Annotation of rpl/lapack/lapack/zhetd2.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZHETD2( UPLO, N, A, LDA, D, E, TAU, 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: CHARACTER UPLO
! 10: INTEGER INFO, LDA, N
! 11: * ..
! 12: * .. Array Arguments ..
! 13: DOUBLE PRECISION D( * ), E( * )
! 14: COMPLEX*16 A( LDA, * ), TAU( * )
! 15: * ..
! 16: *
! 17: * Purpose
! 18: * =======
! 19: *
! 20: * ZHETD2 reduces a complex Hermitian matrix A to real symmetric
! 21: * tridiagonal form T by a unitary similarity transformation:
! 22: * Q' * A * Q = T.
! 23: *
! 24: * Arguments
! 25: * =========
! 26: *
! 27: * UPLO (input) CHARACTER*1
! 28: * Specifies whether the upper or lower triangular part of the
! 29: * Hermitian matrix A is stored:
! 30: * = 'U': Upper triangular
! 31: * = 'L': Lower triangular
! 32: *
! 33: * N (input) INTEGER
! 34: * The order of the matrix A. N >= 0.
! 35: *
! 36: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
! 37: * On entry, the Hermitian matrix A. If UPLO = 'U', the leading
! 38: * n-by-n upper triangular part of A contains the upper
! 39: * triangular part of the matrix A, and the strictly lower
! 40: * triangular part of A is not referenced. If UPLO = 'L', the
! 41: * leading n-by-n lower triangular part of A contains the lower
! 42: * triangular part of the matrix A, and the strictly upper
! 43: * triangular part of A is not referenced.
! 44: * On exit, if UPLO = 'U', the diagonal and first superdiagonal
! 45: * of A are overwritten by the corresponding elements of the
! 46: * tridiagonal matrix T, and the elements above the first
! 47: * superdiagonal, with the array TAU, represent the unitary
! 48: * matrix Q as a product of elementary reflectors; if UPLO
! 49: * = 'L', the diagonal and first subdiagonal of A are over-
! 50: * written by the corresponding elements of the tridiagonal
! 51: * matrix T, and the elements below the first subdiagonal, with
! 52: * the array TAU, represent the unitary matrix Q as a product
! 53: * of elementary reflectors. See Further Details.
! 54: *
! 55: * LDA (input) INTEGER
! 56: * The leading dimension of the array A. LDA >= max(1,N).
! 57: *
! 58: * D (output) DOUBLE PRECISION array, dimension (N)
! 59: * The diagonal elements of the tridiagonal matrix T:
! 60: * D(i) = A(i,i).
! 61: *
! 62: * E (output) DOUBLE PRECISION array, dimension (N-1)
! 63: * The off-diagonal elements of the tridiagonal matrix T:
! 64: * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
! 65: *
! 66: * TAU (output) COMPLEX*16 array, dimension (N-1)
! 67: * The scalar factors of the elementary reflectors (see Further
! 68: * Details).
! 69: *
! 70: * INFO (output) INTEGER
! 71: * = 0: successful exit
! 72: * < 0: if INFO = -i, the i-th argument had an illegal value.
! 73: *
! 74: * Further Details
! 75: * ===============
! 76: *
! 77: * If UPLO = 'U', the matrix Q is represented as a product of elementary
! 78: * reflectors
! 79: *
! 80: * Q = H(n-1) . . . H(2) H(1).
! 81: *
! 82: * Each H(i) has the form
! 83: *
! 84: * H(i) = I - tau * v * v'
! 85: *
! 86: * where tau is a complex scalar, and v is a complex vector with
! 87: * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
! 88: * A(1:i-1,i+1), and tau in TAU(i).
! 89: *
! 90: * If UPLO = 'L', the matrix Q is represented as a product of elementary
! 91: * reflectors
! 92: *
! 93: * Q = H(1) H(2) . . . H(n-1).
! 94: *
! 95: * Each H(i) has the form
! 96: *
! 97: * H(i) = I - tau * v * v'
! 98: *
! 99: * where tau is a complex scalar, and v is a complex vector with
! 100: * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
! 101: * and tau in TAU(i).
! 102: *
! 103: * The contents of A on exit are illustrated by the following examples
! 104: * with n = 5:
! 105: *
! 106: * if UPLO = 'U': if UPLO = 'L':
! 107: *
! 108: * ( d e v2 v3 v4 ) ( d )
! 109: * ( d e v3 v4 ) ( e d )
! 110: * ( d e v4 ) ( v1 e d )
! 111: * ( d e ) ( v1 v2 e d )
! 112: * ( d ) ( v1 v2 v3 e d )
! 113: *
! 114: * where d and e denote diagonal and off-diagonal elements of T, and vi
! 115: * denotes an element of the vector defining H(i).
! 116: *
! 117: * =====================================================================
! 118: *
! 119: * .. Parameters ..
! 120: COMPLEX*16 ONE, ZERO, HALF
! 121: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ),
! 122: $ ZERO = ( 0.0D+0, 0.0D+0 ),
! 123: $ HALF = ( 0.5D+0, 0.0D+0 ) )
! 124: * ..
! 125: * .. Local Scalars ..
! 126: LOGICAL UPPER
! 127: INTEGER I
! 128: COMPLEX*16 ALPHA, TAUI
! 129: * ..
! 130: * .. External Subroutines ..
! 131: EXTERNAL XERBLA, ZAXPY, ZHEMV, ZHER2, ZLARFG
! 132: * ..
! 133: * .. External Functions ..
! 134: LOGICAL LSAME
! 135: COMPLEX*16 ZDOTC
! 136: EXTERNAL LSAME, ZDOTC
! 137: * ..
! 138: * .. Intrinsic Functions ..
! 139: INTRINSIC DBLE, MAX, MIN
! 140: * ..
! 141: * .. Executable Statements ..
! 142: *
! 143: * Test the input parameters
! 144: *
! 145: INFO = 0
! 146: UPPER = LSAME( UPLO, 'U' )
! 147: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 148: INFO = -1
! 149: ELSE IF( N.LT.0 ) THEN
! 150: INFO = -2
! 151: ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
! 152: INFO = -4
! 153: END IF
! 154: IF( INFO.NE.0 ) THEN
! 155: CALL XERBLA( 'ZHETD2', -INFO )
! 156: RETURN
! 157: END IF
! 158: *
! 159: * Quick return if possible
! 160: *
! 161: IF( N.LE.0 )
! 162: $ RETURN
! 163: *
! 164: IF( UPPER ) THEN
! 165: *
! 166: * Reduce the upper triangle of A
! 167: *
! 168: A( N, N ) = DBLE( A( N, N ) )
! 169: DO 10 I = N - 1, 1, -1
! 170: *
! 171: * Generate elementary reflector H(i) = I - tau * v * v'
! 172: * to annihilate A(1:i-1,i+1)
! 173: *
! 174: ALPHA = A( I, I+1 )
! 175: CALL ZLARFG( I, ALPHA, A( 1, I+1 ), 1, TAUI )
! 176: E( I ) = ALPHA
! 177: *
! 178: IF( TAUI.NE.ZERO ) THEN
! 179: *
! 180: * Apply H(i) from both sides to A(1:i,1:i)
! 181: *
! 182: A( I, I+1 ) = ONE
! 183: *
! 184: * Compute x := tau * A * v storing x in TAU(1:i)
! 185: *
! 186: CALL ZHEMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
! 187: $ TAU, 1 )
! 188: *
! 189: * Compute w := x - 1/2 * tau * (x'*v) * v
! 190: *
! 191: ALPHA = -HALF*TAUI*ZDOTC( I, TAU, 1, A( 1, I+1 ), 1 )
! 192: CALL ZAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
! 193: *
! 194: * Apply the transformation as a rank-2 update:
! 195: * A := A - v * w' - w * v'
! 196: *
! 197: CALL ZHER2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
! 198: $ LDA )
! 199: *
! 200: ELSE
! 201: A( I, I ) = DBLE( A( I, I ) )
! 202: END IF
! 203: A( I, I+1 ) = E( I )
! 204: D( I+1 ) = A( I+1, I+1 )
! 205: TAU( I ) = TAUI
! 206: 10 CONTINUE
! 207: D( 1 ) = A( 1, 1 )
! 208: ELSE
! 209: *
! 210: * Reduce the lower triangle of A
! 211: *
! 212: A( 1, 1 ) = DBLE( A( 1, 1 ) )
! 213: DO 20 I = 1, N - 1
! 214: *
! 215: * Generate elementary reflector H(i) = I - tau * v * v'
! 216: * to annihilate A(i+2:n,i)
! 217: *
! 218: ALPHA = A( I+1, I )
! 219: CALL ZLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1, TAUI )
! 220: E( I ) = ALPHA
! 221: *
! 222: IF( TAUI.NE.ZERO ) THEN
! 223: *
! 224: * Apply H(i) from both sides to A(i+1:n,i+1:n)
! 225: *
! 226: A( I+1, I ) = ONE
! 227: *
! 228: * Compute x := tau * A * v storing y in TAU(i:n-1)
! 229: *
! 230: CALL ZHEMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
! 231: $ A( I+1, I ), 1, ZERO, TAU( I ), 1 )
! 232: *
! 233: * Compute w := x - 1/2 * tau * (x'*v) * v
! 234: *
! 235: ALPHA = -HALF*TAUI*ZDOTC( N-I, TAU( I ), 1, A( I+1, I ),
! 236: $ 1 )
! 237: CALL ZAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
! 238: *
! 239: * Apply the transformation as a rank-2 update:
! 240: * A := A - v * w' - w * v'
! 241: *
! 242: CALL ZHER2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
! 243: $ A( I+1, I+1 ), LDA )
! 244: *
! 245: ELSE
! 246: A( I+1, I+1 ) = DBLE( A( I+1, I+1 ) )
! 247: END IF
! 248: A( I+1, I ) = E( I )
! 249: D( I ) = A( I, I )
! 250: TAU( I ) = TAUI
! 251: 20 CONTINUE
! 252: D( N ) = A( N, N )
! 253: END IF
! 254: *
! 255: RETURN
! 256: *
! 257: * End of ZHETD2
! 258: *
! 259: END
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