*> \brief \b DTBSV * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * * Definition: * =========== * * SUBROUTINE DTBSV(UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX) * * .. Scalar Arguments .. * INTEGER INCX,K,LDA,N * CHARACTER DIAG,TRANS,UPLO * .. * .. Array Arguments .. * DOUBLE PRECISION A(LDA,*),X(*) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DTBSV solves one of the systems of equations *> *> A*x = b, or A**T*x = b, *> *> where b and x are n element vectors and A is an n by n unit, or *> non-unit, upper or lower triangular band matrix, with ( k + 1 ) *> diagonals. *> *> No test for singularity or near-singularity is included in this *> routine. Such tests must be performed before calling this routine. *> \endverbatim * * Arguments: * ========== * *> \param[in] UPLO *> \verbatim *> UPLO is CHARACTER*1 *> On entry, UPLO specifies whether the matrix is an upper or *> lower triangular matrix as follows: *> *> UPLO = 'U' or 'u' A is an upper triangular matrix. *> *> UPLO = 'L' or 'l' A is a lower triangular matrix. *> \endverbatim *> *> \param[in] TRANS *> \verbatim *> TRANS is CHARACTER*1 *> On entry, TRANS specifies the equations to be solved as *> follows: *> *> TRANS = 'N' or 'n' A*x = b. *> *> TRANS = 'T' or 't' A**T*x = b. *> *> TRANS = 'C' or 'c' A**T*x = b. *> \endverbatim *> *> \param[in] DIAG *> \verbatim *> DIAG is CHARACTER*1 *> On entry, DIAG specifies whether or not A is unit *> triangular as follows: *> *> DIAG = 'U' or 'u' A is assumed to be unit triangular. *> *> DIAG = 'N' or 'n' A is not assumed to be unit *> triangular. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> On entry, N specifies the order of the matrix A. *> N must be at least zero. *> \endverbatim *> *> \param[in] K *> \verbatim *> K is INTEGER *> On entry with UPLO = 'U' or 'u', K specifies the number of *> super-diagonals of the matrix A. *> On entry with UPLO = 'L' or 'l', K specifies the number of *> sub-diagonals of the matrix A. *> K must satisfy 0 .le. K. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is DOUBLE PRECISION array, dimension ( LDA, N ) *> Before entry with UPLO = 'U' or 'u', the leading ( k + 1 ) *> by n part of the array A must contain the upper triangular *> band part of the matrix of coefficients, supplied column by *> column, with the leading diagonal of the matrix in row *> ( k + 1 ) of the array, the first super-diagonal starting at *> position 2 in row k, and so on. The top left k by k triangle *> of the array A is not referenced. *> The following program segment will transfer an upper *> triangular band matrix from conventional full matrix storage *> to band storage: *> *> DO 20, J = 1, N *> M = K + 1 - J *> DO 10, I = MAX( 1, J - K ), J *> A( M + I, J ) = matrix( I, J ) *> 10 CONTINUE *> 20 CONTINUE *> *> Before entry with UPLO = 'L' or 'l', the leading ( k + 1 ) *> by n part of the array A must contain the lower triangular *> band part of the matrix of coefficients, supplied column by *> column, with the leading diagonal of the matrix in row 1 of *> the array, the first sub-diagonal starting at position 1 in *> row 2, and so on. The bottom right k by k triangle of the *> array A is not referenced. *> The following program segment will transfer a lower *> triangular band matrix from conventional full matrix storage *> to band storage: *> *> DO 20, J = 1, N *> M = 1 - J *> DO 10, I = J, MIN( N, J + K ) *> A( M + I, J ) = matrix( I, J ) *> 10 CONTINUE *> 20 CONTINUE *> *> Note that when DIAG = 'U' or 'u' the elements of the array A *> corresponding to the diagonal elements of the matrix are not *> referenced, but are assumed to be unity. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> On entry, LDA specifies the first dimension of A as declared *> in the calling (sub) program. LDA must be at least *> ( k + 1 ). *> \endverbatim *> *> \param[in,out] X *> \verbatim *> X is DOUBLE PRECISION array, dimension at least *> ( 1 + ( n - 1 )*abs( INCX ) ). *> Before entry, the incremented array X must contain the n *> element right-hand side vector b. On exit, X is overwritten *> with the solution vector x. *> \endverbatim *> *> \param[in] INCX *> \verbatim *> INCX is INTEGER *> On entry, INCX specifies the increment for the elements of *> X. INCX must not be zero. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup double_blas_level2 * *> \par Further Details: * ===================== *> *> \verbatim *> *> Level 2 Blas routine. *> *> -- Written on 22-October-1986. *> Jack Dongarra, Argonne National Lab. *> Jeremy Du Croz, Nag Central Office. *> Sven Hammarling, Nag Central Office. *> Richard Hanson, Sandia National Labs. *> \endverbatim *> * ===================================================================== SUBROUTINE DTBSV(UPLO,TRANS,DIAG,N,K,A,LDA,X,INCX) * * -- Reference BLAS level2 routine -- * -- Reference BLAS is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. INTEGER INCX,K,LDA,N CHARACTER DIAG,TRANS,UPLO * .. * .. Array Arguments .. DOUBLE PRECISION A(LDA,*),X(*) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO PARAMETER (ZERO=0.0D+0) * .. * .. Local Scalars .. DOUBLE PRECISION TEMP INTEGER I,INFO,IX,J,JX,KPLUS1,KX,L LOGICAL NOUNIT * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. External Subroutines .. EXTERNAL XERBLA * .. * .. Intrinsic Functions .. INTRINSIC MAX,MIN * .. * * Test the input parameters. * INFO = 0 IF (.NOT.LSAME(UPLO,'U') .AND. .NOT.LSAME(UPLO,'L')) THEN INFO = 1 ELSE IF (.NOT.LSAME(TRANS,'N') .AND. .NOT.LSAME(TRANS,'T') .AND. + .NOT.LSAME(TRANS,'C')) THEN INFO = 2 ELSE IF (.NOT.LSAME(DIAG,'U') .AND. .NOT.LSAME(DIAG,'N')) THEN INFO = 3 ELSE IF (N.LT.0) THEN INFO = 4 ELSE IF (K.LT.0) THEN INFO = 5 ELSE IF (LDA.LT. (K+1)) THEN INFO = 7 ELSE IF (INCX.EQ.0) THEN INFO = 9 END IF IF (INFO.NE.0) THEN CALL XERBLA('DTBSV ',INFO) RETURN END IF * * Quick return if possible. * IF (N.EQ.0) RETURN * NOUNIT = LSAME(DIAG,'N') * * Set up the start point in X if the increment is not unity. This * will be ( N - 1 )*INCX too small for descending loops. * IF (INCX.LE.0) THEN KX = 1 - (N-1)*INCX ELSE IF (INCX.NE.1) THEN KX = 1 END IF * * Start the operations. In this version the elements of A are * accessed by sequentially with one pass through A. * IF (LSAME(TRANS,'N')) THEN * * Form x := inv( A )*x. * IF (LSAME(UPLO,'U')) THEN KPLUS1 = K + 1 IF (INCX.EQ.1) THEN DO 20 J = N,1,-1 IF (X(J).NE.ZERO) THEN L = KPLUS1 - J IF (NOUNIT) X(J) = X(J)/A(KPLUS1,J) TEMP = X(J) DO 10 I = J - 1,MAX(1,J-K),-1 X(I) = X(I) - TEMP*A(L+I,J) 10 CONTINUE END IF 20 CONTINUE ELSE KX = KX + (N-1)*INCX JX = KX DO 40 J = N,1,-1 KX = KX - INCX IF (X(JX).NE.ZERO) THEN IX = KX L = KPLUS1 - J IF (NOUNIT) X(JX) = X(JX)/A(KPLUS1,J) TEMP = X(JX) DO 30 I = J - 1,MAX(1,J-K),-1 X(IX) = X(IX) - TEMP*A(L+I,J) IX = IX - INCX 30 CONTINUE END IF JX = JX - INCX 40 CONTINUE END IF ELSE IF (INCX.EQ.1) THEN DO 60 J = 1,N IF (X(J).NE.ZERO) THEN L = 1 - J IF (NOUNIT) X(J) = X(J)/A(1,J) TEMP = X(J) DO 50 I = J + 1,MIN(N,J+K) X(I) = X(I) - TEMP*A(L+I,J) 50 CONTINUE END IF 60 CONTINUE ELSE JX = KX DO 80 J = 1,N KX = KX + INCX IF (X(JX).NE.ZERO) THEN IX = KX L = 1 - J IF (NOUNIT) X(JX) = X(JX)/A(1,J) TEMP = X(JX) DO 70 I = J + 1,MIN(N,J+K) X(IX) = X(IX) - TEMP*A(L+I,J) IX = IX + INCX 70 CONTINUE END IF JX = JX + INCX 80 CONTINUE END IF END IF ELSE * * Form x := inv( A**T)*x. * IF (LSAME(UPLO,'U')) THEN KPLUS1 = K + 1 IF (INCX.EQ.1) THEN DO 100 J = 1,N TEMP = X(J) L = KPLUS1 - J DO 90 I = MAX(1,J-K),J - 1 TEMP = TEMP - A(L+I,J)*X(I) 90 CONTINUE IF (NOUNIT) TEMP = TEMP/A(KPLUS1,J) X(J) = TEMP 100 CONTINUE ELSE JX = KX DO 120 J = 1,N TEMP = X(JX) IX = KX L = KPLUS1 - J DO 110 I = MAX(1,J-K),J - 1 TEMP = TEMP - A(L+I,J)*X(IX) IX = IX + INCX 110 CONTINUE IF (NOUNIT) TEMP = TEMP/A(KPLUS1,J) X(JX) = TEMP JX = JX + INCX IF (J.GT.K) KX = KX + INCX 120 CONTINUE END IF ELSE IF (INCX.EQ.1) THEN DO 140 J = N,1,-1 TEMP = X(J) L = 1 - J DO 130 I = MIN(N,J+K),J + 1,-1 TEMP = TEMP - A(L+I,J)*X(I) 130 CONTINUE IF (NOUNIT) TEMP = TEMP/A(1,J) X(J) = TEMP 140 CONTINUE ELSE KX = KX + (N-1)*INCX JX = KX DO 160 J = N,1,-1 TEMP = X(JX) IX = KX L = 1 - J DO 150 I = MIN(N,J+K),J + 1,-1 TEMP = TEMP - A(L+I,J)*X(IX) IX = IX - INCX 150 CONTINUE IF (NOUNIT) TEMP = TEMP/A(1,J) X(JX) = TEMP JX = JX - INCX IF ((N-J).GE.K) KX = KX - INCX 160 CONTINUE END IF END IF END IF * RETURN * * End of DTBSV * END