Annotation of rpl/lapack/lapack/zgehrd.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )
! 2: *
! 3: * -- LAPACK routine (version 3.2.1) --
! 4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 6: * -- April 2009 --
! 7: *
! 8: * .. Scalar Arguments ..
! 9: INTEGER IHI, ILO, INFO, LDA, LWORK, N
! 10: * ..
! 11: * .. Array Arguments ..
! 12: COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * )
! 13: * ..
! 14: *
! 15: * Purpose
! 16: * =======
! 17: *
! 18: * ZGEHRD reduces a complex general matrix A to upper Hessenberg form H by
! 19: * an unitary similarity transformation: Q' * A * Q = H .
! 20: *
! 21: * Arguments
! 22: * =========
! 23: *
! 24: * N (input) INTEGER
! 25: * The order of the matrix A. N >= 0.
! 26: *
! 27: * ILO (input) INTEGER
! 28: * IHI (input) INTEGER
! 29: * It is assumed that A is already upper triangular in rows
! 30: * and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
! 31: * set by a previous call to ZGEBAL; otherwise they should be
! 32: * set to 1 and N respectively. See Further Details.
! 33: * 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
! 34: *
! 35: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
! 36: * On entry, the N-by-N general matrix to be reduced.
! 37: * On exit, the upper triangle and the first subdiagonal of A
! 38: * are overwritten with the upper Hessenberg matrix H, and the
! 39: * elements below the first subdiagonal, with the array TAU,
! 40: * represent the unitary matrix Q as a product of elementary
! 41: * reflectors. See Further Details.
! 42: *
! 43: * LDA (input) INTEGER
! 44: * The leading dimension of the array A. LDA >= max(1,N).
! 45: *
! 46: * TAU (output) COMPLEX*16 array, dimension (N-1)
! 47: * The scalar factors of the elementary reflectors (see Further
! 48: * Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
! 49: * zero.
! 50: *
! 51: * WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
! 52: * On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 53: *
! 54: * LWORK (input) INTEGER
! 55: * The length of the array WORK. LWORK >= max(1,N).
! 56: * For optimum performance LWORK >= N*NB, where NB is the
! 57: * optimal blocksize.
! 58: *
! 59: * If LWORK = -1, then a workspace query is assumed; the routine
! 60: * only calculates the optimal size of the WORK array, returns
! 61: * this value as the first entry of the WORK array, and no error
! 62: * message related to LWORK is issued by XERBLA.
! 63: *
! 64: * INFO (output) INTEGER
! 65: * = 0: successful exit
! 66: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 67: *
! 68: * Further Details
! 69: * ===============
! 70: *
! 71: * The matrix Q is represented as a product of (ihi-ilo) elementary
! 72: * reflectors
! 73: *
! 74: * Q = H(ilo) H(ilo+1) . . . H(ihi-1).
! 75: *
! 76: * Each H(i) has the form
! 77: *
! 78: * H(i) = I - tau * v * v'
! 79: *
! 80: * where tau is a complex scalar, and v is a complex vector with
! 81: * v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
! 82: * exit in A(i+2:ihi,i), and tau in TAU(i).
! 83: *
! 84: * The contents of A are illustrated by the following example, with
! 85: * n = 7, ilo = 2 and ihi = 6:
! 86: *
! 87: * on entry, on exit,
! 88: *
! 89: * ( a a a a a a a ) ( a a h h h h a )
! 90: * ( a a a a a a ) ( a h h h h a )
! 91: * ( a a a a a a ) ( h h h h h h )
! 92: * ( a a a a a a ) ( v2 h h h h h )
! 93: * ( a a a a a a ) ( v2 v3 h h h h )
! 94: * ( a a a a a a ) ( v2 v3 v4 h h h )
! 95: * ( a ) ( a )
! 96: *
! 97: * where a denotes an element of the original matrix A, h denotes a
! 98: * modified element of the upper Hessenberg matrix H, and vi denotes an
! 99: * element of the vector defining H(i).
! 100: *
! 101: * This file is a slight modification of LAPACK-3.0's DGEHRD
! 102: * subroutine incorporating improvements proposed by Quintana-Orti and
! 103: * Van de Geijn (2006). (See DLAHR2.)
! 104: *
! 105: * =====================================================================
! 106: *
! 107: * .. Parameters ..
! 108: INTEGER NBMAX, LDT
! 109: PARAMETER ( NBMAX = 64, LDT = NBMAX+1 )
! 110: COMPLEX*16 ZERO, ONE
! 111: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
! 112: $ ONE = ( 1.0D+0, 0.0D+0 ) )
! 113: * ..
! 114: * .. Local Scalars ..
! 115: LOGICAL LQUERY
! 116: INTEGER I, IB, IINFO, IWS, J, LDWORK, LWKOPT, NB,
! 117: $ NBMIN, NH, NX
! 118: COMPLEX*16 EI
! 119: * ..
! 120: * .. Local Arrays ..
! 121: COMPLEX*16 T( LDT, NBMAX )
! 122: * ..
! 123: * .. External Subroutines ..
! 124: EXTERNAL ZAXPY, ZGEHD2, ZGEMM, ZLAHR2, ZLARFB, ZTRMM,
! 125: $ XERBLA
! 126: * ..
! 127: * .. Intrinsic Functions ..
! 128: INTRINSIC MAX, MIN
! 129: * ..
! 130: * .. External Functions ..
! 131: INTEGER ILAENV
! 132: EXTERNAL ILAENV
! 133: * ..
! 134: * .. Executable Statements ..
! 135: *
! 136: * Test the input parameters
! 137: *
! 138: INFO = 0
! 139: NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) )
! 140: LWKOPT = N*NB
! 141: WORK( 1 ) = LWKOPT
! 142: LQUERY = ( LWORK.EQ.-1 )
! 143: IF( N.LT.0 ) THEN
! 144: INFO = -1
! 145: ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
! 146: INFO = -2
! 147: ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
! 148: INFO = -3
! 149: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 150: INFO = -5
! 151: ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
! 152: INFO = -8
! 153: END IF
! 154: IF( INFO.NE.0 ) THEN
! 155: CALL XERBLA( 'ZGEHRD', -INFO )
! 156: RETURN
! 157: ELSE IF( LQUERY ) THEN
! 158: RETURN
! 159: END IF
! 160: *
! 161: * Set elements 1:ILO-1 and IHI:N-1 of TAU to zero
! 162: *
! 163: DO 10 I = 1, ILO - 1
! 164: TAU( I ) = ZERO
! 165: 10 CONTINUE
! 166: DO 20 I = MAX( 1, IHI ), N - 1
! 167: TAU( I ) = ZERO
! 168: 20 CONTINUE
! 169: *
! 170: * Quick return if possible
! 171: *
! 172: NH = IHI - ILO + 1
! 173: IF( NH.LE.1 ) THEN
! 174: WORK( 1 ) = 1
! 175: RETURN
! 176: END IF
! 177: *
! 178: * Determine the block size
! 179: *
! 180: NB = MIN( NBMAX, ILAENV( 1, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) )
! 181: NBMIN = 2
! 182: IWS = 1
! 183: IF( NB.GT.1 .AND. NB.LT.NH ) THEN
! 184: *
! 185: * Determine when to cross over from blocked to unblocked code
! 186: * (last block is always handled by unblocked code)
! 187: *
! 188: NX = MAX( NB, ILAENV( 3, 'ZGEHRD', ' ', N, ILO, IHI, -1 ) )
! 189: IF( NX.LT.NH ) THEN
! 190: *
! 191: * Determine if workspace is large enough for blocked code
! 192: *
! 193: IWS = N*NB
! 194: IF( LWORK.LT.IWS ) THEN
! 195: *
! 196: * Not enough workspace to use optimal NB: determine the
! 197: * minimum value of NB, and reduce NB or force use of
! 198: * unblocked code
! 199: *
! 200: NBMIN = MAX( 2, ILAENV( 2, 'ZGEHRD', ' ', N, ILO, IHI,
! 201: $ -1 ) )
! 202: IF( LWORK.GE.N*NBMIN ) THEN
! 203: NB = LWORK / N
! 204: ELSE
! 205: NB = 1
! 206: END IF
! 207: END IF
! 208: END IF
! 209: END IF
! 210: LDWORK = N
! 211: *
! 212: IF( NB.LT.NBMIN .OR. NB.GE.NH ) THEN
! 213: *
! 214: * Use unblocked code below
! 215: *
! 216: I = ILO
! 217: *
! 218: ELSE
! 219: *
! 220: * Use blocked code
! 221: *
! 222: DO 40 I = ILO, IHI - 1 - NX, NB
! 223: IB = MIN( NB, IHI-I )
! 224: *
! 225: * Reduce columns i:i+ib-1 to Hessenberg form, returning the
! 226: * matrices V and T of the block reflector H = I - V*T*V'
! 227: * which performs the reduction, and also the matrix Y = A*V*T
! 228: *
! 229: CALL ZLAHR2( IHI, I, IB, A( 1, I ), LDA, TAU( I ), T, LDT,
! 230: $ WORK, LDWORK )
! 231: *
! 232: * Apply the block reflector H to A(1:ihi,i+ib:ihi) from the
! 233: * right, computing A := A - Y * V'. V(i+ib,ib-1) must be set
! 234: * to 1
! 235: *
! 236: EI = A( I+IB, I+IB-1 )
! 237: A( I+IB, I+IB-1 ) = ONE
! 238: CALL ZGEMM( 'No transpose', 'Conjugate transpose',
! 239: $ IHI, IHI-I-IB+1,
! 240: $ IB, -ONE, WORK, LDWORK, A( I+IB, I ), LDA, ONE,
! 241: $ A( 1, I+IB ), LDA )
! 242: A( I+IB, I+IB-1 ) = EI
! 243: *
! 244: * Apply the block reflector H to A(1:i,i+1:i+ib-1) from the
! 245: * right
! 246: *
! 247: CALL ZTRMM( 'Right', 'Lower', 'Conjugate transpose',
! 248: $ 'Unit', I, IB-1,
! 249: $ ONE, A( I+1, I ), LDA, WORK, LDWORK )
! 250: DO 30 J = 0, IB-2
! 251: CALL ZAXPY( I, -ONE, WORK( LDWORK*J+1 ), 1,
! 252: $ A( 1, I+J+1 ), 1 )
! 253: 30 CONTINUE
! 254: *
! 255: * Apply the block reflector H to A(i+1:ihi,i+ib:n) from the
! 256: * left
! 257: *
! 258: CALL ZLARFB( 'Left', 'Conjugate transpose', 'Forward',
! 259: $ 'Columnwise',
! 260: $ IHI-I, N-I-IB+1, IB, A( I+1, I ), LDA, T, LDT,
! 261: $ A( I+1, I+IB ), LDA, WORK, LDWORK )
! 262: 40 CONTINUE
! 263: END IF
! 264: *
! 265: * Use unblocked code to reduce the rest of the matrix
! 266: *
! 267: CALL ZGEHD2( N, I, IHI, A, LDA, TAU, WORK, IINFO )
! 268: WORK( 1 ) = IWS
! 269: *
! 270: RETURN
! 271: *
! 272: * End of ZGEHRD
! 273: *
! 274: END
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