Annotation of rpl/lapack/lapack/zhptrd.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE ZHPTRD( UPLO, N, AP, 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, N
! 11: * ..
! 12: * .. Array Arguments ..
! 13: DOUBLE PRECISION D( * ), E( * )
! 14: COMPLEX*16 AP( * ), TAU( * )
! 15: * ..
! 16: *
! 17: * Purpose
! 18: * =======
! 19: *
! 20: * ZHPTRD reduces a complex Hermitian matrix A stored in packed form to
! 21: * real symmetric tridiagonal form T by a unitary similarity
! 22: * transformation: Q**H * A * Q = T.
! 23: *
! 24: * Arguments
! 25: * =========
! 26: *
! 27: * UPLO (input) CHARACTER*1
! 28: * = 'U': Upper triangle of A is stored;
! 29: * = 'L': Lower triangle of A is stored.
! 30: *
! 31: * N (input) INTEGER
! 32: * The order of the matrix A. N >= 0.
! 33: *
! 34: * AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
! 35: * On entry, the upper or lower triangle of the Hermitian matrix
! 36: * A, packed columnwise in a linear array. The j-th column of A
! 37: * is stored in the array AP as follows:
! 38: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 39: * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
! 40: * On exit, if UPLO = 'U', the diagonal and first superdiagonal
! 41: * of A are overwritten by the corresponding elements of the
! 42: * tridiagonal matrix T, and the elements above the first
! 43: * superdiagonal, with the array TAU, represent the unitary
! 44: * matrix Q as a product of elementary reflectors; if UPLO
! 45: * = 'L', the diagonal and first subdiagonal of A are over-
! 46: * written by the corresponding elements of the tridiagonal
! 47: * matrix T, and the elements below the first subdiagonal, with
! 48: * the array TAU, represent the unitary matrix Q as a product
! 49: * of elementary reflectors. See Further Details.
! 50: *
! 51: * D (output) DOUBLE PRECISION array, dimension (N)
! 52: * The diagonal elements of the tridiagonal matrix T:
! 53: * D(i) = A(i,i).
! 54: *
! 55: * E (output) DOUBLE PRECISION array, dimension (N-1)
! 56: * The off-diagonal elements of the tridiagonal matrix T:
! 57: * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
! 58: *
! 59: * TAU (output) COMPLEX*16 array, dimension (N-1)
! 60: * The scalar factors of the elementary reflectors (see Further
! 61: * Details).
! 62: *
! 63: * INFO (output) INTEGER
! 64: * = 0: successful exit
! 65: * < 0: if INFO = -i, the i-th argument had an illegal value
! 66: *
! 67: * Further Details
! 68: * ===============
! 69: *
! 70: * If UPLO = 'U', the matrix Q is represented as a product of elementary
! 71: * reflectors
! 72: *
! 73: * Q = H(n-1) . . . H(2) H(1).
! 74: *
! 75: * Each H(i) has the form
! 76: *
! 77: * H(i) = I - tau * v * v'
! 78: *
! 79: * where tau is a complex scalar, and v is a complex vector with
! 80: * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
! 81: * overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
! 82: *
! 83: * If UPLO = 'L', the matrix Q is represented as a product of elementary
! 84: * reflectors
! 85: *
! 86: * Q = H(1) H(2) . . . H(n-1).
! 87: *
! 88: * Each H(i) has the form
! 89: *
! 90: * H(i) = I - tau * v * v'
! 91: *
! 92: * where tau is a complex scalar, and v is a complex vector with
! 93: * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
! 94: * overwriting A(i+2:n,i), and tau is stored in TAU(i).
! 95: *
! 96: * =====================================================================
! 97: *
! 98: * .. Parameters ..
! 99: COMPLEX*16 ONE, ZERO, HALF
! 100: PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ),
! 101: $ ZERO = ( 0.0D+0, 0.0D+0 ),
! 102: $ HALF = ( 0.5D+0, 0.0D+0 ) )
! 103: * ..
! 104: * .. Local Scalars ..
! 105: LOGICAL UPPER
! 106: INTEGER I, I1, I1I1, II
! 107: COMPLEX*16 ALPHA, TAUI
! 108: * ..
! 109: * .. External Subroutines ..
! 110: EXTERNAL XERBLA, ZAXPY, ZHPMV, ZHPR2, ZLARFG
! 111: * ..
! 112: * .. External Functions ..
! 113: LOGICAL LSAME
! 114: COMPLEX*16 ZDOTC
! 115: EXTERNAL LSAME, ZDOTC
! 116: * ..
! 117: * .. Intrinsic Functions ..
! 118: INTRINSIC DBLE
! 119: * ..
! 120: * .. Executable Statements ..
! 121: *
! 122: * Test the input parameters
! 123: *
! 124: INFO = 0
! 125: UPPER = LSAME( UPLO, 'U' )
! 126: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 127: INFO = -1
! 128: ELSE IF( N.LT.0 ) THEN
! 129: INFO = -2
! 130: END IF
! 131: IF( INFO.NE.0 ) THEN
! 132: CALL XERBLA( 'ZHPTRD', -INFO )
! 133: RETURN
! 134: END IF
! 135: *
! 136: * Quick return if possible
! 137: *
! 138: IF( N.LE.0 )
! 139: $ RETURN
! 140: *
! 141: IF( UPPER ) THEN
! 142: *
! 143: * Reduce the upper triangle of A.
! 144: * I1 is the index in AP of A(1,I+1).
! 145: *
! 146: I1 = N*( N-1 ) / 2 + 1
! 147: AP( I1+N-1 ) = DBLE( AP( I1+N-1 ) )
! 148: DO 10 I = N - 1, 1, -1
! 149: *
! 150: * Generate elementary reflector H(i) = I - tau * v * v'
! 151: * to annihilate A(1:i-1,i+1)
! 152: *
! 153: ALPHA = AP( I1+I-1 )
! 154: CALL ZLARFG( I, ALPHA, AP( I1 ), 1, TAUI )
! 155: E( I ) = ALPHA
! 156: *
! 157: IF( TAUI.NE.ZERO ) THEN
! 158: *
! 159: * Apply H(i) from both sides to A(1:i,1:i)
! 160: *
! 161: AP( I1+I-1 ) = ONE
! 162: *
! 163: * Compute y := tau * A * v storing y in TAU(1:i)
! 164: *
! 165: CALL ZHPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
! 166: $ 1 )
! 167: *
! 168: * Compute w := y - 1/2 * tau * (y'*v) * v
! 169: *
! 170: ALPHA = -HALF*TAUI*ZDOTC( I, TAU, 1, AP( I1 ), 1 )
! 171: CALL ZAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
! 172: *
! 173: * Apply the transformation as a rank-2 update:
! 174: * A := A - v * w' - w * v'
! 175: *
! 176: CALL ZHPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
! 177: *
! 178: END IF
! 179: AP( I1+I-1 ) = E( I )
! 180: D( I+1 ) = AP( I1+I )
! 181: TAU( I ) = TAUI
! 182: I1 = I1 - I
! 183: 10 CONTINUE
! 184: D( 1 ) = AP( 1 )
! 185: ELSE
! 186: *
! 187: * Reduce the lower triangle of A. II is the index in AP of
! 188: * A(i,i) and I1I1 is the index of A(i+1,i+1).
! 189: *
! 190: II = 1
! 191: AP( 1 ) = DBLE( AP( 1 ) )
! 192: DO 20 I = 1, N - 1
! 193: I1I1 = II + N - I + 1
! 194: *
! 195: * Generate elementary reflector H(i) = I - tau * v * v'
! 196: * to annihilate A(i+2:n,i)
! 197: *
! 198: ALPHA = AP( II+1 )
! 199: CALL ZLARFG( N-I, ALPHA, AP( II+2 ), 1, TAUI )
! 200: E( I ) = ALPHA
! 201: *
! 202: IF( TAUI.NE.ZERO ) THEN
! 203: *
! 204: * Apply H(i) from both sides to A(i+1:n,i+1:n)
! 205: *
! 206: AP( II+1 ) = ONE
! 207: *
! 208: * Compute y := tau * A * v storing y in TAU(i:n-1)
! 209: *
! 210: CALL ZHPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
! 211: $ ZERO, TAU( I ), 1 )
! 212: *
! 213: * Compute w := y - 1/2 * tau * (y'*v) * v
! 214: *
! 215: ALPHA = -HALF*TAUI*ZDOTC( N-I, TAU( I ), 1, AP( II+1 ),
! 216: $ 1 )
! 217: CALL ZAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
! 218: *
! 219: * Apply the transformation as a rank-2 update:
! 220: * A := A - v * w' - w * v'
! 221: *
! 222: CALL ZHPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
! 223: $ AP( I1I1 ) )
! 224: *
! 225: END IF
! 226: AP( II+1 ) = E( I )
! 227: D( I ) = AP( II )
! 228: TAU( I ) = TAUI
! 229: II = I1I1
! 230: 20 CONTINUE
! 231: D( N ) = AP( II )
! 232: END IF
! 233: *
! 234: RETURN
! 235: *
! 236: * End of ZHPTRD
! 237: *
! 238: END
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