Annotation of rpl/lapack/lapack/dlarfgp.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DLARFGP( N, ALPHA, X, INCX, TAU )
! 2: *
! 3: * -- LAPACK auxiliary routine (version 3.2.2) --
! 4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 6: * June 2010
! 7: *
! 8: * .. Scalar Arguments ..
! 9: INTEGER INCX, N
! 10: DOUBLE PRECISION ALPHA, TAU
! 11: * ..
! 12: * .. Array Arguments ..
! 13: DOUBLE PRECISION X( * )
! 14: * ..
! 15: *
! 16: * Purpose
! 17: * =======
! 18: *
! 19: * DLARFGP generates a real elementary reflector H of order n, such
! 20: * that
! 21: *
! 22: * H * ( alpha ) = ( beta ), H' * H = I.
! 23: * ( x ) ( 0 )
! 24: *
! 25: * where alpha and beta are scalars, beta is non-negative, and x is
! 26: * an (n-1)-element real vector. H is represented in the form
! 27: *
! 28: * H = I - tau * ( 1 ) * ( 1 v' ) ,
! 29: * ( v )
! 30: *
! 31: * where tau is a real scalar and v is a real (n-1)-element
! 32: * vector.
! 33: *
! 34: * If the elements of x are all zero, then tau = 0 and H is taken to be
! 35: * the unit matrix.
! 36: *
! 37: * Arguments
! 38: * =========
! 39: *
! 40: * N (input) INTEGER
! 41: * The order of the elementary reflector.
! 42: *
! 43: * ALPHA (input/output) DOUBLE PRECISION
! 44: * On entry, the value alpha.
! 45: * On exit, it is overwritten with the value beta.
! 46: *
! 47: * X (input/output) DOUBLE PRECISION array, dimension
! 48: * (1+(N-2)*abs(INCX))
! 49: * On entry, the vector x.
! 50: * On exit, it is overwritten with the vector v.
! 51: *
! 52: * INCX (input) INTEGER
! 53: * The increment between elements of X. INCX > 0.
! 54: *
! 55: * TAU (output) DOUBLE PRECISION
! 56: * The value tau.
! 57: *
! 58: * =====================================================================
! 59: *
! 60: * .. Parameters ..
! 61: DOUBLE PRECISION TWO, ONE, ZERO
! 62: PARAMETER ( TWO = 2.0D+0, ONE = 1.0D+0, ZERO = 0.0D+0 )
! 63: * ..
! 64: * .. Local Scalars ..
! 65: INTEGER J, KNT
! 66: DOUBLE PRECISION BETA, BIGNUM, SAVEALPHA, SMLNUM, XNORM
! 67: * ..
! 68: * .. External Functions ..
! 69: DOUBLE PRECISION DLAMCH, DLAPY2, DNRM2
! 70: EXTERNAL DLAMCH, DLAPY2, DNRM2
! 71: * ..
! 72: * .. Intrinsic Functions ..
! 73: INTRINSIC ABS, SIGN
! 74: * ..
! 75: * .. External Subroutines ..
! 76: EXTERNAL DSCAL
! 77: * ..
! 78: * .. Executable Statements ..
! 79: *
! 80: IF( N.LE.0 ) THEN
! 81: TAU = ZERO
! 82: RETURN
! 83: END IF
! 84: *
! 85: XNORM = DNRM2( N-1, X, INCX )
! 86: *
! 87: IF( XNORM.EQ.ZERO ) THEN
! 88: *
! 89: * H = [+/-1, 0; I], sign chosen so ALPHA >= 0
! 90: *
! 91: IF( ALPHA.GE.ZERO ) THEN
! 92: * When TAU.eq.ZERO, the vector is special-cased to be
! 93: * all zeros in the application routines. We do not need
! 94: * to clear it.
! 95: TAU = ZERO
! 96: ELSE
! 97: * However, the application routines rely on explicit
! 98: * zero checks when TAU.ne.ZERO, and we must clear X.
! 99: TAU = TWO
! 100: DO J = 1, N-1
! 101: X( 1 + (J-1)*INCX ) = 0
! 102: END DO
! 103: ALPHA = -ALPHA
! 104: END IF
! 105: ELSE
! 106: *
! 107: * general case
! 108: *
! 109: BETA = SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
! 110: SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'E' )
! 111: KNT = 0
! 112: IF( ABS( BETA ).LT.SMLNUM ) THEN
! 113: *
! 114: * XNORM, BETA may be inaccurate; scale X and recompute them
! 115: *
! 116: BIGNUM = ONE / SMLNUM
! 117: 10 CONTINUE
! 118: KNT = KNT + 1
! 119: CALL DSCAL( N-1, BIGNUM, X, INCX )
! 120: BETA = BETA*BIGNUM
! 121: ALPHA = ALPHA*BIGNUM
! 122: IF( ABS( BETA ).LT.SMLNUM )
! 123: $ GO TO 10
! 124: *
! 125: * New BETA is at most 1, at least SMLNUM
! 126: *
! 127: XNORM = DNRM2( N-1, X, INCX )
! 128: BETA = SIGN( DLAPY2( ALPHA, XNORM ), ALPHA )
! 129: END IF
! 130: SAVEALPHA = ALPHA
! 131: ALPHA = ALPHA + BETA
! 132: IF( BETA.LT.ZERO ) THEN
! 133: BETA = -BETA
! 134: TAU = -ALPHA / BETA
! 135: ELSE
! 136: ALPHA = XNORM * (XNORM/ALPHA)
! 137: TAU = ALPHA / BETA
! 138: ALPHA = -ALPHA
! 139: END IF
! 140: *
! 141: IF ( ABS(TAU).LE.SMLNUM ) THEN
! 142: *
! 143: * In the case where the computed TAU ends up being a denormalized number,
! 144: * it loses relative accuracy. This is a BIG problem. Solution: flush TAU
! 145: * to ZERO. This explains the next IF statement.
! 146: *
! 147: * (Bug report provided by Pat Quillen from MathWorks on Jul 29, 2009.)
! 148: * (Thanks Pat. Thanks MathWorks.)
! 149: *
! 150: IF( SAVEALPHA.GE.ZERO ) THEN
! 151: TAU = ZERO
! 152: ELSE
! 153: TAU = TWO
! 154: DO J = 1, N-1
! 155: X( 1 + (J-1)*INCX ) = 0
! 156: END DO
! 157: BETA = -SAVEALPHA
! 158: END IF
! 159: *
! 160: ELSE
! 161: *
! 162: * This is the general case.
! 163: *
! 164: CALL DSCAL( N-1, ONE / ALPHA, X, INCX )
! 165: *
! 166: END IF
! 167: *
! 168: * If BETA is subnormal, it may lose relative accuracy
! 169: *
! 170: DO 20 J = 1, KNT
! 171: BETA = BETA*SMLNUM
! 172: 20 CONTINUE
! 173: ALPHA = BETA
! 174: END IF
! 175: *
! 176: RETURN
! 177: *
! 178: * End of DLARFGP
! 179: *
! 180: END
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