*DECK CTBSV SUBROUTINE CTBSV (UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX) C***BEGIN PROLOGUE CTBSV C***PURPOSE Solve a complex triangular banded system of equations. C***LIBRARY SLATEC (BLAS) C***CATEGORY D1B4 C***TYPE COMPLEX (STBSV-S, DTBSV-D, CTBSV-C) C***KEYWORDS LEVEL 2 BLAS, LINEAR ALGEBRA C***AUTHOR Dongarra, J. J., (ANL) C Du Croz, J., (NAG) C Hammarling, S., (NAG) C Hanson, R. J., (SNLA) C***DESCRIPTION C C CTBSV solves one of the systems of equations C C A*x = b, or A'*x = b, or conjg( A')*x = b, C C where b and x are n element vectors and A is an n by n unit, or C non-unit, upper or lower triangular band matrix, with ( k + 1 ) C diagonals. C C No test for singularity or near-singularity is included in this C routine. Such tests must be performed before calling this routine. C C Parameters C ========== C C UPLO - CHARACTER*1. C On entry, UPLO specifies whether the matrix is an upper or C lower triangular matrix as follows: C C UPLO = 'U' or 'u' A is an upper triangular matrix. C C UPLO = 'L' or 'l' A is a lower triangular matrix. C C Unchanged on exit. C C TRANS - CHARACTER*1. C On entry, TRANS specifies the equations to be solved as C follows: C C TRANS = 'N' or 'n' A*x = b. C C TRANS = 'T' or 't' A'*x = b. C C TRANS = 'C' or 'c' conjg( A' )*x = b. C C Unchanged on exit. C C DIAG - CHARACTER*1. C On entry, DIAG specifies whether or not A is unit C triangular as follows: C C DIAG = 'U' or 'u' A is assumed to be unit triangular. C C DIAG = 'N' or 'n' A is not assumed to be unit C triangular. C C Unchanged on exit. C C N - INTEGER. C On entry, N specifies the order of the matrix A. C N must be at least zero. C Unchanged on exit. C C K - INTEGER. C On entry with UPLO = 'U' or 'u', K specifies the number of C super-diagonals of the matrix A. C On entry with UPLO = 'L' or 'l', K specifies the number of C sub-diagonals of the matrix A. C K must satisfy 0 .le. K. C Unchanged on exit. C C A - COMPLEX array of DIMENSION ( LDA, n ). C Before entry with UPLO = 'U' or 'u', the leading ( k + 1 ) C by n part of the array A must contain the upper triangular C band part of the matrix of coefficients, supplied column by C column, with the leading diagonal of the matrix in row C ( k + 1 ) of the array, the first super-diagonal starting at C position 2 in row k, and so on. The top left k by k triangle C of the array A is not referenced. C The following program segment will transfer an upper C triangular band matrix from conventional full matrix storage C to band storage: C C DO 20, J = 1, N C M = K + 1 - J C DO 10, I = MAX( 1, J - K ), J C A( M + I, J ) = matrix( I, J ) C 10 CONTINUE C 20 CONTINUE C C Before entry with UPLO = 'L' or 'l', the leading ( k + 1 ) C by n part of the array A must contain the lower triangular C band part of the matrix of coefficients, supplied column by C column, with the leading diagonal of the matrix in row 1 of C the array, the first sub-diagonal starting at position 1 in C row 2, and so on. The bottom right k by k triangle of the C array A is not referenced. C The following program segment will transfer a lower C triangular band matrix from conventional full matrix storage C to band storage: C C DO 20, J = 1, N C M = 1 - J C DO 10, I = J, MIN( N, J + K ) C A( M + I, J ) = matrix( I, J ) C 10 CONTINUE C 20 CONTINUE C C Note that when DIAG = 'U' or 'u' the elements of the array A C corresponding to the diagonal elements of the matrix are not C referenced, but are assumed to be unity. C Unchanged on exit. C C LDA - INTEGER. C On entry, LDA specifies the first dimension of A as declared C in the calling (sub) program. LDA must be at least C ( k + 1 ). C Unchanged on exit. C C X - COMPLEX array of dimension at least C ( 1 + ( n - 1 )*abs( INCX ) ). C Before entry, the incremented array X must contain the n C element right-hand side vector b. On exit, X is overwritten C with the solution vector x. C C INCX - INTEGER. C On entry, INCX specifies the increment for the elements of C X. INCX must not be zero. C Unchanged on exit. C C***REFERENCES Dongarra, J. J., Du Croz, J., Hammarling, S., and C Hanson, R. J. An extended set of Fortran basic linear C algebra subprograms. ACM TOMS, Vol. 14, No. 1, C pp. 1-17, March 1988. C***ROUTINES CALLED LSAME, XERBLA C***REVISION HISTORY (YYMMDD) C 861022 DATE WRITTEN C 910605 Modified to meet SLATEC prologue standards. Only comment C lines were modified. (BKS) C***END PROLOGUE CTBSV C .. Scalar Arguments .. INTEGER INCX, K, LDA, N CHARACTER*1 DIAG, TRANS, UPLO C .. Array Arguments .. COMPLEX A( LDA, * ), X( * ) C .. Parameters .. COMPLEX ZERO PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ) ) C .. Local Scalars .. COMPLEX TEMP INTEGER I, INFO, IX, J, JX, KPLUS1, KX, L LOGICAL NOCONJ, NOUNIT C .. External Functions .. LOGICAL LSAME EXTERNAL LSAME C .. External Subroutines .. EXTERNAL XERBLA C .. Intrinsic Functions .. INTRINSIC CONJG, MAX, MIN C***FIRST EXECUTABLE STATEMENT CTBSV C C Test the input parameters. C 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( 'CTBSV ', INFO ) RETURN END IF C C Quick return if possible. C IF( N.EQ.0 ) \$ RETURN C NOCONJ = LSAME( TRANS, 'T' ) NOUNIT = LSAME( DIAG , 'N' ) C C Set up the start point in X if the increment is not unity. This C will be ( N - 1 )*INCX too small for descending loops. C IF( INCX.LE.0 )THEN KX = 1 - ( N - 1 )*INCX ELSE IF( INCX.NE.1 )THEN KX = 1 END IF C C Start the operations. In this version the elements of A are C accessed by sequentially with one pass through A. C IF( LSAME( TRANS, 'N' ) )THEN C C Form x := inv( A )*x. C 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 C C Form x := inv( A' )*x or x := inv( conjg( A') )*x. C IF( LSAME( UPLO, 'U' ) )THEN KPLUS1 = K + 1 IF( INCX.EQ.1 )THEN DO 110, J = 1, N TEMP = X( J ) L = KPLUS1 - J IF( NOCONJ )THEN 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 ) ELSE DO 100, I = MAX( 1, J - K ), J - 1 TEMP = TEMP - CONJG( A( L + I, J ) )*X( I ) 100 CONTINUE IF( NOUNIT ) \$ TEMP = TEMP/CONJG( A( KPLUS1, J ) ) END IF X( J ) = TEMP 110 CONTINUE ELSE JX = KX DO 140, J = 1, N TEMP = X( JX ) IX = KX L = KPLUS1 - J IF( NOCONJ )THEN DO 120, I = MAX( 1, J - K ), J - 1 TEMP = TEMP - A( L + I, J )*X( IX ) IX = IX + INCX 120 CONTINUE IF( NOUNIT ) \$ TEMP = TEMP/A( KPLUS1, J ) ELSE DO 130, I = MAX( 1, J - K ), J - 1 TEMP = TEMP - CONJG( A( L + I, J ) )*X( IX ) IX = IX + INCX 130 CONTINUE IF( NOUNIT ) \$ TEMP = TEMP/CONJG( A( KPLUS1, J ) ) END IF X( JX ) = TEMP JX = JX + INCX IF( J.GT.K ) \$ KX = KX + INCX 140 CONTINUE END IF ELSE IF( INCX.EQ.1 )THEN DO 170, J = N, 1, -1 TEMP = X( J ) L = 1 - J IF( NOCONJ )THEN DO 150, I = MIN( N, J + K ), J + 1, -1 TEMP = TEMP - A( L + I, J )*X( I ) 150 CONTINUE IF( NOUNIT ) \$ TEMP = TEMP/A( 1, J ) ELSE DO 160, I = MIN( N, J + K ), J + 1, -1 TEMP = TEMP - CONJG( A( L + I, J ) )*X( I ) 160 CONTINUE IF( NOUNIT ) \$ TEMP = TEMP/CONJG( A( 1, J ) ) END IF X( J ) = TEMP 170 CONTINUE ELSE KX = KX + ( N - 1 )*INCX JX = KX DO 200, J = N, 1, -1 TEMP = X( JX ) IX = KX L = 1 - J IF( NOCONJ )THEN DO 180, I = MIN( N, J + K ), J + 1, -1 TEMP = TEMP - A( L + I, J )*X( IX ) IX = IX - INCX 180 CONTINUE IF( NOUNIT ) \$ TEMP = TEMP/A( 1, J ) ELSE DO 190, I = MIN( N, J + K ), J + 1, -1 TEMP = TEMP - CONJG( A( L + I, J ) )*X( IX ) IX = IX - INCX 190 CONTINUE IF( NOUNIT ) \$ TEMP = TEMP/CONJG( A( 1, J ) ) END IF X( JX ) = TEMP JX = JX - INCX IF( ( N - J ).GE.K ) \$ KX = KX - INCX 200 CONTINUE END IF END IF END IF C RETURN C C End of CTBSV . C END