Move LAPACK trunk into position.

1 files changed, 341 insertions, 0 deletions

diff --git a/SRC/spbrfs.f b/SRC/spbrfs.f
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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