Annotation of rpl/lapack/lapack/dsptrd.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DSPTRD( 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 AP( * ), D( * ), E( * ), TAU( * )
! 14: * ..
! 15: *
! 16: * Purpose
! 17: * =======
! 18: *
! 19: * DSPTRD reduces a real symmetric matrix A stored in packed form to
! 20: * symmetric tridiagonal form T by an orthogonal similarity
! 21: * transformation: Q**T * A * Q = T.
! 22: *
! 23: * Arguments
! 24: * =========
! 25: *
! 26: * UPLO (input) CHARACTER*1
! 27: * = 'U': Upper triangle of A is stored;
! 28: * = 'L': Lower triangle of A is stored.
! 29: *
! 30: * N (input) INTEGER
! 31: * The order of the matrix A. N >= 0.
! 32: *
! 33: * AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2)
! 34: * On entry, the upper or lower triangle of the symmetric matrix
! 35: * A, packed columnwise in a linear array. The j-th column of A
! 36: * is stored in the array AP as follows:
! 37: * if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
! 38: * if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
! 39: * On exit, if UPLO = 'U', the diagonal and first superdiagonal
! 40: * of A are overwritten by the corresponding elements of the
! 41: * tridiagonal matrix T, and the elements above the first
! 42: * superdiagonal, with the array TAU, represent the orthogonal
! 43: * matrix Q as a product of elementary reflectors; if UPLO
! 44: * = 'L', the diagonal and first subdiagonal of A are over-
! 45: * written by the corresponding elements of the tridiagonal
! 46: * matrix T, and the elements below the first subdiagonal, with
! 47: * the array TAU, represent the orthogonal matrix Q as a product
! 48: * of elementary reflectors. See Further Details.
! 49: *
! 50: * D (output) DOUBLE PRECISION array, dimension (N)
! 51: * The diagonal elements of the tridiagonal matrix T:
! 52: * D(i) = A(i,i).
! 53: *
! 54: * E (output) DOUBLE PRECISION array, dimension (N-1)
! 55: * The off-diagonal elements of the tridiagonal matrix T:
! 56: * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
! 57: *
! 58: * TAU (output) DOUBLE PRECISION array, dimension (N-1)
! 59: * The scalar factors of the elementary reflectors (see Further
! 60: * Details).
! 61: *
! 62: * INFO (output) INTEGER
! 63: * = 0: successful exit
! 64: * < 0: if INFO = -i, the i-th argument had an illegal value
! 65: *
! 66: * Further Details
! 67: * ===============
! 68: *
! 69: * If UPLO = 'U', the matrix Q is represented as a product of elementary
! 70: * reflectors
! 71: *
! 72: * Q = H(n-1) . . . H(2) H(1).
! 73: *
! 74: * Each H(i) has the form
! 75: *
! 76: * H(i) = I - tau * v * v'
! 77: *
! 78: * where tau is a real scalar, and v is a real vector with
! 79: * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
! 80: * overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
! 81: *
! 82: * If UPLO = 'L', the matrix Q is represented as a product of elementary
! 83: * reflectors
! 84: *
! 85: * Q = H(1) H(2) . . . H(n-1).
! 86: *
! 87: * Each H(i) has the form
! 88: *
! 89: * H(i) = I - tau * v * v'
! 90: *
! 91: * where tau is a real scalar, and v is a real vector with
! 92: * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
! 93: * overwriting A(i+2:n,i), and tau is stored in TAU(i).
! 94: *
! 95: * =====================================================================
! 96: *
! 97: * .. Parameters ..
! 98: DOUBLE PRECISION ONE, ZERO, HALF
! 99: PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
! 100: $ HALF = 1.0D0 / 2.0D0 )
! 101: * ..
! 102: * .. Local Scalars ..
! 103: LOGICAL UPPER
! 104: INTEGER I, I1, I1I1, II
! 105: DOUBLE PRECISION ALPHA, TAUI
! 106: * ..
! 107: * .. External Subroutines ..
! 108: EXTERNAL DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
! 109: * ..
! 110: * .. External Functions ..
! 111: LOGICAL LSAME
! 112: DOUBLE PRECISION DDOT
! 113: EXTERNAL LSAME, DDOT
! 114: * ..
! 115: * .. Executable Statements ..
! 116: *
! 117: * Test the input parameters
! 118: *
! 119: INFO = 0
! 120: UPPER = LSAME( UPLO, 'U' )
! 121: IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
! 122: INFO = -1
! 123: ELSE IF( N.LT.0 ) THEN
! 124: INFO = -2
! 125: END IF
! 126: IF( INFO.NE.0 ) THEN
! 127: CALL XERBLA( 'DSPTRD', -INFO )
! 128: RETURN
! 129: END IF
! 130: *
! 131: * Quick return if possible
! 132: *
! 133: IF( N.LE.0 )
! 134: $ RETURN
! 135: *
! 136: IF( UPPER ) THEN
! 137: *
! 138: * Reduce the upper triangle of A.
! 139: * I1 is the index in AP of A(1,I+1).
! 140: *
! 141: I1 = N*( N-1 ) / 2 + 1
! 142: DO 10 I = N - 1, 1, -1
! 143: *
! 144: * Generate elementary reflector H(i) = I - tau * v * v'
! 145: * to annihilate A(1:i-1,i+1)
! 146: *
! 147: CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
! 148: E( I ) = AP( I1+I-1 )
! 149: *
! 150: IF( TAUI.NE.ZERO ) THEN
! 151: *
! 152: * Apply H(i) from both sides to A(1:i,1:i)
! 153: *
! 154: AP( I1+I-1 ) = ONE
! 155: *
! 156: * Compute y := tau * A * v storing y in TAU(1:i)
! 157: *
! 158: CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
! 159: $ 1 )
! 160: *
! 161: * Compute w := y - 1/2 * tau * (y'*v) * v
! 162: *
! 163: ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
! 164: CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
! 165: *
! 166: * Apply the transformation as a rank-2 update:
! 167: * A := A - v * w' - w * v'
! 168: *
! 169: CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
! 170: *
! 171: AP( I1+I-1 ) = E( I )
! 172: END IF
! 173: D( I+1 ) = AP( I1+I )
! 174: TAU( I ) = TAUI
! 175: I1 = I1 - I
! 176: 10 CONTINUE
! 177: D( 1 ) = AP( 1 )
! 178: ELSE
! 179: *
! 180: * Reduce the lower triangle of A. II is the index in AP of
! 181: * A(i,i) and I1I1 is the index of A(i+1,i+1).
! 182: *
! 183: II = 1
! 184: DO 20 I = 1, N - 1
! 185: I1I1 = II + N - I + 1
! 186: *
! 187: * Generate elementary reflector H(i) = I - tau * v * v'
! 188: * to annihilate A(i+2:n,i)
! 189: *
! 190: CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
! 191: E( I ) = AP( II+1 )
! 192: *
! 193: IF( TAUI.NE.ZERO ) THEN
! 194: *
! 195: * Apply H(i) from both sides to A(i+1:n,i+1:n)
! 196: *
! 197: AP( II+1 ) = ONE
! 198: *
! 199: * Compute y := tau * A * v storing y in TAU(i:n-1)
! 200: *
! 201: CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
! 202: $ ZERO, TAU( I ), 1 )
! 203: *
! 204: * Compute w := y - 1/2 * tau * (y'*v) * v
! 205: *
! 206: ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
! 207: $ 1 )
! 208: CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
! 209: *
! 210: * Apply the transformation as a rank-2 update:
! 211: * A := A - v * w' - w * v'
! 212: *
! 213: CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
! 214: $ AP( I1I1 ) )
! 215: *
! 216: AP( II+1 ) = E( I )
! 217: END IF
! 218: D( I ) = AP( II )
! 219: TAU( I ) = TAUI
! 220: II = I1I1
! 221: 20 CONTINUE
! 222: D( N ) = AP( II )
! 223: END IF
! 224: *
! 225: RETURN
! 226: *
! 227: * End of DSPTRD
! 228: *
! 229: END
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