zptrfs.f.html

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      SUBROUTINE <a name="ZPTRFS.1"></a><a href="zptrfs.f.html#ZPTRFS.1">ZPTRFS</a>( UPLO, N, NRHS, D, E, DF, EF, B, LDB, X, LDX,
     $                   FERR, BERR, WORK, RWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      CHARACTER          UPLO
      INTEGER            INFO, LDB, LDX, N, NRHS
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      DOUBLE PRECISION   BERR( * ), D( * ), DF( * ), FERR( * ),
     $                   RWORK( * )
      COMPLEX*16         B( LDB, * ), E( * ), EF( * ), WORK( * ),
     $                   X( LDX, * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="ZPTRFS.22"></a><a href="zptrfs.f.html#ZPTRFS.1">ZPTRFS</a> improves the computed solution to a system of linear
</span><span class="comment">*</span><span class="comment">  equations when the coefficient matrix is Hermitian positive definite
</span><span class="comment">*</span><span class="comment">  and tridiagonal, and provides error bounds and backward error
</span><span class="comment">*</span><span class="comment">  estimates for the solution.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  UPLO    (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          Specifies whether the superdiagonal or the subdiagonal of the
</span><span class="comment">*</span><span class="comment">          tridiagonal matrix A is stored and the form of the
</span><span class="comment">*</span><span class="comment">          factorization:
</span><span class="comment">*</span><span class="comment">          = 'U':  E is the superdiagonal of A, and A = U**H*D*U;
</span><span class="comment">*</span><span class="comment">          = 'L':  E is the subdiagonal of A, and A = L*D*L**H.
</span><span class="comment">*</span><span class="comment">          (The two forms are equivalent if A is real.)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The order of the matrix A.  N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  NRHS    (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of right hand sides, i.e., the number of columns
</span><span class="comment">*</span><span class="comment">          of the matrix B.  NRHS &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  D       (input) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment">          The n real diagonal elements of the tridiagonal matrix A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  E       (input) COMPLEX*16 array, dimension (N-1)
</span><span class="comment">*</span><span class="comment">          The (n-1) off-diagonal elements of the tridiagonal matrix A
</span><span class="comment">*</span><span class="comment">          (see UPLO).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  DF      (input) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment">          The n diagonal elements of the diagonal matrix D from
</span><span class="comment">*</span><span class="comment">          the factorization computed by <a name="ZPTTRF.54"></a><a href="zpttrf.f.html#ZPTTRF.1">ZPTTRF</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  EF      (input) COMPLEX*16 array, dimension (N-1)
</span><span class="comment">*</span><span class="comment">          The (n-1) off-diagonal elements of the unit bidiagonal
</span><span class="comment">*</span><span class="comment">          factor U or L from the factorization computed by <a name="ZPTTRF.58"></a><a href="zpttrf.f.html#ZPTTRF.1">ZPTTRF</a>
</span><span class="comment">*</span><span class="comment">          (see UPLO).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  B       (input) COMPLEX*16 array, dimension (LDB,NRHS)
</span><span class="comment">*</span><span class="comment">          The right hand side matrix B.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDB     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array B.  LDB &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  X       (input/output) COMPLEX*16 array, dimension (LDX,NRHS)
</span><span class="comment">*</span><span class="comment">          On entry, the solution matrix X, as computed by <a name="ZPTTRS.68"></a><a href="zpttrs.f.html#ZPTTRS.1">ZPTTRS</a>.
</span><span class="comment">*</span><span class="comment">          On exit, the improved solution matrix X.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDX     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array X.  LDX &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  FERR    (output) DOUBLE PRECISION array, dimension (NRHS)
</span><span class="comment">*</span><span class="comment">          The forward error bound for each solution vector
</span><span class="comment">*</span><span class="comment">          X(j) (the j-th column of the solution matrix X).
</span><span class="comment">*</span><span class="comment">          If XTRUE is the true solution corresponding to X(j), FERR(j)
</span><span class="comment">*</span><span class="comment">          is an estimated upper bound for the magnitude of the largest
</span><span class="comment">*</span><span class="comment">          element in (X(j) - XTRUE) divided by the magnitude of the
</span><span class="comment">*</span><span class="comment">          largest element in X(j).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  BERR    (output) DOUBLE PRECISION array, dimension (NRHS)
</span><span class="comment">*</span><span class="comment">          The componentwise relative backward error of each solution
</span><span class="comment">*</span><span class="comment">          vector X(j) (i.e., the smallest relative change in
</span><span class="comment">*</span><span class="comment">          any element of A or B that makes X(j) an exact solution).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK    (workspace) COMPLEX*16 array, dimension (N)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RWORK   (workspace) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INFO    (output) INTEGER
</span><span class="comment">*</span><span class="comment">          = 0:  successful exit
</span><span class="comment">*</span><span class="comment">          &lt; 0:  if INFO = -i, the i-th argument had an illegal value
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Internal Parameters
</span><span class="comment">*</span><span class="comment">  ===================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  ITMAX is the maximum number of steps of iterative refinement.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      INTEGER            ITMAX
      PARAMETER          ( ITMAX = 5 )
      DOUBLE PRECISION   ZERO
      PARAMETER          ( ZERO = 0.0D+0 )
      DOUBLE PRECISION   ONE
      PARAMETER          ( ONE = 1.0D+0 )
      DOUBLE PRECISION   TWO
      PARAMETER          ( TWO = 2.0D+0 )
      DOUBLE PRECISION   THREE
      PARAMETER          ( THREE = 3.0D+0 )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      LOGICAL            UPPER
      INTEGER            COUNT, I, IX, J, NZ
      DOUBLE PRECISION   EPS, LSTRES, S, SAFE1, SAFE2, SAFMIN
      COMPLEX*16         BI, CX, DX, EX, ZDUM
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Functions ..
</span>      LOGICAL            <a name="LSAME.121"></a><a href="lsame.f.html#LSAME.1">LSAME</a>
      INTEGER            IDAMAX
      DOUBLE PRECISION   <a name="DLAMCH.123"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>
      EXTERNAL           <a name="LSAME.124"></a><a href="lsame.f.html#LSAME.1">LSAME</a>, IDAMAX, <a name="DLAMCH.124"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Subroutines ..
</span>      EXTERNAL           <a name="XERBLA.127"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>, ZAXPY, <a name="ZPTTRS.127"></a><a href="zpttrs.f.html#ZPTTRS.1">ZPTTRS</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Statement Functions ..
</span>      DOUBLE PRECISION   CABS1
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Statement Function definitions ..
</span>      CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Test the input parameters.
</span><span class="comment">*</span><span class="comment">
</span>      INFO = 0
      UPPER = <a name="LSAME.143"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( UPLO, <span class="string">'U'</span> )
      IF( .NOT.UPPER .AND. .NOT.<a name="LSAME.144"></a><a href="lsame.f.html#LSAME.1">LSAME</a>( UPLO, <span class="string">'L'</span> ) ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
         INFO = -11
      END IF
      IF( INFO.NE.0 ) THEN
         CALL <a name="XERBLA.156"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>( <span class="string">'<a name="ZPTRFS.156"></a><a href="zptrfs.f.html#ZPTRFS.1">ZPTRFS</a>'</span>, -INFO )
         RETURN
      END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Quick return if possible
</span><span class="comment">*</span><span class="comment">
</span>      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
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     NZ = maximum number of nonzero elements in each row of A, plus 1
</span><span class="comment">*</span><span class="comment">
</span>      NZ = 4
      EPS = <a name="DLAMCH.173"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>( <span class="string">'Epsilon'</span> )
      SAFMIN = <a name="DLAMCH.174"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>( <span class="string">'Safe minimum'</span> )
      SAFE1 = NZ*SAFMIN
      SAFE2 = SAFE1 / EPS
<span class="comment">*</span><span class="comment">
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