! 450: *
! 451:
! 452: * Workspace query to dlaqz3
! 453: NW = MAX( NWR, NMIN )
! 454: CALL DLAQZ3( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NW, A, LDA, B, LDB,
! 455: $ Q, LDQ, Z, LDZ, N_UNDEFLATED, N_DEFLATED, ALPHAR,
! 456: $ ALPHAI, BETA, WORK, NW, WORK, NW, WORK, -1, REC,
! 457: $ AED_INFO )
! 458: ITEMP1 = INT( WORK( 1 ) )
! 459: * Workspace query to dlaqz4
! 460: CALL DLAQZ4( ILSCHUR, ILQ, ILZ, N, ILO, IHI, NSR, NBR, ALPHAR,
! 461: $ ALPHAI, BETA, A, LDA, B, LDB, Q, LDQ, Z, LDZ, WORK,
! 462: $ NBR, WORK, NBR, WORK, -1, SWEEP_INFO )
! 463: ITEMP2 = INT( WORK( 1 ) )
! 464:
! 465: LWORKREQ = MAX( ITEMP1+2*NW**2, ITEMP2+2*NBR**2 )
! 466: IF ( LWORK .EQ.-1 ) THEN
! 467: WORK( 1 ) = DBLE( LWORKREQ )
! 468: RETURN
! 469: ELSE IF ( LWORK .LT. LWORKREQ ) THEN
! 470: INFO = -19
! 471: END IF
! 472: IF( INFO.NE.0 ) THEN
! 473: CALL XERBLA( 'DLAQZ0', INFO )
! 474: RETURN
! 475: END IF
! 476: *
! 477: * Initialize Q and Z
! 478: *
! 479: IF( IWANTQ.EQ.3 ) CALL DLASET( 'FULL', N, N, ZERO, ONE, Q, LDQ )
! 480: IF( IWANTZ.EQ.3 ) CALL DLASET( 'FULL', N, N, ZERO, ONE, Z, LDZ )
! 481:
! 482: * Get machine constants
! 483: SAFMIN = DLAMCH( 'SAFE MINIMUM' )
! 484: SAFMAX = ONE/SAFMIN
! 485: CALL DLABAD( SAFMIN, SAFMAX )
! 486: ULP = DLAMCH( 'PRECISION' )
! 487: SMLNUM = SAFMIN*( DBLE( N )/ULP )
! 488:
! 489: BNORM = DLANHS( 'F', IHI-ILO+1, B( ILO, ILO ), LDB, WORK )
! 490: BTOL = MAX( SAFMIN, ULP*BNORM )
! 491:
! 492: ISTART = ILO
! 493: ISTOP = IHI
! 494: MAXIT = 3*( IHI-ILO+1 )
! 495: LD = 0
! 496:
! 497: DO IITER = 1, MAXIT
! 498: IF( IITER .GE. MAXIT ) THEN
! 499: INFO = ISTOP+1
! 500: GOTO 80
! 501: END IF
! 502: IF ( ISTART+1 .GE. ISTOP ) THEN
! 503: ISTOP = ISTART
! 504: EXIT
! 505: END IF
! 506:
! 507: * Check deflations at the end
! 508: IF ( ABS( A( ISTOP-1, ISTOP-2 ) ) .LE. MAX( SMLNUM,
! 509: $ ULP*( ABS( A( ISTOP-1, ISTOP-1 ) )+ABS( A( ISTOP-2,
! 510: $ ISTOP-2 ) ) ) ) ) THEN
! 511: A( ISTOP-1, ISTOP-2 ) = ZERO
! 512: ISTOP = ISTOP-2
! 513: LD = 0
! 514: ESHIFT = ZERO
! 515: ELSE IF ( ABS( A( ISTOP, ISTOP-1 ) ) .LE. MAX( SMLNUM,
! 516: $ ULP*( ABS( A( ISTOP, ISTOP ) )+ABS( A( ISTOP-1,
! 517: $ ISTOP-1 ) ) ) ) ) THEN
! 518: A( ISTOP, ISTOP-1 ) = ZERO
! 519: ISTOP = ISTOP-1
! 520: LD = 0
! 521: ESHIFT = ZERO
! 522: END IF
! 523: * Check deflations at the start
! 524: IF ( ABS( A( ISTART+2, ISTART+1 ) ) .LE. MAX( SMLNUM,
! 525: $ ULP*( ABS( A( ISTART+1, ISTART+1 ) )+ABS( A( ISTART+2,
! 526: $ ISTART+2 ) ) ) ) ) THEN
! 527: A( ISTART+2, ISTART+1 ) = ZERO
! 528: ISTART = ISTART+2
! 529: LD = 0
! 530: ESHIFT = ZERO
! 531: ELSE IF ( ABS( A( ISTART+1, ISTART ) ) .LE. MAX( SMLNUM,
! 532: $ ULP*( ABS( A( ISTART, ISTART ) )+ABS( A( ISTART+1,
! 533: $ ISTART+1 ) ) ) ) ) THEN
! 534: A( ISTART+1, ISTART ) = ZERO
! 535: ISTART = ISTART+1
! 536: LD = 0
! 537: ESHIFT = ZERO
! 538: END IF
! 539:
! 540: IF ( ISTART+1 .GE. ISTOP ) THEN
! 541: EXIT
! 542: END IF
! 543:
! 544: * Check interior deflations
! 545: ISTART2 = ISTART
! 546: DO K = ISTOP, ISTART+1, -1
! 547: IF ( ABS( A( K, K-1 ) ) .LE. MAX( SMLNUM, ULP*( ABS( A( K,
! 548: $ K ) )+ABS( A( K-1, K-1 ) ) ) ) ) THEN
! 549: A( K, K-1 ) = ZERO
! 550: ISTART2 = K
! 551: EXIT
! 552: END IF
! 553: END DO
! 554:
! 555: * Get range to apply rotations to
! 556: IF ( ILSCHUR ) THEN
! 557: ISTARTM = 1
! 558: ISTOPM = N
! 559: ELSE
! 560: ISTARTM = ISTART2
! 561: ISTOPM = ISTOP
! 562: END IF
! 563:
! 564: * Check infinite eigenvalues, this is done without blocking so might
! 565: * slow down the method when many infinite eigenvalues are present
! 566: K = ISTOP
! 567: DO WHILE ( K.GE.ISTART2 )
! 568:
! 569: IF( ABS( B( K, K ) ) .LT. BTOL ) THEN
! 570: * A diagonal element of B is negligable, move it
! 571: * to the top and deflate it
! 572:
! 573: DO K2 = K, ISTART2+1, -1
! 574: CALL DLARTG( B( K2-1, K2 ), B( K2-1, K2-1 ), C1, S1,
! 575: $ TEMP )
! 576: B( K2-1, K2 ) = TEMP
! 577: B( K2-1, K2-1 ) = ZERO
! 578:
! 579: CALL DROT( K2-2-ISTARTM+1, B( ISTARTM, K2 ), 1,
! 580: $ B( ISTARTM, K2-1 ), 1, C1, S1 )
! 581: CALL DROT( MIN( K2+1, ISTOP )-ISTARTM+1, A( ISTARTM,
! 582: $ K2 ), 1, A( ISTARTM, K2-1 ), 1, C1, S1 )
! 583: IF ( ILZ ) THEN
! 584: CALL DROT( N, Z( 1, K2 ), 1, Z( 1, K2-1 ), 1, C1,
! 585: $ S1 )
! 586: END IF
! 587:
! 588: IF( K2.LT.ISTOP ) THEN
! 589: CALL DLARTG( A( K2, K2-1 ), A( K2+1, K2-1 ), C1,
! 590: $ S1, TEMP )
! 591: A( K2, K2-1 ) = TEMP
! 592: A( K2+1, K2-1 ) = ZERO
! 593:
! 594: CALL DROT( ISTOPM-K2+1, A( K2, K2 ), LDA, A( K2+1,
! 595: $ K2 ), LDA, C1, S1 )
! 596: CALL DROT( ISTOPM-K2+1, B( K2, K2 ), LDB, B( K2+1,
! 597: $ K2 ), LDB, C1, S1 )
! 598: IF( ILQ ) THEN
! 599: CALL DROT( N, Q( 1, K2 ), 1, Q( 1, K2+1 ), 1,
! 600: $ C1, S1 )
! 601: END IF
! 602: END IF
! 603:
! 604: END DO
! 605:
! 606: IF( ISTART2.LT.ISTOP )THEN
! 607: CALL DLARTG( A( ISTART2, ISTART2 ), A( ISTART2+1,
! 608: $ ISTART2 ), C1, S1, TEMP )
! 609: A( ISTART2, ISTART2 ) = TEMP
! 610: A( ISTART2+1, ISTART2 ) = ZERO
! 611:
! 612: CALL DROT( ISTOPM-( ISTART2+1 )+1, A( ISTART2,
! 613: $ ISTART2+1 ), LDA, A( ISTART2+1,
! 614: $ ISTART2+1 ), LDA, C1, S1 )
! 615: CALL DROT( ISTOPM-( ISTART2+1 )+1, B( ISTART2,
! 616: $ ISTART2+1 ), LDB, B( ISTART2+1,
! 617: $ ISTART2+1 ), LDB, C1, S1 )
! 618: IF( ILQ ) THEN
! 619: CALL DROT( N, Q( 1, ISTART2 ), 1, Q( 1,
! 620: $ ISTART2+1 ), 1, C1, S1 )
! 621: END IF
! 622: END IF
! 623:
! 624: ISTART2 = ISTART2+1
! 625:
! 626: END IF
! 627: K = K-1
! 628: END DO
! 629:
! 630: * istart2 now points to the top of the bottom right
! 631: * unreduced Hessenberg block
! 632: IF ( ISTART2 .GE. ISTOP ) THEN
! 633: ISTOP = ISTART2-1
! 634: LD = 0
! 635: ESHIFT = ZERO
! 636: CYCLE
! 637: END IF
! 638:
! 639: NW = NWR
! 640: NSHIFTS = NSR
! 641: NBLOCK = NBR
! 642:
! 643: IF ( ISTOP-ISTART2+1 .LT. NMIN ) THEN
! 644: * Setting nw to the size of the subblock will make AED deflate
! 645: * all the eigenvalues. This is slightly more efficient than just
! 646: * using DHGEQZ because the off diagonal part gets updated via BLAS.
! 647: IF ( ISTOP-ISTART+1 .LT. NMIN ) THEN
! 648: NW = ISTOP-ISTART+1
! 649: ISTART2 = ISTART
! 650: ELSE
! 651: NW = ISTOP-ISTART2+1
! 652: END IF
! 653: END IF
! 654:
! 655: *
! 656: * Time for AED
! 657: *
! 658: CALL DLAQZ3( ILSCHUR, ILQ, ILZ, N, ISTART2, ISTOP, NW, A, LDA,
! 659: $ B, LDB, Q, LDQ, Z, LDZ, N_UNDEFLATED, N_DEFLATED,
! 660: $ ALPHAR, ALPHAI, BETA, WORK, NW, WORK( NW**2+1 ),
! 661: $ NW, WORK( 2*NW**2+1 ), LWORK-2*NW**2, REC,
! 662: $ AED_INFO )
! 663:
! 664: IF ( N_DEFLATED > 0 ) THEN
! 665: ISTOP = ISTOP-N_DEFLATED
! 666: LD = 0
! 667: ESHIFT = ZERO
! 668: END IF
! 669:
! 670: IF ( 100*N_DEFLATED > NIBBLE*( N_DEFLATED+N_UNDEFLATED ) .OR.
! 671: $ ISTOP-ISTART2+1 .LT. NMIN ) THEN
! 672: * AED has uncovered many eigenvalues. Skip a QZ sweep and run
! 673: * AED again.
! 674: CYCLE
! 675: END IF
! 676:
! 677: LD = LD+1
! 678:
! 679: NS = MIN( NSHIFTS, ISTOP-ISTART2 )
! 680: NS = MIN( NS, N_UNDEFLATED )
! 681: SHIFTPOS = ISTOP-N_DEFLATED-N_UNDEFLATED+1
! 682: *
! 683: * Shuffle shifts to put double shifts in front
! 684: * This ensures that we don't split up a double shift
! 685: *
! 686: DO I = SHIFTPOS, SHIFTPOS+N_UNDEFLATED-1, 2
! 687: IF( ALPHAI( I ).NE.-ALPHAI( I+1 ) ) THEN
! 688: *
! 689: SWAP = ALPHAR( I )
! 690: ALPHAR( I ) = ALPHAR( I+1 )
! 691: ALPHAR( I+1 ) = ALPHAR( I+2 )
! 692: ALPHAR( I+2 ) = SWAP
! 693:
! 694: SWAP = ALPHAI( I )
! 695: ALPHAI( I ) = ALPHAI( I+1 )
! 696: ALPHAI( I+1 ) = ALPHAI( I+2 )
! 697: ALPHAI( I+2 ) = SWAP
! 698:
! 699: SWAP = BETA( I )
! 700: BETA( I ) = BETA( I+1 )
! 701: BETA( I+1 ) = BETA( I+2 )
! 702: BETA( I+2 ) = SWAP
! 703: END IF
! 704: END DO
! 705:
! 706: IF ( MOD( LD, 6 ) .EQ. 0 ) THEN
! 707: *
! 708: * Exceptional shift. Chosen for no particularly good reason.
! 709: *
! 710: IF( ( DBLE( MAXIT )*SAFMIN )*ABS( A( ISTOP,
! 711: $ ISTOP-1 ) ).LT.ABS( A( ISTOP-1, ISTOP-1 ) ) ) THEN
! 712: ESHIFT = A( ISTOP, ISTOP-1 )/B( ISTOP-1, ISTOP-1 )
! 713: ELSE
! 714: ESHIFT = ESHIFT+ONE/( SAFMIN*DBLE( MAXIT ) )
! 715: END IF
! 716: ALPHAR( SHIFTPOS ) = ONE
! 717: ALPHAR( SHIFTPOS+1 ) = ZERO
! 718: ALPHAI( SHIFTPOS ) = ZERO
! 719: ALPHAI( SHIFTPOS+1 ) = ZERO
! 720: BETA( SHIFTPOS ) = ESHIFT
! 721: BETA( SHIFTPOS+1 ) = ESHIFT
! 722: NS = 2
! 723: END IF
! 724:
! 725: *
! 726: * Time for a QZ sweep
! 727: *
! 728: CALL DLAQZ4( ILSCHUR, ILQ, ILZ, N, ISTART2, ISTOP, NS, NBLOCK,
! 729: $ ALPHAR( SHIFTPOS ), ALPHAI( SHIFTPOS ),
! 730: $ BETA( SHIFTPOS ), A, LDA, B, LDB, Q, LDQ, Z, LDZ,
! 731: $ WORK, NBLOCK, WORK( NBLOCK**2+1 ), NBLOCK,
! 732: $ WORK( 2*NBLOCK**2+1 ), LWORK-2*NBLOCK**2,
! 733: $ SWEEP_INFO )
! 734:
! 735: END DO
! 736:
! 737: *
! 738: * Call DHGEQZ to normalize the eigenvalue blocks and set the eigenvalues
! 739: * If all the eigenvalues have been found, DHGEQZ will not do any iterations
! 740: * and only normalize the blocks. In case of a rare convergence failure,
! 741: * the single shift might perform better.
! 742: *
! 743: 80 CALL DHGEQZ( WANTS, WANTQ, WANTZ, N, ILO, IHI, A, LDA, B, LDB,
! 744: $ ALPHAR, ALPHAI, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
! 745: $ NORM_INFO )
! 746:
! 747: INFO = NORM_INFO
! 748:
! 749: END SUBROUTINE
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