Annotation of rpl/lapack/lapack/zlatrd.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
! 2: *
! 3: * -- LAPACK auxiliary 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 LDA, LDW, N, NB
! 11: * ..
! 12: * .. Array Arguments ..
! 13: DOUBLE PRECISION E( * )
! 14: COMPLEX*16 A( LDA, * ), TAU( * ), W( LDW, * )
! 15: * ..
! 16: *
! 17: * Purpose
! 18: * =======
! 19: *
! 20: * ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to
! 21: * Hermitian tridiagonal form by a unitary similarity
! 22: * transformation Q' * A * Q, and returns the matrices V and W which are
! 23: * needed to apply the transformation to the unreduced part of A.
! 24: *
! 25: * If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a
! 26: * matrix, of which the upper triangle is supplied;
! 27: * if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a
! 28: * matrix, of which the lower triangle is supplied.
! 29: *
! 30: * This is an auxiliary routine called by ZHETRD.
! 31: *
! 32: * Arguments
! 33: * =========
! 34: *
! 35: * UPLO (input) CHARACTER*1
! 36: * Specifies whether the upper or lower triangular part of the
! 37: * Hermitian matrix A is stored:
! 38: * = 'U': Upper triangular
! 39: * = 'L': Lower triangular
! 40: *
! 41: * N (input) INTEGER
! 42: * The order of the matrix A.
! 43: *
! 44: * NB (input) INTEGER
! 45: * The number of rows and columns to be reduced.
! 46: *
! 47: * A (input/output) COMPLEX*16 array, dimension (LDA,N)
! 48: * On entry, the Hermitian matrix A. If UPLO = 'U', the leading
! 49: * n-by-n upper triangular part of A contains the upper
! 50: * triangular part of the matrix A, and the strictly lower
! 51: * triangular part of A is not referenced. If UPLO = 'L', the
! 52: * leading n-by-n lower triangular part of A contains the lower
! 53: * triangular part of the matrix A, and the strictly upper
! 54: * triangular part of A is not referenced.
! 55: * On exit:
! 56: * if UPLO = 'U', the last NB columns have been reduced to
! 57: * tridiagonal form, with the diagonal elements overwriting
! 58: * the diagonal elements of A; the elements above the diagonal
! 59: * with the array TAU, represent the unitary matrix Q as a
! 60: * product of elementary reflectors;
! 61: * if UPLO = 'L', the first NB columns have been reduced to
! 62: * tridiagonal form, with the diagonal elements overwriting
! 63: * the diagonal elements of A; the elements below the diagonal
! 64: * with the array TAU, represent the unitary matrix Q as a
! 65: * product of elementary reflectors.
! 66: * See Further Details.
! 67: *
! 68: * LDA (input) INTEGER
! 69: * The leading dimension of the array A. LDA >= max(1,N).
! 70: *
! 71: * E (output) DOUBLE PRECISION array, dimension (N-1)
! 72: * If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
! 73: * elements of the last NB columns of the reduced matrix;
! 74: * if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
! 75: * the first NB columns of the reduced matrix.
! 76: *
! 77: * TAU (output) COMPLEX*16 array, dimension (N-1)
! 78: * The scalar factors of the elementary reflectors, stored in
! 79: * TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
! 80: * See Further Details.
! 81: *
! 82: * W (output) COMPLEX*16 array, dimension (LDW,NB)
! 83: * The n-by-nb matrix W required to update the unreduced part
! 84: * of A.
! 85: *
! 86: * LDW (input) INTEGER
! 87: * The leading dimension of the array W. LDW >= max(1,N).
! 88: *
! 89: * Further Details
! 90: * ===============
! 91: *
! 92: * If UPLO = 'U', the matrix Q is represented as a product of elementary
! 93: * reflectors
! 94: *
! 95: * Q = H(n) H(n-1) . . . H(n-nb+1).
! 96: *
! 97: * Each H(i) has the form
! 98: *
! 99: * H(i) = I - tau * v * v'
! 100: *
! 101: * where tau is a complex scalar, and v is a complex vector with
! 102: * v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
! 103: * and tau in TAU(i-1).
! 104: *
! 105: * If UPLO = 'L', the matrix Q is represented as a product of elementary
! 106: * reflectors
! 107: *
! 108: * Q = H(1) H(2) . . . H(nb).
! 109: *
! 110: * Each H(i) has the form
! 111: *
! 112: * H(i) = I - tau * v * v'
! 113: *
! 114: * where tau is a complex scalar, and v is a complex vector with
! 115: * v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
! 116: * and tau in TAU(i).
! 117: *
! 118: * The elements of the vectors v together form the n-by-nb matrix V
! 119: * which is needed, with W, to apply the transformation to the unreduced
! 120: * part of the matrix, using a Hermitian rank-2k update of the form:
! 121: * A := A - V*W' - W*V'.
! 122: *
! 123: * The contents of A on exit are illustrated by the following examples
! 124: * with n = 5 and nb = 2:
! 125: *
! 126: * if UPLO = 'U': if UPLO = 'L':
! 127: *
! 128: * ( a a a v4 v5 ) ( d )
! 129: * ( a a v4 v5 ) ( 1 d )
! 130: * ( a 1 v5 ) ( v1 1 a )
! 131: * ( d 1 ) ( v1 v2 a a )
! 132: * ( d ) ( v1 v2 a a a )
! 133: *
! 134: * where d denotes a diagonal element of the reduced matrix, a denotes
! 135: * an element of the original matrix that is unchanged, and vi denotes
! 136: * an element of the vector defining H(i).
! 137: *
! 138: * =====================================================================
! 139: *
! 140: * .. Parameters ..
! 141: COMPLEX*16 ZERO, ONE, HALF
! 142: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
! 143: $ ONE = ( 1.0D+0, 0.0D+0 ),
! 144: $ HALF = ( 0.5D+0, 0.0D+0 ) )
! 145: * ..
! 146: * .. Local Scalars ..
! 147: INTEGER I, IW
! 148: COMPLEX*16 ALPHA
! 149: * ..
! 150: * .. External Subroutines ..
! 151: EXTERNAL ZAXPY, ZGEMV, ZHEMV, ZLACGV, ZLARFG, ZSCAL
! 152: * ..
! 153: * .. External Functions ..
! 154: LOGICAL LSAME
! 155: COMPLEX*16 ZDOTC
! 156: EXTERNAL LSAME, ZDOTC
! 157: * ..
! 158: * .. Intrinsic Functions ..
! 159: INTRINSIC DBLE, MIN
! 160: * ..
! 161: * .. Executable Statements ..
! 162: *
! 163: * Quick return if possible
! 164: *
! 165: IF( N.LE.0 )
! 166: $ RETURN
! 167: *
! 168: IF( LSAME( UPLO, 'U' ) ) THEN
! 169: *
! 170: * Reduce last NB columns of upper triangle
! 171: *
! 172: DO 10 I = N, N - NB + 1, -1
! 173: IW = I - N + NB
! 174: IF( I.LT.N ) THEN
! 175: *
! 176: * Update A(1:i,i)
! 177: *
! 178: A( I, I ) = DBLE( A( I, I ) )
! 179: CALL ZLACGV( N-I, W( I, IW+1 ), LDW )
! 180: CALL ZGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
! 181: $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
! 182: CALL ZLACGV( N-I, W( I, IW+1 ), LDW )
! 183: CALL ZLACGV( N-I, A( I, I+1 ), LDA )
! 184: CALL ZGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ),
! 185: $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
! 186: CALL ZLACGV( N-I, A( I, I+1 ), LDA )
! 187: A( I, I ) = DBLE( A( I, I ) )
! 188: END IF
! 189: IF( I.GT.1 ) THEN
! 190: *
! 191: * Generate elementary reflector H(i) to annihilate
! 192: * A(1:i-2,i)
! 193: *
! 194: ALPHA = A( I-1, I )
! 195: CALL ZLARFG( I-1, ALPHA, A( 1, I ), 1, TAU( I-1 ) )
! 196: E( I-1 ) = ALPHA
! 197: A( I-1, I ) = ONE
! 198: *
! 199: * Compute W(1:i-1,i)
! 200: *
! 201: CALL ZHEMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
! 202: $ ZERO, W( 1, IW ), 1 )
! 203: IF( I.LT.N ) THEN
! 204: CALL ZGEMV( 'Conjugate transpose', I-1, N-I, ONE,
! 205: $ W( 1, IW+1 ), LDW, A( 1, I ), 1, ZERO,
! 206: $ W( I+1, IW ), 1 )
! 207: CALL ZGEMV( 'No transpose', I-1, N-I, -ONE,
! 208: $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,
! 209: $ W( 1, IW ), 1 )
! 210: CALL ZGEMV( 'Conjugate transpose', I-1, N-I, ONE,
! 211: $ A( 1, I+1 ), LDA, A( 1, I ), 1, ZERO,
! 212: $ W( I+1, IW ), 1 )
! 213: CALL ZGEMV( 'No transpose', I-1, N-I, -ONE,
! 214: $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,
! 215: $ W( 1, IW ), 1 )
! 216: END IF
! 217: CALL ZSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
! 218: ALPHA = -HALF*TAU( I-1 )*ZDOTC( I-1, W( 1, IW ), 1,
! 219: $ A( 1, I ), 1 )
! 220: CALL ZAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )
! 221: END IF
! 222: *
! 223: 10 CONTINUE
! 224: ELSE
! 225: *
! 226: * Reduce first NB columns of lower triangle
! 227: *
! 228: DO 20 I = 1, NB
! 229: *
! 230: * Update A(i:n,i)
! 231: *
! 232: A( I, I ) = DBLE( A( I, I ) )
! 233: CALL ZLACGV( I-1, W( I, 1 ), LDW )
! 234: CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
! 235: $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
! 236: CALL ZLACGV( I-1, W( I, 1 ), LDW )
! 237: CALL ZLACGV( I-1, A( I, 1 ), LDA )
! 238: CALL ZGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
! 239: $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
! 240: CALL ZLACGV( I-1, A( I, 1 ), LDA )
! 241: A( I, I ) = DBLE( A( I, I ) )
! 242: IF( I.LT.N ) THEN
! 243: *
! 244: * Generate elementary reflector H(i) to annihilate
! 245: * A(i+2:n,i)
! 246: *
! 247: ALPHA = A( I+1, I )
! 248: CALL ZLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1,
! 249: $ TAU( I ) )
! 250: E( I ) = ALPHA
! 251: A( I+1, I ) = ONE
! 252: *
! 253: * Compute W(i+1:n,i)
! 254: *
! 255: CALL ZHEMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
! 256: $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
! 257: CALL ZGEMV( 'Conjugate transpose', N-I, I-1, ONE,
! 258: $ W( I+1, 1 ), LDW, A( I+1, I ), 1, ZERO,
! 259: $ W( 1, I ), 1 )
! 260: CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ),
! 261: $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
! 262: CALL ZGEMV( 'Conjugate transpose', N-I, I-1, ONE,
! 263: $ A( I+1, 1 ), LDA, A( I+1, I ), 1, ZERO,
! 264: $ W( 1, I ), 1 )
! 265: CALL ZGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ),
! 266: $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
! 267: CALL ZSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
! 268: ALPHA = -HALF*TAU( I )*ZDOTC( N-I, W( I+1, I ), 1,
! 269: $ A( I+1, I ), 1 )
! 270: CALL ZAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 )
! 271: END IF
! 272: *
! 273: 20 CONTINUE
! 274: END IF
! 275: *
! 276: RETURN
! 277: *
! 278: * End of ZLATRD
! 279: *
! 280: END
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