LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
spteqr.f
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00001 *> \brief \b SPTEQR
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00009 *> Download SPTEQR + dependencies 
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00011 *> [TGZ]</a> 
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00013 *> [ZIP]</a> 
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       CHARACTER          COMPZ
00025 *       INTEGER            INFO, LDZ, N
00026 *       ..
00027 *       .. Array Arguments ..
00028 *       REAL               D( * ), E( * ), WORK( * ), Z( LDZ, * )
00029 *       ..
00030 *  
00031 *
00032 *> \par Purpose:
00033 *  =============
00034 *>
00035 *> \verbatim
00036 *>
00037 *> SPTEQR computes all eigenvalues and, optionally, eigenvectors of a
00038 *> symmetric positive definite tridiagonal matrix by first factoring the
00039 *> matrix using SPTTRF, and then calling SBDSQR to compute the singular
00040 *> values of the bidiagonal factor.
00041 *>
00042 *> This routine computes the eigenvalues of the positive definite
00043 *> tridiagonal matrix to high relative accuracy.  This means that if the
00044 *> eigenvalues range over many orders of magnitude in size, then the
00045 *> small eigenvalues and corresponding eigenvectors will be computed
00046 *> more accurately than, for example, with the standard QR method.
00047 *>
00048 *> The eigenvectors of a full or band symmetric positive definite matrix
00049 *> can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to
00050 *> reduce this matrix to tridiagonal form. (The reduction to tridiagonal
00051 *> form, however, may preclude the possibility of obtaining high
00052 *> relative accuracy in the small eigenvalues of the original matrix, if
00053 *> these eigenvalues range over many orders of magnitude.)
00054 *> \endverbatim
00055 *
00056 *  Arguments:
00057 *  ==========
00058 *
00059 *> \param[in] COMPZ
00060 *> \verbatim
00061 *>          COMPZ is CHARACTER*1
00062 *>          = 'N':  Compute eigenvalues only.
00063 *>          = 'V':  Compute eigenvectors of original symmetric
00064 *>                  matrix also.  Array Z contains the orthogonal
00065 *>                  matrix used to reduce the original matrix to
00066 *>                  tridiagonal form.
00067 *>          = 'I':  Compute eigenvectors of tridiagonal matrix also.
00068 *> \endverbatim
00069 *>
00070 *> \param[in] N
00071 *> \verbatim
00072 *>          N is INTEGER
00073 *>          The order of the matrix.  N >= 0.
00074 *> \endverbatim
00075 *>
00076 *> \param[in,out] D
00077 *> \verbatim
00078 *>          D is REAL array, dimension (N)
00079 *>          On entry, the n diagonal elements of the tridiagonal
00080 *>          matrix.
00081 *>          On normal exit, D contains the eigenvalues, in descending
00082 *>          order.
00083 *> \endverbatim
00084 *>
00085 *> \param[in,out] E
00086 *> \verbatim
00087 *>          E is REAL array, dimension (N-1)
00088 *>          On entry, the (n-1) subdiagonal elements of the tridiagonal
00089 *>          matrix.
00090 *>          On exit, E has been destroyed.
00091 *> \endverbatim
00092 *>
00093 *> \param[in,out] Z
00094 *> \verbatim
00095 *>          Z is REAL array, dimension (LDZ, N)
00096 *>          On entry, if COMPZ = 'V', the orthogonal matrix used in the
00097 *>          reduction to tridiagonal form.
00098 *>          On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
00099 *>          original symmetric matrix;
00100 *>          if COMPZ = 'I', the orthonormal eigenvectors of the
00101 *>          tridiagonal matrix.
00102 *>          If INFO > 0 on exit, Z contains the eigenvectors associated
00103 *>          with only the stored eigenvalues.
00104 *>          If  COMPZ = 'N', then Z is not referenced.
00105 *> \endverbatim
00106 *>
00107 *> \param[in] LDZ
00108 *> \verbatim
00109 *>          LDZ is INTEGER
00110 *>          The leading dimension of the array Z.  LDZ >= 1, and if
00111 *>          COMPZ = 'V' or 'I', LDZ >= max(1,N).
00112 *> \endverbatim
00113 *>
00114 *> \param[out] WORK
00115 *> \verbatim
00116 *>          WORK is REAL array, dimension (4*N)
00117 *> \endverbatim
00118 *>
00119 *> \param[out] INFO
00120 *> \verbatim
00121 *>          INFO is INTEGER
00122 *>          = 0:  successful exit.
00123 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
00124 *>          > 0:  if INFO = i, and i is:
00125 *>                <= N  the Cholesky factorization of the matrix could
00126 *>                      not be performed because the i-th principal minor
00127 *>                      was not positive definite.
00128 *>                > N   the SVD algorithm failed to converge;
00129 *>                      if INFO = N+i, i off-diagonal elements of the
00130 *>                      bidiagonal factor did not converge to zero.
00131 *> \endverbatim
00132 *
00133 *  Authors:
00134 *  ========
00135 *
00136 *> \author Univ. of Tennessee 
00137 *> \author Univ. of California Berkeley 
00138 *> \author Univ. of Colorado Denver 
00139 *> \author NAG Ltd. 
00140 *
00141 *> \date November 2011
00142 *
00143 *> \ingroup realOTHERcomputational
00144 *
00145 *  =====================================================================
00146       SUBROUTINE SPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
00147 *
00148 *  -- LAPACK computational routine (version 3.4.0) --
00149 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00150 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00151 *     November 2011
00152 *
00153 *     .. Scalar Arguments ..
00154       CHARACTER          COMPZ
00155       INTEGER            INFO, LDZ, N
00156 *     ..
00157 *     .. Array Arguments ..
00158       REAL               D( * ), E( * ), WORK( * ), Z( LDZ, * )
00159 *     ..
00160 *
00161 *  =====================================================================
00162 *
00163 *     .. Parameters ..
00164       REAL               ZERO, ONE
00165       PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
00166 *     ..
00167 *     .. External Functions ..
00168       LOGICAL            LSAME
00169       EXTERNAL           LSAME
00170 *     ..
00171 *     .. External Subroutines ..
00172       EXTERNAL           SBDSQR, SLASET, SPTTRF, XERBLA
00173 *     ..
00174 *     .. Local Arrays ..
00175       REAL               C( 1, 1 ), VT( 1, 1 )
00176 *     ..
00177 *     .. Local Scalars ..
00178       INTEGER            I, ICOMPZ, NRU
00179 *     ..
00180 *     .. Intrinsic Functions ..
00181       INTRINSIC          MAX, SQRT
00182 *     ..
00183 *     .. Executable Statements ..
00184 *
00185 *     Test the input parameters.
00186 *
00187       INFO = 0
00188 *
00189       IF( LSAME( COMPZ, 'N' ) ) THEN
00190          ICOMPZ = 0
00191       ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
00192          ICOMPZ = 1
00193       ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
00194          ICOMPZ = 2
00195       ELSE
00196          ICOMPZ = -1
00197       END IF
00198       IF( ICOMPZ.LT.0 ) THEN
00199          INFO = -1
00200       ELSE IF( N.LT.0 ) THEN
00201          INFO = -2
00202       ELSE IF( ( LDZ.LT.1 ) .OR. ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1,
00203      $         N ) ) ) THEN
00204          INFO = -6
00205       END IF
00206       IF( INFO.NE.0 ) THEN
00207          CALL XERBLA( 'SPTEQR', -INFO )
00208          RETURN
00209       END IF
00210 *
00211 *     Quick return if possible
00212 *
00213       IF( N.EQ.0 )
00214      $   RETURN
00215 *
00216       IF( N.EQ.1 ) THEN
00217          IF( ICOMPZ.GT.0 )
00218      $      Z( 1, 1 ) = ONE
00219          RETURN
00220       END IF
00221       IF( ICOMPZ.EQ.2 )
00222      $   CALL SLASET( 'Full', N, N, ZERO, ONE, Z, LDZ )
00223 *
00224 *     Call SPTTRF to factor the matrix.
00225 *
00226       CALL SPTTRF( N, D, E, INFO )
00227       IF( INFO.NE.0 )
00228      $   RETURN
00229       DO 10 I = 1, N
00230          D( I ) = SQRT( D( I ) )
00231    10 CONTINUE
00232       DO 20 I = 1, N - 1
00233          E( I ) = E( I )*D( I )
00234    20 CONTINUE
00235 *
00236 *     Call SBDSQR to compute the singular values/vectors of the
00237 *     bidiagonal factor.
00238 *
00239       IF( ICOMPZ.GT.0 ) THEN
00240          NRU = N
00241       ELSE
00242          NRU = 0
00243       END IF
00244       CALL SBDSQR( 'Lower', N, 0, NRU, 0, D, E, VT, 1, Z, LDZ, C, 1,
00245      $             WORK, INFO )
00246 *
00247 *     Square the singular values.
00248 *
00249       IF( INFO.EQ.0 ) THEN
00250          DO 30 I = 1, N
00251             D( I ) = D( I )*D( I )
00252    30    CONTINUE
00253       ELSE
00254          INFO = N + INFO
00255       END IF
00256 *
00257       RETURN
00258 *
00259 *     End of SPTEQR
00260 *
00261       END
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