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authorjason <jason@8a072113-8704-0410-8d35-dd094bca7971>2008-10-28 01:38:50 +0000
committerjason <jason@8a072113-8704-0410-8d35-dd094bca7971>2008-10-28 01:38:50 +0000
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Move LAPACK trunk into position.
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+ SUBROUTINE SPBRFS( UPLO, N, KD, NRHS, AB, LDAB, AFB, LDAFB, B,
+ $ LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO )
+*
+* -- LAPACK routine (version 3.1) --
+* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
+* November 2006
+*
+* Modified to call SLACN2 in place of SLACON, 7 Feb 03, SJH.
+*
+* .. Scalar Arguments ..
+ CHARACTER UPLO
+ INTEGER INFO, KD, LDAB, LDAFB, LDB, LDX, N, NRHS
+* ..
+* .. Array Arguments ..
+ INTEGER IWORK( * )
+ REAL AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
+ $ BERR( * ), FERR( * ), WORK( * ), X( LDX, * )
+* ..
+*
+* Purpose
+* =======
+*
+* SPBRFS improves the computed solution to a system of linear
+* equations when the coefficient matrix is symmetric positive definite
+* and banded, and provides error bounds and backward error estimates
+* for the solution.
+*
+* Arguments
+* =========
+*
+* UPLO (input) CHARACTER*1
+* = 'U': Upper triangle of A is stored;
+* = 'L': Lower triangle of A is stored.
+*
+* N (input) INTEGER
+* The order of the matrix A. N >= 0.
+*
+* KD (input) INTEGER
+* The number of superdiagonals of the matrix A if UPLO = 'U',
+* or the number of subdiagonals if UPLO = 'L'. KD >= 0.
+*
+* NRHS (input) INTEGER
+* The number of right hand sides, i.e., the number of columns
+* of the matrices B and X. NRHS >= 0.
+*
+* AB (input) REAL array, dimension (LDAB,N)
+* 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).
+*
+* LDAB (input) INTEGER
+* The leading dimension of the array AB. LDAB >= KD+1.
+*
+* AFB (input) REAL array, dimension (LDAFB,N)
+* The triangular factor U or L from the Cholesky factorization
+* A = U**T*U or A = L*L**T of the band matrix A as computed by
+* SPBTRF, in the same storage format as A (see AB).
+*
+* LDAFB (input) INTEGER
+* The leading dimension of the array AFB. LDAFB >= KD+1.
+*
+* B (input) REAL array, dimension (LDB,NRHS)
+* The right hand side matrix B.
+*
+* LDB (input) INTEGER
+* The leading dimension of the array B. LDB >= max(1,N).
+*
+* X (input/output) REAL array, dimension (LDX,NRHS)
+* On entry, the solution matrix X, as computed by SPBTRS.
+* On exit, the improved solution matrix X.
+*
+* LDX (input) INTEGER
+* The leading dimension of the array X. LDX >= max(1,N).
+*
+* FERR (output) REAL array, dimension (NRHS)
+* The estimated forward error bound for each solution vector
+* X(j) (the j-th column of the solution matrix X).
+* If XTRUE is the true solution corresponding to X(j), FERR(j)
+* is an estimated upper bound for the magnitude of the largest
+* element in (X(j) - XTRUE) divided by the magnitude of the
+* largest element in X(j). The estimate is as reliable as
+* the estimate for RCOND, and is almost always a slight
+* overestimate of the true error.
+*
+* BERR (output) REAL array, dimension (NRHS)
+* The componentwise relative backward error of each solution
+* vector X(j) (i.e., the smallest relative change in
+* any element of A or B that makes X(j) an exact solution).
+*
+* WORK (workspace) REAL array, dimension (3*N)
+*
+* IWORK (workspace) INTEGER array, dimension (N)
+*
+* INFO (output) INTEGER
+* = 0: successful exit
+* < 0: if INFO = -i, the i-th argument had an illegal value
+*
+* Internal Parameters
+* ===================
+*
+* ITMAX is the maximum number of steps of iterative refinement.
+*
+* =====================================================================
+*
+* .. Parameters ..
+ INTEGER ITMAX
+ PARAMETER ( ITMAX = 5 )
+ REAL ZERO
+ PARAMETER ( ZERO = 0.0E+0 )
+ REAL ONE
+ PARAMETER ( ONE = 1.0E+0 )
+ REAL TWO
+ PARAMETER ( TWO = 2.0E+0 )
+ REAL THREE
+ PARAMETER ( THREE = 3.0E+0 )
+* ..
+* .. Local Scalars ..
+ LOGICAL UPPER
+ INTEGER COUNT, I, J, K, KASE, L, NZ
+ REAL EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN, XK
+* ..
+* .. Local Arrays ..
+ INTEGER ISAVE( 3 )
+* ..
+* .. External Subroutines ..
+ EXTERNAL SAXPY, SCOPY, SLACN2, SPBTRS, SSBMV, XERBLA
+* ..
+* .. Intrinsic Functions ..
+ INTRINSIC ABS, MAX, MIN
+* ..
+* .. External Functions ..
+ LOGICAL LSAME
+ REAL SLAMCH
+ EXTERNAL LSAME, SLAMCH
+* ..
+* .. Executable Statements ..
+*
+* Test the input parameters.
+*
+ INFO = 0
+ UPPER = LSAME( UPLO, 'U' )
+ IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
+ INFO = -1
+ ELSE IF( N.LT.0 ) THEN
+ INFO = -2
+ ELSE IF( KD.LT.0 ) THEN
+ INFO = -3
+ ELSE IF( NRHS.LT.0 ) THEN
+ INFO = -4
+ ELSE IF( LDAB.LT.KD+1 ) THEN
+ INFO = -6
+ ELSE IF( LDAFB.LT.KD+1 ) THEN
+ INFO = -8
+ ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
+ INFO = -10
+ ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
+ INFO = -12
+ END IF
+ IF( INFO.NE.0 ) THEN
+ CALL XERBLA( 'SPBRFS', -INFO )
+ RETURN
+ END IF
+*
+* Quick return if possible
+*
+ IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN
+ DO 10 J = 1, NRHS
+ FERR( J ) = ZERO
+ BERR( J ) = ZERO
+ 10 CONTINUE
+ RETURN
+ END IF
+*
+* NZ = maximum number of nonzero elements in each row of A, plus 1
+*
+ NZ = MIN( N+1, 2*KD+2 )
+ EPS = SLAMCH( 'Epsilon' )
+ SAFMIN = SLAMCH( 'Safe minimum' )
+ SAFE1 = NZ*SAFMIN
+ SAFE2 = SAFE1 / EPS
+*
+* Do for each right hand side
+*
+ DO 140 J = 1, NRHS
+*
+ COUNT = 1
+ LSTRES = THREE
+ 20 CONTINUE
+*
+* Loop until stopping criterion is satisfied.
+*
+* Compute residual R = B - A * X
+*
+ CALL SCOPY( N, B( 1, J ), 1, WORK( N+1 ), 1 )
+ CALL SSBMV( UPLO, N, KD, -ONE, AB, LDAB, X( 1, J ), 1, ONE,
+ $ WORK( N+1 ), 1 )
+*
+* Compute componentwise relative backward error from formula
+*
+* max(i) ( abs(R(i)) / ( abs(A)*abs(X) + abs(B) )(i) )
+*
+* where abs(Z) is the componentwise absolute value of the matrix
+* or vector Z. If the i-th component of the denominator is less
+* than SAFE2, then SAFE1 is added to the i-th components of the
+* numerator and denominator before dividing.
+*
+ DO 30 I = 1, N
+ WORK( I ) = ABS( B( I, J ) )
+ 30 CONTINUE
+*
+* Compute abs(A)*abs(X) + abs(B).
+*
+ IF( UPPER ) THEN
+ DO 50 K = 1, N
+ S = ZERO
+ XK = ABS( X( K, J ) )
+ L = KD + 1 - K
+ DO 40 I = MAX( 1, K-KD ), K - 1
+ WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
+ S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
+ 40 CONTINUE
+ WORK( K ) = WORK( K ) + ABS( AB( KD+1, K ) )*XK + S
+ 50 CONTINUE
+ ELSE
+ DO 70 K = 1, N
+ S = ZERO
+ XK = ABS( X( K, J ) )
+ WORK( K ) = WORK( K ) + ABS( AB( 1, K ) )*XK
+ L = 1 - K
+ DO 60 I = K + 1, MIN( N, K+KD )
+ WORK( I ) = WORK( I ) + ABS( AB( L+I, K ) )*XK
+ S = S + ABS( AB( L+I, K ) )*ABS( X( I, J ) )
+ 60 CONTINUE
+ WORK( K ) = WORK( K ) + S
+ 70 CONTINUE
+ END IF
+ S = ZERO
+ DO 80 I = 1, N
+ IF( WORK( I ).GT.SAFE2 ) THEN
+ S = MAX( S, ABS( WORK( N+I ) ) / WORK( I ) )
+ ELSE
+ S = MAX( S, ( ABS( WORK( N+I ) )+SAFE1 ) /
+ $ ( WORK( I )+SAFE1 ) )
+ END IF
+ 80 CONTINUE
+ BERR( J ) = S
+*
+* Test stopping criterion. Continue iterating if
+* 1) The residual BERR(J) is larger than machine epsilon, and
+* 2) BERR(J) decreased by at least a factor of 2 during the
+* last iteration, and
+* 3) At most ITMAX iterations tried.
+*
+ IF( BERR( J ).GT.EPS .AND. TWO*BERR( J ).LE.LSTRES .AND.
+ $ COUNT.LE.ITMAX ) THEN
+*
+* Update solution and try again.
+*
+ CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
+ $ INFO )
+ CALL SAXPY( N, ONE, WORK( N+1 ), 1, X( 1, J ), 1 )
+ LSTRES = BERR( J )
+ COUNT = COUNT + 1
+ GO TO 20
+ END IF
+*
+* Bound error from formula
+*
+* norm(X - XTRUE) / norm(X) .le. FERR =
+* norm( abs(inv(A))*
+* ( abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) ))) / norm(X)
+*
+* where
+* norm(Z) is the magnitude of the largest component of Z
+* inv(A) is the inverse of A
+* abs(Z) is the componentwise absolute value of the matrix or
+* vector Z
+* NZ is the maximum number of nonzeros in any row of A, plus 1
+* EPS is machine epsilon
+*
+* The i-th component of abs(R)+NZ*EPS*(abs(A)*abs(X)+abs(B))
+* is incremented by SAFE1 if the i-th component of
+* abs(A)*abs(X) + abs(B) is less than SAFE2.
+*
+* Use SLACN2 to estimate the infinity-norm of the matrix
+* inv(A) * diag(W),
+* where W = abs(R) + NZ*EPS*( abs(A)*abs(X)+abs(B) )))
+*
+ DO 90 I = 1, N
+ IF( WORK( I ).GT.SAFE2 ) THEN
+ WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I )
+ ELSE
+ WORK( I ) = ABS( WORK( N+I ) ) + NZ*EPS*WORK( I ) + SAFE1
+ END IF
+ 90 CONTINUE
+*
+ KASE = 0
+ 100 CONTINUE
+ CALL SLACN2( N, WORK( 2*N+1 ), WORK( N+1 ), IWORK, FERR( J ),
+ $ KASE, ISAVE )
+ IF( KASE.NE.0 ) THEN
+ IF( KASE.EQ.1 ) THEN
+*
+* Multiply by diag(W)*inv(A').
+*
+ CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
+ $ INFO )
+ DO 110 I = 1, N
+ WORK( N+I ) = WORK( N+I )*WORK( I )
+ 110 CONTINUE
+ ELSE IF( KASE.EQ.2 ) THEN
+*
+* Multiply by inv(A)*diag(W).
+*
+ DO 120 I = 1, N
+ WORK( N+I ) = WORK( N+I )*WORK( I )
+ 120 CONTINUE
+ CALL SPBTRS( UPLO, N, KD, 1, AFB, LDAFB, WORK( N+1 ), N,
+ $ INFO )
+ END IF
+ GO TO 100
+ END IF
+*
+* Normalize error.
+*
+ LSTRES = ZERO
+ DO 130 I = 1, N
+ LSTRES = MAX( LSTRES, ABS( X( I, J ) ) )
+ 130 CONTINUE
+ IF( LSTRES.NE.ZERO )
+ $ FERR( J ) = FERR( J ) / LSTRES
+*
+ 140 CONTINUE
+*
+ RETURN
+*
+* End of SPBRFS
+*
+ END