Annotation of rpl/lapack/lapack/dgehd2.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DGEHD2( N, ILO, IHI, A, LDA, TAU, WORK, INFO )
! 2: *
! 3: * -- LAPACK routine (version 3.2) --
! 4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 6: * November 2006
! 7: *
! 8: * .. Scalar Arguments ..
! 9: INTEGER IHI, ILO, INFO, LDA, N
! 10: * ..
! 11: * .. Array Arguments ..
! 12: DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
! 13: * ..
! 14: *
! 15: * Purpose
! 16: * =======
! 17: *
! 18: * DGEHD2 reduces a real general matrix A to upper Hessenberg form H by
! 19: * an orthogonal 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 DGEBAL; otherwise they should be
! 32: * set to 1 and N respectively. See Further Details.
! 33: * 1 <= ILO <= IHI <= max(1,N).
! 34: *
! 35: * A (input/output) DOUBLE PRECISION 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 orthogonal 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) DOUBLE PRECISION array, dimension (N-1)
! 47: * The scalar factors of the elementary reflectors (see Further
! 48: * Details).
! 49: *
! 50: * WORK (workspace) DOUBLE PRECISION array, dimension (N)
! 51: *
! 52: * INFO (output) INTEGER
! 53: * = 0: successful exit.
! 54: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 55: *
! 56: * Further Details
! 57: * ===============
! 58: *
! 59: * The matrix Q is represented as a product of (ihi-ilo) elementary
! 60: * reflectors
! 61: *
! 62: * Q = H(ilo) H(ilo+1) . . . H(ihi-1).
! 63: *
! 64: * Each H(i) has the form
! 65: *
! 66: * H(i) = I - tau * v * v'
! 67: *
! 68: * where tau is a real scalar, and v is a real vector with
! 69: * v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
! 70: * exit in A(i+2:ihi,i), and tau in TAU(i).
! 71: *
! 72: * The contents of A are illustrated by the following example, with
! 73: * n = 7, ilo = 2 and ihi = 6:
! 74: *
! 75: * on entry, on exit,
! 76: *
! 77: * ( a a a a a a a ) ( a a h h h h a )
! 78: * ( a a a a a a ) ( a h h h h a )
! 79: * ( a a a a a a ) ( h h h h h h )
! 80: * ( a a a a a a ) ( v2 h h h h h )
! 81: * ( a a a a a a ) ( v2 v3 h h h h )
! 82: * ( a a a a a a ) ( v2 v3 v4 h h h )
! 83: * ( a ) ( a )
! 84: *
! 85: * where a denotes an element of the original matrix A, h denotes a
! 86: * modified element of the upper Hessenberg matrix H, and vi denotes an
! 87: * element of the vector defining H(i).
! 88: *
! 89: * =====================================================================
! 90: *
! 91: * .. Parameters ..
! 92: DOUBLE PRECISION ONE
! 93: PARAMETER ( ONE = 1.0D+0 )
! 94: * ..
! 95: * .. Local Scalars ..
! 96: INTEGER I
! 97: DOUBLE PRECISION AII
! 98: * ..
! 99: * .. External Subroutines ..
! 100: EXTERNAL DLARF, DLARFG, XERBLA
! 101: * ..
! 102: * .. Intrinsic Functions ..
! 103: INTRINSIC MAX, MIN
! 104: * ..
! 105: * .. Executable Statements ..
! 106: *
! 107: * Test the input parameters
! 108: *
! 109: INFO = 0
! 110: IF( N.LT.0 ) THEN
! 111: INFO = -1
! 112: ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
! 113: INFO = -2
! 114: ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
! 115: INFO = -3
! 116: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 117: INFO = -5
! 118: END IF
! 119: IF( INFO.NE.0 ) THEN
! 120: CALL XERBLA( 'DGEHD2', -INFO )
! 121: RETURN
! 122: END IF
! 123: *
! 124: DO 10 I = ILO, IHI - 1
! 125: *
! 126: * Compute elementary reflector H(i) to annihilate A(i+2:ihi,i)
! 127: *
! 128: CALL DLARFG( IHI-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
! 129: $ TAU( I ) )
! 130: AII = A( I+1, I )
! 131: A( I+1, I ) = ONE
! 132: *
! 133: * Apply H(i) to A(1:ihi,i+1:ihi) from the right
! 134: *
! 135: CALL DLARF( 'Right', IHI, IHI-I, A( I+1, I ), 1, TAU( I ),
! 136: $ A( 1, I+1 ), LDA, WORK )
! 137: *
! 138: * Apply H(i) to A(i+1:ihi,i+1:n) from the left
! 139: *
! 140: CALL DLARF( 'Left', IHI-I, N-I, A( I+1, I ), 1, TAU( I ),
! 141: $ A( I+1, I+1 ), LDA, WORK )
! 142: *
! 143: A( I+1, I ) = AII
! 144: 10 CONTINUE
! 145: *
! 146: RETURN
! 147: *
! 148: * End of DGEHD2
! 149: *
! 150: END
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