SUBROUTINE SLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
$ CNORM, INFO )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER DIAG, NORMIN, TRANS, UPLO
INTEGER INFO, LDA, N
REAL SCALE
* ..
* .. Array Arguments ..
REAL A( LDA, * ), CNORM( * ), X( * )
* ..
*
* Purpose
* =======
*
* SLATRS solves one of the triangular systems
*
* A *x = s*b or A'*x = s*b
*
* with scaling to prevent overflow. Here A is an upper or lower
* triangular matrix, A' denotes the transpose of A, x and b are
* n-element vectors, and s is a scaling factor, usually less than
* or equal to 1, chosen so that the components of x will be less than
* the overflow threshold. If the unscaled problem will not cause
* overflow, the Level 2 BLAS routine STRSV is called. If the matrix A
* is singular (A(j,j) = 0 for some j), then s is set to 0 and a
* non-trivial solution to A*x = 0 is returned.
*
* Arguments
* =========
*
* UPLO (input) CHARACTER*1
* Specifies whether the matrix A is upper or lower triangular.
* = 'U': Upper triangular
* = 'L': Lower triangular
*
* TRANS (input) CHARACTER*1
* Specifies the operation applied to A.
* = 'N': Solve A * x = s*b (No transpose)
* = 'T': Solve A'* x = s*b (Transpose)
* = 'C': Solve A'* x = s*b (Conjugate transpose = Transpose)
*
* DIAG (input) CHARACTER*1
* Specifies whether or not the matrix A is unit triangular.
* = 'N': Non-unit triangular
* = 'U': Unit triangular
*
* NORMIN (input) CHARACTER*1
* Specifies whether CNORM has been set or not.
* = 'Y': CNORM contains the column norms on entry
* = 'N': CNORM is not set on entry. On exit, the norms will
* be computed and stored in CNORM.
*
* N (input) INTEGER
* The order of the matrix A. N >= 0.
*
* A (input) REAL array, dimension (LDA,N)
* The triangular matrix A. If UPLO = 'U', the leading n by n
* upper triangular part of the array A contains the upper
* triangular matrix, and the strictly lower triangular part of
* A is not referenced. If UPLO = 'L', the leading n by n lower
* triangular part of the array A contains the lower triangular
* matrix, and the strictly upper triangular part of A is not
* referenced. If DIAG = 'U', the diagonal elements of A are
* also not referenced and are assumed to be 1.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max (1,N).
*
* X (input/output) REAL array, dimension (N)
* On entry, the right hand side b of the triangular system.
* On exit, X is overwritten by the solution vector x.
*
* SCALE (output) REAL
* The scaling factor s for the triangular system
* A * x = s*b or A'* x = s*b.
* If SCALE = 0, the matrix A is singular or badly scaled, and
* the vector x is an exact or approximate solution to A*x = 0.
*
* CNORM (input or output) REAL array, dimension (N)
*
* If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
* contains the norm of the off-diagonal part of the j-th column
* of A. If TRANS = 'N', CNORM(j) must be greater than or equal
* to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
* must be greater than or equal to the 1-norm.
*
* If NORMIN = 'N', CNORM is an output argument and CNORM(j)
* returns the 1-norm of the offdiagonal part of the j-th column
* of A.
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -k, the k-th argument had an illegal value
*
* Further Details
* ======= =======
*
* A rough bound on x is computed; if that is less than overflow, STRSV
* is called, otherwise, specific code is used which checks for possible
* overflow or divide-by-zero at every operation.
*
* A columnwise scheme is used for solving A*x = b. The basic algorithm
* if A is lower triangular is
*
* x[1:n] := b[1:n]
* for j = 1, ..., n
* x(j) := x(j) / A(j,j)
* x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
* end
*
* Define bounds on the components of x after j iterations of the loop:
* M(j) = bound on x[1:j]
* G(j) = bound on x[j+1:n]
* Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
*
* Then for iteration j+1 we have
* M(j+1) <= G(j) / | A(j+1,j+1) |
* G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
* <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
*
* where CNORM(j+1) is greater than or equal to the infinity-norm of
* column j+1 of A, not counting the diagonal. Hence
*
* G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
* 1<=i<=j
* and
*
* |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
* 1<=i< j
*
* Since |x(j)| <= M(j), we use the Level 2 BLAS routine STRSV if the
* reciprocal of the largest M(j), j=1,..,n, is larger than
* max(underflow, 1/overflow).
*
* The bound on x(j) is also used to determine when a step in the
* columnwise method can be performed without fear of overflow. If
* the computed bound is greater than a large constant, x is scaled to
* prevent overflow, but if the bound overflows, x is set to 0, x(j) to
* 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
*
* Similarly, a row-wise scheme is used to solve A'*x = b. The basic
* algorithm for A upper triangular is
*
* for j = 1, ..., n
* x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
* end
*
* We simultaneously compute two bounds
* G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
* M(j) = bound on x(i), 1<=i<=j
*
* The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
* add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
* Then the bound on x(j) is
*
* M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
*
* <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
* 1<=i<=j
*
* and we can safely call STRSV if 1/M(n) and 1/G(n) are both greater
* than max(underflow, 1/overflow).
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0E+0, HALF = 0.5E+0, ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL NOTRAN, NOUNIT, UPPER
INTEGER I, IMAX, J, JFIRST, JINC, JLAST
REAL BIGNUM, GROW, REC, SMLNUM, SUMJ, TJJ, TJJS,
$ TMAX, TSCAL, USCAL, XBND, XJ, XMAX
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ISAMAX
REAL SASUM, SDOT, SLAMCH
EXTERNAL LSAME, ISAMAX, SASUM, SDOT, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SSCAL, STRSV, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
NOTRAN = LSAME( TRANS, 'N' )
NOUNIT = LSAME( DIAG, 'N' )
*
* Test the input parameters.
*
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( .NOT.NOTRAN .AND. .NOT.LSAME( TRANS, 'T' ) .AND. .NOT.
$ LSAME( TRANS, 'C' ) ) THEN
INFO = -2
ELSE IF( .NOT.NOUNIT .AND. .NOT.LSAME( DIAG, 'U' ) ) THEN
INFO = -3
ELSE IF( .NOT.LSAME( NORMIN, 'Y' ) .AND. .NOT.
$ LSAME( NORMIN, 'N' ) ) THEN
INFO = -4
ELSE IF( N.LT.0 ) THEN
INFO = -5
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLATRS', -INFO )
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Determine machine dependent parameters to control overflow.
*
SMLNUM = SLAMCH( 'Safe minimum' ) / SLAMCH( 'Precision' )
BIGNUM = ONE / SMLNUM
SCALE = ONE
*
IF( LSAME( NORMIN, 'N' ) ) THEN
*
* Compute the 1-norm of each column, not including the diagonal.
*
IF( UPPER ) THEN
*
* A is upper triangular.
*
DO 10 J = 1, N
CNORM( J ) = SASUM( J-1, A( 1, J ), 1 )
10 CONTINUE
ELSE
*
* A is lower triangular.
*
DO 20 J = 1, N - 1
CNORM( J ) = SASUM( N-J, A( J+1, J ), 1 )
20 CONTINUE
CNORM( N ) = ZERO
END IF
END IF
*
* Scale the column norms by TSCAL if the maximum element in CNORM is
* greater than BIGNUM.
*
IMAX = ISAMAX( N, CNORM, 1 )
TMAX = CNORM( IMAX )
IF( TMAX.LE.BIGNUM ) THEN
TSCAL = ONE
ELSE
TSCAL = ONE / ( SMLNUM*TMAX )
CALL SSCAL( N, TSCAL, CNORM, 1 )
END IF
*
* Compute a bound on the computed solution vector to see if the
* Level 2 BLAS routine STRSV can be used.
*
J = ISAMAX( N, X, 1 )
XMAX = ABS( X( J ) )
XBND = XMAX
IF( NOTRAN ) THEN
*
* Compute the growth in A * x = b.
*
IF( UPPER ) THEN
JFIRST = N
JLAST = 1
JINC = -1
ELSE
JFIRST = 1
JLAST = N
JINC = 1
END IF
*
IF( TSCAL.NE.ONE ) THEN
GROW = ZERO
GO TO 50
END IF
*
IF( NOUNIT ) THEN
*
* A is non-unit triangular.
*
* Compute GROW = 1/G(j) and XBND = 1/M(j).
* Initially, G(0) = max{x(i), i=1,...,n}.
*
GROW = ONE / MAX( XBND, SMLNUM )
XBND = GROW
DO 30 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 50
*
* M(j) = G(j-1) / abs(A(j,j))
*
TJJ = ABS( A( J, J ) )
XBND = MIN( XBND, MIN( ONE, TJJ )*GROW )
IF( TJJ+CNORM( J ).GE.SMLNUM ) THEN
*
* G(j) = G(j-1)*( 1 + CNORM(j) / abs(A(j,j)) )
*
GROW = GROW*( TJJ / ( TJJ+CNORM( J ) ) )
ELSE
*
* G(j) could overflow, set GROW to 0.
*
GROW = ZERO
END IF
30 CONTINUE
GROW = XBND
ELSE
*
* A is unit triangular.
*
* Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
DO 40 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 50
*
* G(j) = G(j-1)*( 1 + CNORM(j) )
*
GROW = GROW*( ONE / ( ONE+CNORM( J ) ) )
40 CONTINUE
END IF
50 CONTINUE
*
ELSE
*
* Compute the growth in A' * x = b.
*
IF( UPPER ) THEN
JFIRST = 1
JLAST = N
JINC = 1
ELSE
JFIRST = N
JLAST = 1
JINC = -1
END IF
*
IF( TSCAL.NE.ONE ) THEN
GROW = ZERO
GO TO 80
END IF
*
IF( NOUNIT ) THEN
*
* A is non-unit triangular.
*
* Compute GROW = 1/G(j) and XBND = 1/M(j).
* Initially, M(0) = max{x(i), i=1,...,n}.
*
GROW = ONE / MAX( XBND, SMLNUM )
XBND = GROW
DO 60 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 80
*
* G(j) = max( G(j-1), M(j-1)*( 1 + CNORM(j) ) )
*
XJ = ONE + CNORM( J )
GROW = MIN( GROW, XBND / XJ )
*
* M(j) = M(j-1)*( 1 + CNORM(j) ) / abs(A(j,j))
*
TJJ = ABS( A( J, J ) )
IF( XJ.GT.TJJ )
$ XBND = XBND*( TJJ / XJ )
60 CONTINUE
GROW = MIN( GROW, XBND )
ELSE
*
* A is unit triangular.
*
* Compute GROW = 1/G(j), where G(0) = max{x(i), i=1,...,n}.
*
GROW = MIN( ONE, ONE / MAX( XBND, SMLNUM ) )
DO 70 J = JFIRST, JLAST, JINC
*
* Exit the loop if the growth factor is too small.
*
IF( GROW.LE.SMLNUM )
$ GO TO 80
*
* G(j) = ( 1 + CNORM(j) )*G(j-1)
*
XJ = ONE + CNORM( J )
GROW = GROW / XJ
70 CONTINUE
END IF
80 CONTINUE
END IF
*
IF( ( GROW*TSCAL ).GT.SMLNUM ) THEN
*
* Use the Level 2 BLAS solve if the reciprocal of the bound on
* elements of X is not too small.
*
CALL STRSV( UPLO, TRANS, DIAG, N, A, LDA, X, 1 )
ELSE
*
* Use a Level 1 BLAS solve, scaling intermediate results.
*
IF( XMAX.GT.BIGNUM ) THEN
*
* Scale X so that its components are less than or equal to
* BIGNUM in absolute value.
*
SCALE = BIGNUM / XMAX
CALL SSCAL( N, SCALE, X, 1 )
XMAX = BIGNUM
END IF
*
IF( NOTRAN ) THEN
*
* Solve A * x = b
*
DO 100 J = JFIRST, JLAST, JINC
*
* Compute x(j) = b(j) / A(j,j), scaling x if necessary.
*
XJ = ABS( X( J ) )
IF( NOUNIT ) THEN
TJJS = A( J, J )*TSCAL
ELSE
TJJS = TSCAL
IF( TSCAL.EQ.ONE )
$ GO TO 95
END IF
TJJ = ABS( TJJS )
IF( TJJ.GT.SMLNUM ) THEN
*
* abs(A(j,j)) > SMLNUM:
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by 1/b(j).
*
REC = ONE / XJ
CALL SSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
X( J ) = X( J ) / TJJS
XJ = ABS( X( J ) )
ELSE IF( TJJ.GT.ZERO ) THEN
*
* 0 < abs(A(j,j)) <= SMLNUM:
*
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM
* to avoid overflow when dividing by A(j,j).
*
REC = ( TJJ*BIGNUM ) / XJ
IF( CNORM( J ).GT.ONE ) THEN
*
* Scale by 1/CNORM(j) to avoid overflow when
* multiplying x(j) times column j.
*
REC = REC / CNORM( J )
END IF
CALL SSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
X( J ) = X( J ) / TJJS
XJ = ABS( X( J ) )
ELSE
*
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
* scale = 0, and compute a solution to A*x = 0.
*
DO 90 I = 1, N
X( I ) = ZERO
90 CONTINUE
X( J ) = ONE
XJ = ONE
SCALE = ZERO
XMAX = ZERO
END IF
95 CONTINUE
*
* Scale x if necessary to avoid overflow when adding a
* multiple of column j of A.
*
IF( XJ.GT.ONE ) THEN
REC = ONE / XJ
IF( CNORM( J ).GT.( BIGNUM-XMAX )*REC ) THEN
*
* Scale x by 1/(2*abs(x(j))).
*
REC = REC*HALF
CALL SSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
END IF
ELSE IF( XJ*CNORM( J ).GT.( BIGNUM-XMAX ) ) THEN
*
* Scale x by 1/2.
*
CALL SSCAL( N, HALF, X, 1 )
SCALE = SCALE*HALF
END IF
*
IF( UPPER ) THEN
IF( J.GT.1 ) THEN
*
* Compute the update
* x(1:j-1) := x(1:j-1) - x(j) * A(1:j-1,j)
*
CALL SAXPY( J-1, -X( J )*TSCAL, A( 1, J ), 1, X,
$ 1 )
I = ISAMAX( J-1, X, 1 )
XMAX = ABS( X( I ) )
END IF
ELSE
IF( J.LT.N ) THEN
*
* Compute the update
* x(j+1:n) := x(j+1:n) - x(j) * A(j+1:n,j)
*
CALL SAXPY( N-J, -X( J )*TSCAL, A( J+1, J ), 1,
$ X( J+1 ), 1 )
I = J + ISAMAX( N-J, X( J+1 ), 1 )
XMAX = ABS( X( I ) )
END IF
END IF
100 CONTINUE
*
ELSE
*
* Solve A' * x = b
*
DO 140 J = JFIRST, JLAST, JINC
*
* Compute x(j) = b(j) - sum A(k,j)*x(k).
* k<>j
*
XJ = ABS( X( J ) )
USCAL = TSCAL
REC = ONE / MAX( XMAX, ONE )
IF( CNORM( J ).GT.( BIGNUM-XJ )*REC ) THEN
*
* If x(j) could overflow, scale x by 1/(2*XMAX).
*
REC = REC*HALF
IF( NOUNIT ) THEN
TJJS = A( J, J )*TSCAL
ELSE
TJJS = TSCAL
END IF
TJJ = ABS( TJJS )
IF( TJJ.GT.ONE ) THEN
*
* Divide by A(j,j) when scaling x if A(j,j) > 1.
*
REC = MIN( ONE, REC*TJJ )
USCAL = USCAL / TJJS
END IF
IF( REC.LT.ONE ) THEN
CALL SSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
*
SUMJ = ZERO
IF( USCAL.EQ.ONE ) THEN
*
* If the scaling needed for A in the dot product is 1,
* call SDOT to perform the dot product.
*
IF( UPPER ) THEN
SUMJ = SDOT( J-1, A( 1, J ), 1, X, 1 )
ELSE IF( J.LT.N ) THEN
SUMJ = SDOT( N-J, A( J+1, J ), 1, X( J+1 ), 1 )
END IF
ELSE
*
* Otherwise, use in-line code for the dot product.
*
IF( UPPER ) THEN
DO 110 I = 1, J - 1
SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I )
110 CONTINUE
ELSE IF( J.LT.N ) THEN
DO 120 I = J + 1, N
SUMJ = SUMJ + ( A( I, J )*USCAL )*X( I )
120 CONTINUE
END IF
END IF
*
IF( USCAL.EQ.TSCAL ) THEN
*
* Compute x(j) := ( x(j) - sumj ) / A(j,j) if 1/A(j,j)
* was not used to scale the dotproduct.
*
X( J ) = X( J ) - SUMJ
XJ = ABS( X( J ) )
IF( NOUNIT ) THEN
TJJS = A( J, J )*TSCAL
ELSE
TJJS = TSCAL
IF( TSCAL.EQ.ONE )
$ GO TO 135
END IF
*
* Compute x(j) = x(j) / A(j,j), scaling if necessary.
*
TJJ = ABS( TJJS )
IF( TJJ.GT.SMLNUM ) THEN
*
* abs(A(j,j)) > SMLNUM:
*
IF( TJJ.LT.ONE ) THEN
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale X by 1/abs(x(j)).
*
REC = ONE / XJ
CALL SSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
END IF
X( J ) = X( J ) / TJJS
ELSE IF( TJJ.GT.ZERO ) THEN
*
* 0 < abs(A(j,j)) <= SMLNUM:
*
IF( XJ.GT.TJJ*BIGNUM ) THEN
*
* Scale x by (1/abs(x(j)))*abs(A(j,j))*BIGNUM.
*
REC = ( TJJ*BIGNUM ) / XJ
CALL SSCAL( N, REC, X, 1 )
SCALE = SCALE*REC
XMAX = XMAX*REC
END IF
X( J ) = X( J ) / TJJS
ELSE
*
* A(j,j) = 0: Set x(1:n) = 0, x(j) = 1, and
* scale = 0, and compute a solution to A'*x = 0.
*
DO 130 I = 1, N
X( I ) = ZERO
130 CONTINUE
X( J ) = ONE
SCALE = ZERO
XMAX = ZERO
END IF
135 CONTINUE
ELSE
*
* Compute x(j) := x(j) / A(j,j) - sumj if the dot
* product has already been divided by 1/A(j,j).
*
X( J ) = X( J ) / TJJS - SUMJ
END IF
XMAX = MAX( XMAX, ABS( X( J ) ) )
140 CONTINUE
END IF
SCALE = SCALE / TSCAL
END IF
*
* Scale the column norms by 1/TSCAL for return.
*
IF( TSCAL.NE.ONE ) THEN
CALL SSCAL( N, ONE / TSCAL, CNORM, 1 )
END IF
*
RETURN
*
* End of SLATRS
*
END