*> \brief \b DSBTRD * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DSBTRD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, * WORK, INFO ) * * .. Scalar Arguments .. * CHARACTER UPLO, VECT * INTEGER INFO, KD, LDAB, LDQ, N * .. * .. Array Arguments .. * DOUBLE PRECISION AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ), * $ WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DSBTRD reduces a real symmetric band matrix A to symmetric *> tridiagonal form T by an orthogonal similarity transformation: *> Q**T * A * Q = T. *> \endverbatim * * Arguments: * ========== * *> \param[in] VECT *> \verbatim *> VECT is CHARACTER*1 *> = 'N': do not form Q; *> = 'V': form Q; *> = 'U': update a matrix X, by forming X*Q. *> \endverbatim *> *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> = 'U': Upper triangle of A is stored; *> = 'L': Lower triangle of A is stored. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] KD *> \verbatim *> KD is INTEGER *> The number of superdiagonals of the matrix A if UPLO = 'U', *> or the number of subdiagonals if UPLO = 'L'. KD >= 0. *> \endverbatim *> *> \param[in,out] AB *> \verbatim *> AB is DOUBLE PRECISION array, dimension (LDAB,N) *> On entry, the upper or lower triangle of the symmetric band *> matrix A, stored in the first KD+1 rows of the array. The *> j-th column of A is stored in the j-th column of the array AB *> as follows: *> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; *> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). *> On exit, the diagonal elements of AB are overwritten by the *> diagonal elements of the tridiagonal matrix T; if KD > 0, the *> elements on the first superdiagonal (if UPLO = 'U') or the *> first subdiagonal (if UPLO = 'L') are overwritten by the *> off-diagonal elements of T; the rest of AB is overwritten by *> values generated during the reduction. *> \endverbatim *> *> \param[in] LDAB *> \verbatim *> LDAB is INTEGER *> The leading dimension of the array AB. LDAB >= KD+1. *> \endverbatim *> *> \param[out] D *> \verbatim *> D is DOUBLE PRECISION array, dimension (N) *> The diagonal elements of the tridiagonal matrix T. *> \endverbatim *> *> \param[out] E *> \verbatim *> E is DOUBLE PRECISION array, dimension (N-1) *> The off-diagonal elements of the tridiagonal matrix T: *> E(i) = T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'. *> \endverbatim *> *> \param[in,out] Q *> \verbatim *> Q is DOUBLE PRECISION array, dimension (LDQ,N) *> On entry, if VECT = 'U', then Q must contain an N-by-N *> matrix X; if VECT = 'N' or 'V', then Q need not be set. *> *> On exit: *> if VECT = 'V', Q contains the N-by-N orthogonal matrix Q; *> if VECT = 'U', Q contains the product X*Q; *> if VECT = 'N', the array Q is not referenced. *> \endverbatim *> *> \param[in] LDQ *> \verbatim *> LDQ is INTEGER *> The leading dimension of the array Q. *> LDQ >= 1, and LDQ >= N if VECT = 'V' or 'U'. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (N) *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit *> < 0: if INFO = -i, the i-th argument had an illegal value *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup doubleOTHERcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> Modified by Linda Kaufman, Bell Labs. *> \endverbatim *> * ===================================================================== SUBROUTINE DSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, $ WORK, INFO ) * * -- LAPACK computational routine (version 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. Scalar Arguments .. CHARACTER UPLO, VECT INTEGER INFO, KD, LDAB, LDQ, N * .. * .. Array Arguments .. DOUBLE PRECISION AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ), $ WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL INITQ, UPPER, WANTQ INTEGER I, I2, IBL, INCA, INCX, IQAEND, IQB, IQEND, J, $ J1, J1END, J1INC, J2, JEND, JIN, JINC, K, KD1, $ KDM1, KDN, L, LAST, LEND, NQ, NR, NRT DOUBLE PRECISION TEMP * .. * .. External Subroutines .. EXTERNAL DLAR2V, DLARGV, DLARTG, DLARTV, DLASET, DROT, $ XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX, MIN * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. Executable Statements .. * * Test the input parameters * INITQ = LSAME( VECT, 'V' ) WANTQ = INITQ .OR. LSAME( VECT, 'U' ) UPPER = LSAME( UPLO, 'U' ) KD1 = KD + 1 KDM1 = KD - 1 INCX = LDAB - 1 IQEND = 1 * INFO = 0 IF( .NOT.WANTQ .AND. .NOT.LSAME( VECT, 'N' ) ) THEN INFO = -1 ELSE IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KD.LT.0 ) THEN INFO = -4 ELSE IF( LDAB.LT.KD1 ) THEN INFO = -6 ELSE IF( LDQ.LT.MAX( 1, N ) .AND. WANTQ ) THEN INFO = -10 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSBTRD', -INFO ) RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * Initialize Q to the unit matrix, if needed * IF( INITQ ) $ CALL DLASET( 'Full', N, N, ZERO, ONE, Q, LDQ ) * * Wherever possible, plane rotations are generated and applied in * vector operations of length NR over the index set J1:J2:KD1. * * The cosines and sines of the plane rotations are stored in the * arrays D and WORK. * INCA = KD1*LDAB KDN = MIN( N-1, KD ) IF( UPPER ) THEN * IF( KD.GT.1 ) THEN * * Reduce to tridiagonal form, working with upper triangle * NR = 0 J1 = KDN + 2 J2 = 1 * DO 90 I = 1, N - 2 * * Reduce i-th row of matrix to tridiagonal form * DO 80 K = KDN + 1, 2, -1 J1 = J1 + KDN J2 = J2 + KDN * IF( NR.GT.0 ) THEN * * generate plane rotations to annihilate nonzero * elements which have been created outside the band * CALL DLARGV( NR, AB( 1, J1-1 ), INCA, WORK( J1 ), $ KD1, D( J1 ), KD1 ) * * apply rotations from the right * * * Dependent on the the number of diagonals either * DLARTV or DROT is used * IF( NR.GE.2*KD-1 ) THEN DO 10 L = 1, KD - 1 CALL DLARTV( NR, AB( L+1, J1-1 ), INCA, $ AB( L, J1 ), INCA, D( J1 ), $ WORK( J1 ), KD1 ) 10 CONTINUE * ELSE JEND = J1 + ( NR-1 )*KD1 DO 20 JINC = J1, JEND, KD1 CALL DROT( KDM1, AB( 2, JINC-1 ), 1, $ AB( 1, JINC ), 1, D( JINC ), $ WORK( JINC ) ) 20 CONTINUE END IF END IF * * IF( K.GT.2 ) THEN IF( K.LE.N-I+1 ) THEN * * generate plane rotation to annihilate a(i,i+k-1) * within the band * CALL DLARTG( AB( KD-K+3, I+K-2 ), $ AB( KD-K+2, I+K-1 ), D( I+K-1 ), $ WORK( I+K-1 ), TEMP ) AB( KD-K+3, I+K-2 ) = TEMP * * apply rotation from the right * CALL DROT( K-3, AB( KD-K+4, I+K-2 ), 1, $ AB( KD-K+3, I+K-1 ), 1, D( I+K-1 ), $ WORK( I+K-1 ) ) END IF NR = NR + 1 J1 = J1 - KDN - 1 END IF * * apply plane rotations from both sides to diagonal * blocks * IF( NR.GT.0 ) $ CALL DLAR2V( NR, AB( KD1, J1-1 ), AB( KD1, J1 ), $ AB( KD, J1 ), INCA, D( J1 ), $ WORK( J1 ), KD1 ) * * apply plane rotations from the left * IF( NR.GT.0 ) THEN IF( 2*KD-1.LT.NR ) THEN * * Dependent on the the number of diagonals either * DLARTV or DROT is used * DO 30 L = 1, KD - 1 IF( J2+L.GT.N ) THEN NRT = NR - 1 ELSE NRT = NR END IF IF( NRT.GT.0 ) $ CALL DLARTV( NRT, AB( KD-L, J1+L ), INCA, $ AB( KD-L+1, J1+L ), INCA, $ D( J1 ), WORK( J1 ), KD1 ) 30 CONTINUE ELSE J1END = J1 + KD1*( NR-2 ) IF( J1END.GE.J1 ) THEN DO 40 JIN = J1, J1END, KD1 CALL DROT( KD-1, AB( KD-1, JIN+1 ), INCX, $ AB( KD, JIN+1 ), INCX, $ D( JIN ), WORK( JIN ) ) 40 CONTINUE END IF LEND = MIN( KDM1, N-J2 ) LAST = J1END + KD1 IF( LEND.GT.0 ) $ CALL DROT( LEND, AB( KD-1, LAST+1 ), INCX, $ AB( KD, LAST+1 ), INCX, D( LAST ), $ WORK( LAST ) ) END IF END IF * IF( WANTQ ) THEN * * accumulate product of plane rotations in Q * IF( INITQ ) THEN * * take advantage of the fact that Q was * initially the Identity matrix * IQEND = MAX( IQEND, J2 ) I2 = MAX( 0, K-3 ) IQAEND = 1 + I*KD IF( K.EQ.2 ) $ IQAEND = IQAEND + KD IQAEND = MIN( IQAEND, IQEND ) DO 50 J = J1, J2, KD1 IBL = I - I2 / KDM1 I2 = I2 + 1 IQB = MAX( 1, J-IBL ) NQ = 1 + IQAEND - IQB IQAEND = MIN( IQAEND+KD, IQEND ) CALL DROT( NQ, Q( IQB, J-1 ), 1, Q( IQB, J ), $ 1, D( J ), WORK( J ) ) 50 CONTINUE ELSE * DO 60 J = J1, J2, KD1 CALL DROT( N, Q( 1, J-1 ), 1, Q( 1, J ), 1, $ D( J ), WORK( J ) ) 60 CONTINUE END IF * END IF * IF( J2+KDN.GT.N ) THEN * * adjust J2 to keep within the bounds of the matrix * NR = NR - 1 J2 = J2 - KDN - 1 END IF * DO 70 J = J1, J2, KD1 * * create nonzero element a(j-1,j+kd) outside the band * and store it in WORK * WORK( J+KD ) = WORK( J )*AB( 1, J+KD ) AB( 1, J+KD ) = D( J )*AB( 1, J+KD ) 70 CONTINUE 80 CONTINUE 90 CONTINUE END IF * IF( KD.GT.0 ) THEN * * copy off-diagonal elements to E * DO 100 I = 1, N - 1 E( I ) = AB( KD, I+1 ) 100 CONTINUE ELSE * * set E to zero if original matrix was diagonal * DO 110 I = 1, N - 1 E( I ) = ZERO 110 CONTINUE END IF * * copy diagonal elements to D * DO 120 I = 1, N D( I ) = AB( KD1, I ) 120 CONTINUE * ELSE * IF( KD.GT.1 ) THEN * * Reduce to tridiagonal form, working with lower triangle * NR = 0 J1 = KDN + 2 J2 = 1 * DO 210 I = 1, N - 2 * * Reduce i-th column of matrix to tridiagonal form * DO 200 K = KDN + 1, 2, -1 J1 = J1 + KDN J2 = J2 + KDN * IF( NR.GT.0 ) THEN * * generate plane rotations to annihilate nonzero * elements which have been created outside the band * CALL DLARGV( NR, AB( KD1, J1-KD1 ), INCA, $ WORK( J1 ), KD1, D( J1 ), KD1 ) * * apply plane rotations from one side * * * Dependent on the the number of diagonals either * DLARTV or DROT is used * IF( NR.GT.2*KD-1 ) THEN DO 130 L = 1, KD - 1 CALL DLARTV( NR, AB( KD1-L, J1-KD1+L ), INCA, $ AB( KD1-L+1, J1-KD1+L ), INCA, $ D( J1 ), WORK( J1 ), KD1 ) 130 CONTINUE ELSE JEND = J1 + KD1*( NR-1 ) DO 140 JINC = J1, JEND, KD1 CALL DROT( KDM1, AB( KD, JINC-KD ), INCX, $ AB( KD1, JINC-KD ), INCX, $ D( JINC ), WORK( JINC ) ) 140 CONTINUE END IF * END IF * IF( K.GT.2 ) THEN IF( K.LE.N-I+1 ) THEN * * generate plane rotation to annihilate a(i+k-1,i) * within the band * CALL DLARTG( AB( K-1, I ), AB( K, I ), $ D( I+K-1 ), WORK( I+K-1 ), TEMP ) AB( K-1, I ) = TEMP * * apply rotation from the left * CALL DROT( K-3, AB( K-2, I+1 ), LDAB-1, $ AB( K-1, I+1 ), LDAB-1, D( I+K-1 ), $ WORK( I+K-1 ) ) END IF NR = NR + 1 J1 = J1 - KDN - 1 END IF * * apply plane rotations from both sides to diagonal * blocks * IF( NR.GT.0 ) $ CALL DLAR2V( NR, AB( 1, J1-1 ), AB( 1, J1 ), $ AB( 2, J1-1 ), INCA, D( J1 ), $ WORK( J1 ), KD1 ) * * apply plane rotations from the right * * * Dependent on the the number of diagonals either * DLARTV or DROT is used * IF( NR.GT.0 ) THEN IF( NR.GT.2*KD-1 ) THEN DO 150 L = 1, KD - 1 IF( J2+L.GT.N ) THEN NRT = NR - 1 ELSE NRT = NR END IF IF( NRT.GT.0 ) $ CALL DLARTV( NRT, AB( L+2, J1-1 ), INCA, $ AB( L+1, J1 ), INCA, D( J1 ), $ WORK( J1 ), KD1 ) 150 CONTINUE ELSE J1END = J1 + KD1*( NR-2 ) IF( J1END.GE.J1 ) THEN DO 160 J1INC = J1, J1END, KD1 CALL DROT( KDM1, AB( 3, J1INC-1 ), 1, $ AB( 2, J1INC ), 1, D( J1INC ), $ WORK( J1INC ) ) 160 CONTINUE END IF LEND = MIN( KDM1, N-J2 ) LAST = J1END + KD1 IF( LEND.GT.0 ) $ CALL DROT( LEND, AB( 3, LAST-1 ), 1, $ AB( 2, LAST ), 1, D( LAST ), $ WORK( LAST ) ) END IF END IF * * * IF( WANTQ ) THEN * * accumulate product of plane rotations in Q * IF( INITQ ) THEN * * take advantage of the fact that Q was * initially the Identity matrix * IQEND = MAX( IQEND, J2 ) I2 = MAX( 0, K-3 ) IQAEND = 1 + I*KD IF( K.EQ.2 ) $ IQAEND = IQAEND + KD IQAEND = MIN( IQAEND, IQEND ) DO 170 J = J1, J2, KD1 IBL = I - I2 / KDM1 I2 = I2 + 1 IQB = MAX( 1, J-IBL ) NQ = 1 + IQAEND - IQB IQAEND = MIN( IQAEND+KD, IQEND ) CALL DROT( NQ, Q( IQB, J-1 ), 1, Q( IQB, J ), $ 1, D( J ), WORK( J ) ) 170 CONTINUE ELSE * DO 180 J = J1, J2, KD1 CALL DROT( N, Q( 1, J-1 ), 1, Q( 1, J ), 1, $ D( J ), WORK( J ) ) 180 CONTINUE END IF END IF * IF( J2+KDN.GT.N ) THEN * * adjust J2 to keep within the bounds of the matrix * NR = NR - 1 J2 = J2 - KDN - 1 END IF * DO 190 J = J1, J2, KD1 * * create nonzero element a(j+kd,j-1) outside the * band and store it in WORK * WORK( J+KD ) = WORK( J )*AB( KD1, J ) AB( KD1, J ) = D( J )*AB( KD1, J ) 190 CONTINUE 200 CONTINUE 210 CONTINUE END IF * IF( KD.GT.0 ) THEN * * copy off-diagonal elements to E * DO 220 I = 1, N - 1 E( I ) = AB( 2, I ) 220 CONTINUE ELSE * * set E to zero if original matrix was diagonal * DO 230 I = 1, N - 1 E( I ) = ZERO 230 CONTINUE END IF * * copy diagonal elements to D * DO 240 I = 1, N D( I ) = AB( 1, I ) 240 CONTINUE END IF * RETURN * * End of DSBTRD * END