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SUBROUTINE DTBSV ( UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX )
* .. Scalar Arguments ..
INTEGER INCX, K, LDA, N
CHARACTER*1 DIAG, TRANS, UPLO
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), X( * )
* ..
*
* Purpose
* =======
*
* DTBSV solves one of the systems of equations
*
* A*x = b, or A'*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.
*
* Parameters
* ==========
*
* UPLO - 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.
*
* Unchanged on exit.
*
* TRANS - 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'*x = b.
*
* TRANS = 'C' or 'c' A'*x = b.
*
* Unchanged on exit.
*
* DIAG - 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.
*
* Unchanged on exit.
*
* N - INTEGER.
* On entry, N specifies the order of the matrix A.
* N must be at least zero.
* Unchanged on exit.
*
* K - 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.
* Unchanged on exit.
*
* A - DOUBLE PRECISION array of 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.
* Unchanged on exit.
*
* LDA - INTEGER.
* On entry, LDA specifies the first dimension of A as declared
* in the calling (sub) program. LDA must be at least
* ( k + 1 ).
* Unchanged on exit.
*
* X - DOUBLE PRECISION array of 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.
*
* INCX - INTEGER.
* On entry, INCX specifies the increment for the elements of
* X. INCX must not be zero.
* Unchanged on exit.
*
*
* 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.
*
*
* .. 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
* ..
* .. Executable Statements ..
*
* 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')*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
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