Annotation of rpl/lapack/lapack/zhgeqz.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, H, LDH, T, LDT,
! 2: $ ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK,
! 3: $ RWORK, INFO )
! 4: *
! 5: * -- LAPACK routine (version 3.2) --
! 6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 8: * November 2006
! 9: *
! 10: * .. Scalar Arguments ..
! 11: CHARACTER COMPQ, COMPZ, JOB
! 12: INTEGER IHI, ILO, INFO, LDH, LDQ, LDT, LDZ, LWORK, N
! 13: * ..
! 14: * .. Array Arguments ..
! 15: DOUBLE PRECISION RWORK( * )
! 16: COMPLEX*16 ALPHA( * ), BETA( * ), H( LDH, * ),
! 17: $ Q( LDQ, * ), T( LDT, * ), WORK( * ),
! 18: $ Z( LDZ, * )
! 19: * ..
! 20: *
! 21: * Purpose
! 22: * =======
! 23: *
! 24: * ZHGEQZ computes the eigenvalues of a complex matrix pair (H,T),
! 25: * where H is an upper Hessenberg matrix and T is upper triangular,
! 26: * using the single-shift QZ method.
! 27: * Matrix pairs of this type are produced by the reduction to
! 28: * generalized upper Hessenberg form of a complex matrix pair (A,B):
! 29: *
! 30: * A = Q1*H*Z1**H, B = Q1*T*Z1**H,
! 31: *
! 32: * as computed by ZGGHRD.
! 33: *
! 34: * If JOB='S', then the Hessenberg-triangular pair (H,T) is
! 35: * also reduced to generalized Schur form,
! 36: *
! 37: * H = Q*S*Z**H, T = Q*P*Z**H,
! 38: *
! 39: * where Q and Z are unitary matrices and S and P are upper triangular.
! 40: *
! 41: * Optionally, the unitary matrix Q from the generalized Schur
! 42: * factorization may be postmultiplied into an input matrix Q1, and the
! 43: * unitary matrix Z may be postmultiplied into an input matrix Z1.
! 44: * If Q1 and Z1 are the unitary matrices from ZGGHRD that reduced
! 45: * the matrix pair (A,B) to generalized Hessenberg form, then the output
! 46: * matrices Q1*Q and Z1*Z are the unitary factors from the generalized
! 47: * Schur factorization of (A,B):
! 48: *
! 49: * A = (Q1*Q)*S*(Z1*Z)**H, B = (Q1*Q)*P*(Z1*Z)**H.
! 50: *
! 51: * To avoid overflow, eigenvalues of the matrix pair (H,T)
! 52: * (equivalently, of (A,B)) are computed as a pair of complex values
! 53: * (alpha,beta). If beta is nonzero, lambda = alpha / beta is an
! 54: * eigenvalue of the generalized nonsymmetric eigenvalue problem (GNEP)
! 55: * A*x = lambda*B*x
! 56: * and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
! 57: * alternate form of the GNEP
! 58: * mu*A*y = B*y.
! 59: * The values of alpha and beta for the i-th eigenvalue can be read
! 60: * directly from the generalized Schur form: alpha = S(i,i),
! 61: * beta = P(i,i).
! 62: *
! 63: * Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrix
! 64: * Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
! 65: * pp. 241--256.
! 66: *
! 67: * Arguments
! 68: * =========
! 69: *
! 70: * JOB (input) CHARACTER*1
! 71: * = 'E': Compute eigenvalues only;
! 72: * = 'S': Computer eigenvalues and the Schur form.
! 73: *
! 74: * COMPQ (input) CHARACTER*1
! 75: * = 'N': Left Schur vectors (Q) are not computed;
! 76: * = 'I': Q is initialized to the unit matrix and the matrix Q
! 77: * of left Schur vectors of (H,T) is returned;
! 78: * = 'V': Q must contain a unitary matrix Q1 on entry and
! 79: * the product Q1*Q is returned.
! 80: *
! 81: * COMPZ (input) CHARACTER*1
! 82: * = 'N': Right Schur vectors (Z) are not computed;
! 83: * = 'I': Q is initialized to the unit matrix and the matrix Z
! 84: * of right Schur vectors of (H,T) is returned;
! 85: * = 'V': Z must contain a unitary matrix Z1 on entry and
! 86: * the product Z1*Z is returned.
! 87: *
! 88: * N (input) INTEGER
! 89: * The order of the matrices H, T, Q, and Z. N >= 0.
! 90: *
! 91: * ILO (input) INTEGER
! 92: * IHI (input) INTEGER
! 93: * ILO and IHI mark the rows and columns of H which are in
! 94: * Hessenberg form. It is assumed that A is already upper
! 95: * triangular in rows and columns 1:ILO-1 and IHI+1:N.
! 96: * If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0.
! 97: *
! 98: * H (input/output) COMPLEX*16 array, dimension (LDH, N)
! 99: * On entry, the N-by-N upper Hessenberg matrix H.
! 100: * On exit, if JOB = 'S', H contains the upper triangular
! 101: * matrix S from the generalized Schur factorization.
! 102: * If JOB = 'E', the diagonal of H matches that of S, but
! 103: * the rest of H is unspecified.
! 104: *
! 105: * LDH (input) INTEGER
! 106: * The leading dimension of the array H. LDH >= max( 1, N ).
! 107: *
! 108: * T (input/output) COMPLEX*16 array, dimension (LDT, N)
! 109: * On entry, the N-by-N upper triangular matrix T.
! 110: * On exit, if JOB = 'S', T contains the upper triangular
! 111: * matrix P from the generalized Schur factorization.
! 112: * If JOB = 'E', the diagonal of T matches that of P, but
! 113: * the rest of T is unspecified.
! 114: *
! 115: * LDT (input) INTEGER
! 116: * The leading dimension of the array T. LDT >= max( 1, N ).
! 117: *
! 118: * ALPHA (output) COMPLEX*16 array, dimension (N)
! 119: * The complex scalars alpha that define the eigenvalues of
! 120: * GNEP. ALPHA(i) = S(i,i) in the generalized Schur
! 121: * factorization.
! 122: *
! 123: * BETA (output) COMPLEX*16 array, dimension (N)
! 124: * The real non-negative scalars beta that define the
! 125: * eigenvalues of GNEP. BETA(i) = P(i,i) in the generalized
! 126: * Schur factorization.
! 127: *
! 128: * Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
! 129: * represent the j-th eigenvalue of the matrix pair (A,B), in
! 130: * one of the forms lambda = alpha/beta or mu = beta/alpha.
! 131: * Since either lambda or mu may overflow, they should not,
! 132: * in general, be computed.
! 133: *
! 134: * Q (input/output) COMPLEX*16 array, dimension (LDQ, N)
! 135: * On entry, if COMPZ = 'V', the unitary matrix Q1 used in the
! 136: * reduction of (A,B) to generalized Hessenberg form.
! 137: * On exit, if COMPZ = 'I', the unitary matrix of left Schur
! 138: * vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
! 139: * left Schur vectors of (A,B).
! 140: * Not referenced if COMPZ = 'N'.
! 141: *
! 142: * LDQ (input) INTEGER
! 143: * The leading dimension of the array Q. LDQ >= 1.
! 144: * If COMPQ='V' or 'I', then LDQ >= N.
! 145: *
! 146: * Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
! 147: * On entry, if COMPZ = 'V', the unitary matrix Z1 used in the
! 148: * reduction of (A,B) to generalized Hessenberg form.
! 149: * On exit, if COMPZ = 'I', the unitary matrix of right Schur
! 150: * vectors of (H,T), and if COMPZ = 'V', the unitary matrix of
! 151: * right Schur vectors of (A,B).
! 152: * Not referenced if COMPZ = 'N'.
! 153: *
! 154: * LDZ (input) INTEGER
! 155: * The leading dimension of the array Z. LDZ >= 1.
! 156: * If COMPZ='V' or 'I', then LDZ >= N.
! 157: *
! 158: * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
! 159: * On exit, if INFO >= 0, WORK(1) returns the optimal LWORK.
! 160: *
! 161: * LWORK (input) INTEGER
! 162: * The dimension of the array WORK. LWORK >= max(1,N).
! 163: *
! 164: * If LWORK = -1, then a workspace query is assumed; the routine
! 165: * only calculates the optimal size of the WORK array, returns
! 166: * this value as the first entry of the WORK array, and no error
! 167: * message related to LWORK is issued by XERBLA.
! 168: *
! 169: * RWORK (workspace) DOUBLE PRECISION array, dimension (N)
! 170: *
! 171: * INFO (output) INTEGER
! 172: * = 0: successful exit
! 173: * < 0: if INFO = -i, the i-th argument had an illegal value
! 174: * = 1,...,N: the QZ iteration did not converge. (H,T) is not
! 175: * in Schur form, but ALPHA(i) and BETA(i),
! 176: * i=INFO+1,...,N should be correct.
! 177: * = N+1,...,2*N: the shift calculation failed. (H,T) is not
! 178: * in Schur form, but ALPHA(i) and BETA(i),
! 179: * i=INFO-N+1,...,N should be correct.
! 180: *
! 181: * Further Details
! 182: * ===============
! 183: *
! 184: * We assume that complex ABS works as long as its value is less than
! 185: * overflow.
! 186: *
! 187: * =====================================================================
! 188: *
! 189: * .. Parameters ..
! 190: COMPLEX*16 CZERO, CONE
! 191: PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
! 192: $ CONE = ( 1.0D+0, 0.0D+0 ) )
! 193: DOUBLE PRECISION ZERO, ONE
! 194: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
! 195: DOUBLE PRECISION HALF
! 196: PARAMETER ( HALF = 0.5D+0 )
! 197: * ..
! 198: * .. Local Scalars ..
! 199: LOGICAL ILAZR2, ILAZRO, ILQ, ILSCHR, ILZ, LQUERY
! 200: INTEGER ICOMPQ, ICOMPZ, IFIRST, IFRSTM, IITER, ILAST,
! 201: $ ILASTM, IN, ISCHUR, ISTART, J, JC, JCH, JITER,
! 202: $ JR, MAXIT
! 203: DOUBLE PRECISION ABSB, ANORM, ASCALE, ATOL, BNORM, BSCALE, BTOL,
! 204: $ C, SAFMIN, TEMP, TEMP2, TEMPR, ULP
! 205: COMPLEX*16 ABI22, AD11, AD12, AD21, AD22, CTEMP, CTEMP2,
! 206: $ CTEMP3, ESHIFT, RTDISC, S, SHIFT, SIGNBC, T1,
! 207: $ U12, X
! 208: * ..
! 209: * .. External Functions ..
! 210: LOGICAL LSAME
! 211: DOUBLE PRECISION DLAMCH, ZLANHS
! 212: EXTERNAL LSAME, DLAMCH, ZLANHS
! 213: * ..
! 214: * .. External Subroutines ..
! 215: EXTERNAL XERBLA, ZLARTG, ZLASET, ZROT, ZSCAL
! 216: * ..
! 217: * .. Intrinsic Functions ..
! 218: INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN,
! 219: $ SQRT
! 220: * ..
! 221: * .. Statement Functions ..
! 222: DOUBLE PRECISION ABS1
! 223: * ..
! 224: * .. Statement Function definitions ..
! 225: ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
! 226: * ..
! 227: * .. Executable Statements ..
! 228: *
! 229: * Decode JOB, COMPQ, COMPZ
! 230: *
! 231: IF( LSAME( JOB, 'E' ) ) THEN
! 232: ILSCHR = .FALSE.
! 233: ISCHUR = 1
! 234: ELSE IF( LSAME( JOB, 'S' ) ) THEN
! 235: ILSCHR = .TRUE.
! 236: ISCHUR = 2
! 237: ELSE
! 238: ISCHUR = 0
! 239: END IF
! 240: *
! 241: IF( LSAME( COMPQ, 'N' ) ) THEN
! 242: ILQ = .FALSE.
! 243: ICOMPQ = 1
! 244: ELSE IF( LSAME( COMPQ, 'V' ) ) THEN
! 245: ILQ = .TRUE.
! 246: ICOMPQ = 2
! 247: ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
! 248: ILQ = .TRUE.
! 249: ICOMPQ = 3
! 250: ELSE
! 251: ICOMPQ = 0
! 252: END IF
! 253: *
! 254: IF( LSAME( COMPZ, 'N' ) ) THEN
! 255: ILZ = .FALSE.
! 256: ICOMPZ = 1
! 257: ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
! 258: ILZ = .TRUE.
! 259: ICOMPZ = 2
! 260: ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
! 261: ILZ = .TRUE.
! 262: ICOMPZ = 3
! 263: ELSE
! 264: ICOMPZ = 0
! 265: END IF
! 266: *
! 267: * Check Argument Values
! 268: *
! 269: INFO = 0
! 270: WORK( 1 ) = MAX( 1, N )
! 271: LQUERY = ( LWORK.EQ.-1 )
! 272: IF( ISCHUR.EQ.0 ) THEN
! 273: INFO = -1
! 274: ELSE IF( ICOMPQ.EQ.0 ) THEN
! 275: INFO = -2
! 276: ELSE IF( ICOMPZ.EQ.0 ) THEN
! 277: INFO = -3
! 278: ELSE IF( N.LT.0 ) THEN
! 279: INFO = -4
! 280: ELSE IF( ILO.LT.1 ) THEN
! 281: INFO = -5
! 282: ELSE IF( IHI.GT.N .OR. IHI.LT.ILO-1 ) THEN
! 283: INFO = -6
! 284: ELSE IF( LDH.LT.N ) THEN
! 285: INFO = -8
! 286: ELSE IF( LDT.LT.N ) THEN
! 287: INFO = -10
! 288: ELSE IF( LDQ.LT.1 .OR. ( ILQ .AND. LDQ.LT.N ) ) THEN
! 289: INFO = -14
! 290: ELSE IF( LDZ.LT.1 .OR. ( ILZ .AND. LDZ.LT.N ) ) THEN
! 291: INFO = -16
! 292: ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
! 293: INFO = -18
! 294: END IF
! 295: IF( INFO.NE.0 ) THEN
! 296: CALL XERBLA( 'ZHGEQZ', -INFO )
! 297: RETURN
! 298: ELSE IF( LQUERY ) THEN
! 299: RETURN
! 300: END IF
! 301: *
! 302: * Quick return if possible
! 303: *
! 304: * WORK( 1 ) = CMPLX( 1 )
! 305: IF( N.LE.0 ) THEN
! 306: WORK( 1 ) = DCMPLX( 1 )
! 307: RETURN
! 308: END IF
! 309: *
! 310: * Initialize Q and Z
! 311: *
! 312: IF( ICOMPQ.EQ.3 )
! 313: $ CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
! 314: IF( ICOMPZ.EQ.3 )
! 315: $ CALL ZLASET( 'Full', N, N, CZERO, CONE, Z, LDZ )
! 316: *
! 317: * Machine Constants
! 318: *
! 319: IN = IHI + 1 - ILO
! 320: SAFMIN = DLAMCH( 'S' )
! 321: ULP = DLAMCH( 'E' )*DLAMCH( 'B' )
! 322: ANORM = ZLANHS( 'F', IN, H( ILO, ILO ), LDH, RWORK )
! 323: BNORM = ZLANHS( 'F', IN, T( ILO, ILO ), LDT, RWORK )
! 324: ATOL = MAX( SAFMIN, ULP*ANORM )
! 325: BTOL = MAX( SAFMIN, ULP*BNORM )
! 326: ASCALE = ONE / MAX( SAFMIN, ANORM )
! 327: BSCALE = ONE / MAX( SAFMIN, BNORM )
! 328: *
! 329: *
! 330: * Set Eigenvalues IHI+1:N
! 331: *
! 332: DO 10 J = IHI + 1, N
! 333: ABSB = ABS( T( J, J ) )
! 334: IF( ABSB.GT.SAFMIN ) THEN
! 335: SIGNBC = DCONJG( T( J, J ) / ABSB )
! 336: T( J, J ) = ABSB
! 337: IF( ILSCHR ) THEN
! 338: CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
! 339: CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
! 340: ELSE
! 341: H( J, J ) = H( J, J )*SIGNBC
! 342: END IF
! 343: IF( ILZ )
! 344: $ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
! 345: ELSE
! 346: T( J, J ) = CZERO
! 347: END IF
! 348: ALPHA( J ) = H( J, J )
! 349: BETA( J ) = T( J, J )
! 350: 10 CONTINUE
! 351: *
! 352: * If IHI < ILO, skip QZ steps
! 353: *
! 354: IF( IHI.LT.ILO )
! 355: $ GO TO 190
! 356: *
! 357: * MAIN QZ ITERATION LOOP
! 358: *
! 359: * Initialize dynamic indices
! 360: *
! 361: * Eigenvalues ILAST+1:N have been found.
! 362: * Column operations modify rows IFRSTM:whatever
! 363: * Row operations modify columns whatever:ILASTM
! 364: *
! 365: * If only eigenvalues are being computed, then
! 366: * IFRSTM is the row of the last splitting row above row ILAST;
! 367: * this is always at least ILO.
! 368: * IITER counts iterations since the last eigenvalue was found,
! 369: * to tell when to use an extraordinary shift.
! 370: * MAXIT is the maximum number of QZ sweeps allowed.
! 371: *
! 372: ILAST = IHI
! 373: IF( ILSCHR ) THEN
! 374: IFRSTM = 1
! 375: ILASTM = N
! 376: ELSE
! 377: IFRSTM = ILO
! 378: ILASTM = IHI
! 379: END IF
! 380: IITER = 0
! 381: ESHIFT = CZERO
! 382: MAXIT = 30*( IHI-ILO+1 )
! 383: *
! 384: DO 170 JITER = 1, MAXIT
! 385: *
! 386: * Check for too many iterations.
! 387: *
! 388: IF( JITER.GT.MAXIT )
! 389: $ GO TO 180
! 390: *
! 391: * Split the matrix if possible.
! 392: *
! 393: * Two tests:
! 394: * 1: H(j,j-1)=0 or j=ILO
! 395: * 2: T(j,j)=0
! 396: *
! 397: * Special case: j=ILAST
! 398: *
! 399: IF( ILAST.EQ.ILO ) THEN
! 400: GO TO 60
! 401: ELSE
! 402: IF( ABS1( H( ILAST, ILAST-1 ) ).LE.ATOL ) THEN
! 403: H( ILAST, ILAST-1 ) = CZERO
! 404: GO TO 60
! 405: END IF
! 406: END IF
! 407: *
! 408: IF( ABS( T( ILAST, ILAST ) ).LE.BTOL ) THEN
! 409: T( ILAST, ILAST ) = CZERO
! 410: GO TO 50
! 411: END IF
! 412: *
! 413: * General case: j<ILAST
! 414: *
! 415: DO 40 J = ILAST - 1, ILO, -1
! 416: *
! 417: * Test 1: for H(j,j-1)=0 or j=ILO
! 418: *
! 419: IF( J.EQ.ILO ) THEN
! 420: ILAZRO = .TRUE.
! 421: ELSE
! 422: IF( ABS1( H( J, J-1 ) ).LE.ATOL ) THEN
! 423: H( J, J-1 ) = CZERO
! 424: ILAZRO = .TRUE.
! 425: ELSE
! 426: ILAZRO = .FALSE.
! 427: END IF
! 428: END IF
! 429: *
! 430: * Test 2: for T(j,j)=0
! 431: *
! 432: IF( ABS( T( J, J ) ).LT.BTOL ) THEN
! 433: T( J, J ) = CZERO
! 434: *
! 435: * Test 1a: Check for 2 consecutive small subdiagonals in A
! 436: *
! 437: ILAZR2 = .FALSE.
! 438: IF( .NOT.ILAZRO ) THEN
! 439: IF( ABS1( H( J, J-1 ) )*( ASCALE*ABS1( H( J+1,
! 440: $ J ) ) ).LE.ABS1( H( J, J ) )*( ASCALE*ATOL ) )
! 441: $ ILAZR2 = .TRUE.
! 442: END IF
! 443: *
! 444: * If both tests pass (1 & 2), i.e., the leading diagonal
! 445: * element of B in the block is zero, split a 1x1 block off
! 446: * at the top. (I.e., at the J-th row/column) The leading
! 447: * diagonal element of the remainder can also be zero, so
! 448: * this may have to be done repeatedly.
! 449: *
! 450: IF( ILAZRO .OR. ILAZR2 ) THEN
! 451: DO 20 JCH = J, ILAST - 1
! 452: CTEMP = H( JCH, JCH )
! 453: CALL ZLARTG( CTEMP, H( JCH+1, JCH ), C, S,
! 454: $ H( JCH, JCH ) )
! 455: H( JCH+1, JCH ) = CZERO
! 456: CALL ZROT( ILASTM-JCH, H( JCH, JCH+1 ), LDH,
! 457: $ H( JCH+1, JCH+1 ), LDH, C, S )
! 458: CALL ZROT( ILASTM-JCH, T( JCH, JCH+1 ), LDT,
! 459: $ T( JCH+1, JCH+1 ), LDT, C, S )
! 460: IF( ILQ )
! 461: $ CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
! 462: $ C, DCONJG( S ) )
! 463: IF( ILAZR2 )
! 464: $ H( JCH, JCH-1 ) = H( JCH, JCH-1 )*C
! 465: ILAZR2 = .FALSE.
! 466: IF( ABS1( T( JCH+1, JCH+1 ) ).GE.BTOL ) THEN
! 467: IF( JCH+1.GE.ILAST ) THEN
! 468: GO TO 60
! 469: ELSE
! 470: IFIRST = JCH + 1
! 471: GO TO 70
! 472: END IF
! 473: END IF
! 474: T( JCH+1, JCH+1 ) = CZERO
! 475: 20 CONTINUE
! 476: GO TO 50
! 477: ELSE
! 478: *
! 479: * Only test 2 passed -- chase the zero to T(ILAST,ILAST)
! 480: * Then process as in the case T(ILAST,ILAST)=0
! 481: *
! 482: DO 30 JCH = J, ILAST - 1
! 483: CTEMP = T( JCH, JCH+1 )
! 484: CALL ZLARTG( CTEMP, T( JCH+1, JCH+1 ), C, S,
! 485: $ T( JCH, JCH+1 ) )
! 486: T( JCH+1, JCH+1 ) = CZERO
! 487: IF( JCH.LT.ILASTM-1 )
! 488: $ CALL ZROT( ILASTM-JCH-1, T( JCH, JCH+2 ), LDT,
! 489: $ T( JCH+1, JCH+2 ), LDT, C, S )
! 490: CALL ZROT( ILASTM-JCH+2, H( JCH, JCH-1 ), LDH,
! 491: $ H( JCH+1, JCH-1 ), LDH, C, S )
! 492: IF( ILQ )
! 493: $ CALL ZROT( N, Q( 1, JCH ), 1, Q( 1, JCH+1 ), 1,
! 494: $ C, DCONJG( S ) )
! 495: CTEMP = H( JCH+1, JCH )
! 496: CALL ZLARTG( CTEMP, H( JCH+1, JCH-1 ), C, S,
! 497: $ H( JCH+1, JCH ) )
! 498: H( JCH+1, JCH-1 ) = CZERO
! 499: CALL ZROT( JCH+1-IFRSTM, H( IFRSTM, JCH ), 1,
! 500: $ H( IFRSTM, JCH-1 ), 1, C, S )
! 501: CALL ZROT( JCH-IFRSTM, T( IFRSTM, JCH ), 1,
! 502: $ T( IFRSTM, JCH-1 ), 1, C, S )
! 503: IF( ILZ )
! 504: $ CALL ZROT( N, Z( 1, JCH ), 1, Z( 1, JCH-1 ), 1,
! 505: $ C, S )
! 506: 30 CONTINUE
! 507: GO TO 50
! 508: END IF
! 509: ELSE IF( ILAZRO ) THEN
! 510: *
! 511: * Only test 1 passed -- work on J:ILAST
! 512: *
! 513: IFIRST = J
! 514: GO TO 70
! 515: END IF
! 516: *
! 517: * Neither test passed -- try next J
! 518: *
! 519: 40 CONTINUE
! 520: *
! 521: * (Drop-through is "impossible")
! 522: *
! 523: INFO = 2*N + 1
! 524: GO TO 210
! 525: *
! 526: * T(ILAST,ILAST)=0 -- clear H(ILAST,ILAST-1) to split off a
! 527: * 1x1 block.
! 528: *
! 529: 50 CONTINUE
! 530: CTEMP = H( ILAST, ILAST )
! 531: CALL ZLARTG( CTEMP, H( ILAST, ILAST-1 ), C, S,
! 532: $ H( ILAST, ILAST ) )
! 533: H( ILAST, ILAST-1 ) = CZERO
! 534: CALL ZROT( ILAST-IFRSTM, H( IFRSTM, ILAST ), 1,
! 535: $ H( IFRSTM, ILAST-1 ), 1, C, S )
! 536: CALL ZROT( ILAST-IFRSTM, T( IFRSTM, ILAST ), 1,
! 537: $ T( IFRSTM, ILAST-1 ), 1, C, S )
! 538: IF( ILZ )
! 539: $ CALL ZROT( N, Z( 1, ILAST ), 1, Z( 1, ILAST-1 ), 1, C, S )
! 540: *
! 541: * H(ILAST,ILAST-1)=0 -- Standardize B, set ALPHA and BETA
! 542: *
! 543: 60 CONTINUE
! 544: ABSB = ABS( T( ILAST, ILAST ) )
! 545: IF( ABSB.GT.SAFMIN ) THEN
! 546: SIGNBC = DCONJG( T( ILAST, ILAST ) / ABSB )
! 547: T( ILAST, ILAST ) = ABSB
! 548: IF( ILSCHR ) THEN
! 549: CALL ZSCAL( ILAST-IFRSTM, SIGNBC, T( IFRSTM, ILAST ), 1 )
! 550: CALL ZSCAL( ILAST+1-IFRSTM, SIGNBC, H( IFRSTM, ILAST ),
! 551: $ 1 )
! 552: ELSE
! 553: H( ILAST, ILAST ) = H( ILAST, ILAST )*SIGNBC
! 554: END IF
! 555: IF( ILZ )
! 556: $ CALL ZSCAL( N, SIGNBC, Z( 1, ILAST ), 1 )
! 557: ELSE
! 558: T( ILAST, ILAST ) = CZERO
! 559: END IF
! 560: ALPHA( ILAST ) = H( ILAST, ILAST )
! 561: BETA( ILAST ) = T( ILAST, ILAST )
! 562: *
! 563: * Go to next block -- exit if finished.
! 564: *
! 565: ILAST = ILAST - 1
! 566: IF( ILAST.LT.ILO )
! 567: $ GO TO 190
! 568: *
! 569: * Reset counters
! 570: *
! 571: IITER = 0
! 572: ESHIFT = CZERO
! 573: IF( .NOT.ILSCHR ) THEN
! 574: ILASTM = ILAST
! 575: IF( IFRSTM.GT.ILAST )
! 576: $ IFRSTM = ILO
! 577: END IF
! 578: GO TO 160
! 579: *
! 580: * QZ step
! 581: *
! 582: * This iteration only involves rows/columns IFIRST:ILAST. We
! 583: * assume IFIRST < ILAST, and that the diagonal of B is non-zero.
! 584: *
! 585: 70 CONTINUE
! 586: IITER = IITER + 1
! 587: IF( .NOT.ILSCHR ) THEN
! 588: IFRSTM = IFIRST
! 589: END IF
! 590: *
! 591: * Compute the Shift.
! 592: *
! 593: * At this point, IFIRST < ILAST, and the diagonal elements of
! 594: * T(IFIRST:ILAST,IFIRST,ILAST) are larger than BTOL (in
! 595: * magnitude)
! 596: *
! 597: IF( ( IITER / 10 )*10.NE.IITER ) THEN
! 598: *
! 599: * The Wilkinson shift (AEP p.512), i.e., the eigenvalue of
! 600: * the bottom-right 2x2 block of A inv(B) which is nearest to
! 601: * the bottom-right element.
! 602: *
! 603: * We factor B as U*D, where U has unit diagonals, and
! 604: * compute (A*inv(D))*inv(U).
! 605: *
! 606: U12 = ( BSCALE*T( ILAST-1, ILAST ) ) /
! 607: $ ( BSCALE*T( ILAST, ILAST ) )
! 608: AD11 = ( ASCALE*H( ILAST-1, ILAST-1 ) ) /
! 609: $ ( BSCALE*T( ILAST-1, ILAST-1 ) )
! 610: AD21 = ( ASCALE*H( ILAST, ILAST-1 ) ) /
! 611: $ ( BSCALE*T( ILAST-1, ILAST-1 ) )
! 612: AD12 = ( ASCALE*H( ILAST-1, ILAST ) ) /
! 613: $ ( BSCALE*T( ILAST, ILAST ) )
! 614: AD22 = ( ASCALE*H( ILAST, ILAST ) ) /
! 615: $ ( BSCALE*T( ILAST, ILAST ) )
! 616: ABI22 = AD22 - U12*AD21
! 617: *
! 618: T1 = HALF*( AD11+ABI22 )
! 619: RTDISC = SQRT( T1**2+AD12*AD21-AD11*AD22 )
! 620: TEMP = DBLE( T1-ABI22 )*DBLE( RTDISC ) +
! 621: $ DIMAG( T1-ABI22 )*DIMAG( RTDISC )
! 622: IF( TEMP.LE.ZERO ) THEN
! 623: SHIFT = T1 + RTDISC
! 624: ELSE
! 625: SHIFT = T1 - RTDISC
! 626: END IF
! 627: ELSE
! 628: *
! 629: * Exceptional shift. Chosen for no particularly good reason.
! 630: *
! 631: ESHIFT = ESHIFT + DCONJG( ( ASCALE*H( ILAST-1, ILAST ) ) /
! 632: $ ( BSCALE*T( ILAST-1, ILAST-1 ) ) )
! 633: SHIFT = ESHIFT
! 634: END IF
! 635: *
! 636: * Now check for two consecutive small subdiagonals.
! 637: *
! 638: DO 80 J = ILAST - 1, IFIRST + 1, -1
! 639: ISTART = J
! 640: CTEMP = ASCALE*H( J, J ) - SHIFT*( BSCALE*T( J, J ) )
! 641: TEMP = ABS1( CTEMP )
! 642: TEMP2 = ASCALE*ABS1( H( J+1, J ) )
! 643: TEMPR = MAX( TEMP, TEMP2 )
! 644: IF( TEMPR.LT.ONE .AND. TEMPR.NE.ZERO ) THEN
! 645: TEMP = TEMP / TEMPR
! 646: TEMP2 = TEMP2 / TEMPR
! 647: END IF
! 648: IF( ABS1( H( J, J-1 ) )*TEMP2.LE.TEMP*ATOL )
! 649: $ GO TO 90
! 650: 80 CONTINUE
! 651: *
! 652: ISTART = IFIRST
! 653: CTEMP = ASCALE*H( IFIRST, IFIRST ) -
! 654: $ SHIFT*( BSCALE*T( IFIRST, IFIRST ) )
! 655: 90 CONTINUE
! 656: *
! 657: * Do an implicit-shift QZ sweep.
! 658: *
! 659: * Initial Q
! 660: *
! 661: CTEMP2 = ASCALE*H( ISTART+1, ISTART )
! 662: CALL ZLARTG( CTEMP, CTEMP2, C, S, CTEMP3 )
! 663: *
! 664: * Sweep
! 665: *
! 666: DO 150 J = ISTART, ILAST - 1
! 667: IF( J.GT.ISTART ) THEN
! 668: CTEMP = H( J, J-1 )
! 669: CALL ZLARTG( CTEMP, H( J+1, J-1 ), C, S, H( J, J-1 ) )
! 670: H( J+1, J-1 ) = CZERO
! 671: END IF
! 672: *
! 673: DO 100 JC = J, ILASTM
! 674: CTEMP = C*H( J, JC ) + S*H( J+1, JC )
! 675: H( J+1, JC ) = -DCONJG( S )*H( J, JC ) + C*H( J+1, JC )
! 676: H( J, JC ) = CTEMP
! 677: CTEMP2 = C*T( J, JC ) + S*T( J+1, JC )
! 678: T( J+1, JC ) = -DCONJG( S )*T( J, JC ) + C*T( J+1, JC )
! 679: T( J, JC ) = CTEMP2
! 680: 100 CONTINUE
! 681: IF( ILQ ) THEN
! 682: DO 110 JR = 1, N
! 683: CTEMP = C*Q( JR, J ) + DCONJG( S )*Q( JR, J+1 )
! 684: Q( JR, J+1 ) = -S*Q( JR, J ) + C*Q( JR, J+1 )
! 685: Q( JR, J ) = CTEMP
! 686: 110 CONTINUE
! 687: END IF
! 688: *
! 689: CTEMP = T( J+1, J+1 )
! 690: CALL ZLARTG( CTEMP, T( J+1, J ), C, S, T( J+1, J+1 ) )
! 691: T( J+1, J ) = CZERO
! 692: *
! 693: DO 120 JR = IFRSTM, MIN( J+2, ILAST )
! 694: CTEMP = C*H( JR, J+1 ) + S*H( JR, J )
! 695: H( JR, J ) = -DCONJG( S )*H( JR, J+1 ) + C*H( JR, J )
! 696: H( JR, J+1 ) = CTEMP
! 697: 120 CONTINUE
! 698: DO 130 JR = IFRSTM, J
! 699: CTEMP = C*T( JR, J+1 ) + S*T( JR, J )
! 700: T( JR, J ) = -DCONJG( S )*T( JR, J+1 ) + C*T( JR, J )
! 701: T( JR, J+1 ) = CTEMP
! 702: 130 CONTINUE
! 703: IF( ILZ ) THEN
! 704: DO 140 JR = 1, N
! 705: CTEMP = C*Z( JR, J+1 ) + S*Z( JR, J )
! 706: Z( JR, J ) = -DCONJG( S )*Z( JR, J+1 ) + C*Z( JR, J )
! 707: Z( JR, J+1 ) = CTEMP
! 708: 140 CONTINUE
! 709: END IF
! 710: 150 CONTINUE
! 711: *
! 712: 160 CONTINUE
! 713: *
! 714: 170 CONTINUE
! 715: *
! 716: * Drop-through = non-convergence
! 717: *
! 718: 180 CONTINUE
! 719: INFO = ILAST
! 720: GO TO 210
! 721: *
! 722: * Successful completion of all QZ steps
! 723: *
! 724: 190 CONTINUE
! 725: *
! 726: * Set Eigenvalues 1:ILO-1
! 727: *
! 728: DO 200 J = 1, ILO - 1
! 729: ABSB = ABS( T( J, J ) )
! 730: IF( ABSB.GT.SAFMIN ) THEN
! 731: SIGNBC = DCONJG( T( J, J ) / ABSB )
! 732: T( J, J ) = ABSB
! 733: IF( ILSCHR ) THEN
! 734: CALL ZSCAL( J-1, SIGNBC, T( 1, J ), 1 )
! 735: CALL ZSCAL( J, SIGNBC, H( 1, J ), 1 )
! 736: ELSE
! 737: H( J, J ) = H( J, J )*SIGNBC
! 738: END IF
! 739: IF( ILZ )
! 740: $ CALL ZSCAL( N, SIGNBC, Z( 1, J ), 1 )
! 741: ELSE
! 742: T( J, J ) = CZERO
! 743: END IF
! 744: ALPHA( J ) = H( J, J )
! 745: BETA( J ) = T( J, J )
! 746: 200 CONTINUE
! 747: *
! 748: * Normal Termination
! 749: *
! 750: INFO = 0
! 751: *
! 752: * Exit (other than argument error) -- return optimal workspace size
! 753: *
! 754: 210 CONTINUE
! 755: WORK( 1 ) = DCMPLX( N )
! 756: RETURN
! 757: *
! 758: * End of ZHGEQZ
! 759: *
! 760: END
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