LAPACK 3.3.1
Linear Algebra PACKage

ssptrd.f

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00001       SUBROUTINE SSPTRD( UPLO, N, AP, D, E, TAU, INFO )
00002 *
00003 *  -- LAPACK routine (version 3.3.1) --
00004 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00005 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00006 *  -- April 2011                                                      --
00007 *
00008 *     .. Scalar Arguments ..
00009       CHARACTER          UPLO
00010       INTEGER            INFO, N
00011 *     ..
00012 *     .. Array Arguments ..
00013       REAL               AP( * ), D( * ), E( * ), TAU( * )
00014 *     ..
00015 *
00016 *  Purpose
00017 *  =======
00018 *
00019 *  SSPTRD reduces a real symmetric matrix A stored in packed form to
00020 *  symmetric tridiagonal form T by an orthogonal similarity
00021 *  transformation: Q**T * A * Q = T.
00022 *
00023 *  Arguments
00024 *  =========
00025 *
00026 *  UPLO    (input) CHARACTER*1
00027 *          = 'U':  Upper triangle of A is stored;
00028 *          = 'L':  Lower triangle of A is stored.
00029 *
00030 *  N       (input) INTEGER
00031 *          The order of the matrix A.  N >= 0.
00032 *
00033 *  AP      (input/output) REAL array, dimension (N*(N+1)/2)
00034 *          On entry, the upper or lower triangle of the symmetric matrix
00035 *          A, packed columnwise in a linear array.  The j-th column of A
00036 *          is stored in the array AP as follows:
00037 *          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00038 *          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
00039 *          On exit, if UPLO = 'U', the diagonal and first superdiagonal
00040 *          of A are overwritten by the corresponding elements of the
00041 *          tridiagonal matrix T, and the elements above the first
00042 *          superdiagonal, with the array TAU, represent the orthogonal
00043 *          matrix Q as a product of elementary reflectors; if UPLO
00044 *          = 'L', the diagonal and first subdiagonal of A are over-
00045 *          written by the corresponding elements of the tridiagonal
00046 *          matrix T, and the elements below the first subdiagonal, with
00047 *          the array TAU, represent the orthogonal matrix Q as a product
00048 *          of elementary reflectors. See Further Details.
00049 *
00050 *  D       (output) REAL array, dimension (N)
00051 *          The diagonal elements of the tridiagonal matrix T:
00052 *          D(i) = A(i,i).
00053 *
00054 *  E       (output) REAL array, dimension (N-1)
00055 *          The off-diagonal elements of the tridiagonal matrix T:
00056 *          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
00057 *
00058 *  TAU     (output) REAL array, dimension (N-1)
00059 *          The scalar factors of the elementary reflectors (see Further
00060 *          Details).
00061 *
00062 *  INFO    (output) INTEGER
00063 *          = 0:  successful exit
00064 *          < 0:  if INFO = -i, the i-th argument had an illegal value
00065 *
00066 *  Further Details
00067 *  ===============
00068 *
00069 *  If UPLO = 'U', the matrix Q is represented as a product of elementary
00070 *  reflectors
00071 *
00072 *     Q = H(n-1) . . . H(2) H(1).
00073 *
00074 *  Each H(i) has the form
00075 *
00076 *     H(i) = I - tau * v * v**T
00077 *
00078 *  where tau is a real scalar, and v is a real vector with
00079 *  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in AP,
00080 *  overwriting A(1:i-1,i+1), and tau is stored in TAU(i).
00081 *
00082 *  If UPLO = 'L', the matrix Q is represented as a product of elementary
00083 *  reflectors
00084 *
00085 *     Q = H(1) H(2) . . . H(n-1).
00086 *
00087 *  Each H(i) has the form
00088 *
00089 *     H(i) = I - tau * v * v**T
00090 *
00091 *  where tau is a real scalar, and v is a real vector with
00092 *  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in AP,
00093 *  overwriting A(i+2:n,i), and tau is stored in TAU(i).
00094 *
00095 *  =====================================================================
00096 *
00097 *     .. Parameters ..
00098       REAL               ONE, ZERO, HALF
00099       PARAMETER          ( ONE = 1.0, ZERO = 0.0, HALF = 1.0 / 2.0 )
00100 *     ..
00101 *     .. Local Scalars ..
00102       LOGICAL            UPPER
00103       INTEGER            I, I1, I1I1, II
00104       REAL               ALPHA, TAUI
00105 *     ..
00106 *     .. External Subroutines ..
00107       EXTERNAL           SAXPY, SLARFG, SSPMV, SSPR2, XERBLA
00108 *     ..
00109 *     .. External Functions ..
00110       LOGICAL            LSAME
00111       REAL               SDOT
00112       EXTERNAL           LSAME, SDOT
00113 *     ..
00114 *     .. Executable Statements ..
00115 *
00116 *     Test the input parameters
00117 *
00118       INFO = 0
00119       UPPER = LSAME( UPLO, 'U' )
00120       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00121          INFO = -1
00122       ELSE IF( N.LT.0 ) THEN
00123          INFO = -2
00124       END IF
00125       IF( INFO.NE.0 ) THEN
00126          CALL XERBLA( 'SSPTRD', -INFO )
00127          RETURN
00128       END IF
00129 *
00130 *     Quick return if possible
00131 *
00132       IF( N.LE.0 )
00133      $   RETURN
00134 *
00135       IF( UPPER ) THEN
00136 *
00137 *        Reduce the upper triangle of A.
00138 *        I1 is the index in AP of A(1,I+1).
00139 *
00140          I1 = N*( N-1 ) / 2 + 1
00141          DO 10 I = N - 1, 1, -1
00142 *
00143 *           Generate elementary reflector H(i) = I - tau * v * v**T
00144 *           to annihilate A(1:i-1,i+1)
00145 *
00146             CALL SLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
00147             E( I ) = AP( I1+I-1 )
00148 *
00149             IF( TAUI.NE.ZERO ) THEN
00150 *
00151 *              Apply H(i) from both sides to A(1:i,1:i)
00152 *
00153                AP( I1+I-1 ) = ONE
00154 *
00155 *              Compute  y := tau * A * v  storing y in TAU(1:i)
00156 *
00157                CALL SSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
00158      $                     1 )
00159 *
00160 *              Compute  w := y - 1/2 * tau * (y**T *v) * v
00161 *
00162                ALPHA = -HALF*TAUI*SDOT( I, TAU, 1, AP( I1 ), 1 )
00163                CALL SAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
00164 *
00165 *              Apply the transformation as a rank-2 update:
00166 *                 A := A - v * w**T - w * v**T
00167 *
00168                CALL SSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
00169 *
00170                AP( I1+I-1 ) = E( I )
00171             END IF
00172             D( I+1 ) = AP( I1+I )
00173             TAU( I ) = TAUI
00174             I1 = I1 - I
00175    10    CONTINUE
00176          D( 1 ) = AP( 1 )
00177       ELSE
00178 *
00179 *        Reduce the lower triangle of A. II is the index in AP of
00180 *        A(i,i) and I1I1 is the index of A(i+1,i+1).
00181 *
00182          II = 1
00183          DO 20 I = 1, N - 1
00184             I1I1 = II + N - I + 1
00185 *
00186 *           Generate elementary reflector H(i) = I - tau * v * v**T
00187 *           to annihilate A(i+2:n,i)
00188 *
00189             CALL SLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
00190             E( I ) = AP( II+1 )
00191 *
00192             IF( TAUI.NE.ZERO ) THEN
00193 *
00194 *              Apply H(i) from both sides to A(i+1:n,i+1:n)
00195 *
00196                AP( II+1 ) = ONE
00197 *
00198 *              Compute  y := tau * A * v  storing y in TAU(i:n-1)
00199 *
00200                CALL SSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
00201      $                     ZERO, TAU( I ), 1 )
00202 *
00203 *              Compute  w := y - 1/2 * tau * (y**T *v) * v
00204 *
00205                ALPHA = -HALF*TAUI*SDOT( N-I, TAU( I ), 1, AP( II+1 ),
00206      $                 1 )
00207                CALL SAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
00208 *
00209 *              Apply the transformation as a rank-2 update:
00210 *                 A := A - v * w**T - w * v**T
00211 *
00212                CALL SSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
00213      $                     AP( I1I1 ) )
00214 *
00215                AP( II+1 ) = E( I )
00216             END IF
00217             D( I ) = AP( II )
00218             TAU( I ) = TAUI
00219             II = I1I1
00220    20    CONTINUE
00221          D( N ) = AP( II )
00222       END IF
00223 *
00224       RETURN
00225 *
00226 *     End of SSPTRD
00227 *
00228       END
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