dsysvx.f.html

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      SUBROUTINE <a name="DSYSVX.1"></a><a href="dsysvx.f.html#DSYSVX.1">DSYSVX</a>( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
     $                   LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
     $                   IWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK driver 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          FACT, UPLO
      INTEGER            INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
      DOUBLE PRECISION   RCOND
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      INTEGER            IPIV( * ), IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
     $                   BERR( * ), FERR( * ), 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="DSYSVX.23"></a><a href="dsysvx.f.html#DSYSVX.1">DSYSVX</a> uses the diagonal pivoting factorization to compute the
</span><span class="comment">*</span><span class="comment">  solution to a real system of linear equations A * X = B,
</span><span class="comment">*</span><span class="comment">  where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
</span><span class="comment">*</span><span class="comment">  matrices.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Error bounds on the solution and a condition estimate are also
</span><span class="comment">*</span><span class="comment">  provided.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Description
</span><span class="comment">*</span><span class="comment">  ===========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The following steps are performed:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  1. If FACT = 'N', the diagonal pivoting method is used to factor A.
</span><span class="comment">*</span><span class="comment">     The form of the factorization is
</span><span class="comment">*</span><span class="comment">        A = U * D * U**T,  if UPLO = 'U', or
</span><span class="comment">*</span><span class="comment">        A = L * D * L**T,  if UPLO = 'L',
</span><span class="comment">*</span><span class="comment">     where U (or L) is a product of permutation and unit upper (lower)
</span><span class="comment">*</span><span class="comment">     triangular matrices, and D is symmetric and block diagonal with
</span><span class="comment">*</span><span class="comment">     1-by-1 and 2-by-2 diagonal blocks.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  2. If some D(i,i)=0, so that D is exactly singular, then the routine
</span><span class="comment">*</span><span class="comment">     returns with INFO = i. Otherwise, the factored form of A is used
</span><span class="comment">*</span><span class="comment">     to estimate the condition number of the matrix A.  If the
</span><span class="comment">*</span><span class="comment">     reciprocal of the condition number is less than machine precision,
</span><span class="comment">*</span><span class="comment">     INFO = N+1 is returned as a warning, but the routine still goes on
</span><span class="comment">*</span><span class="comment">     to solve for X and compute error bounds as described below.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  3. The system of equations is solved for X using the factored form
</span><span class="comment">*</span><span class="comment">     of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  4. Iterative refinement is applied to improve the computed solution
</span><span class="comment">*</span><span class="comment">     matrix and calculate error bounds and backward error estimates
</span><span class="comment">*</span><span class="comment">     for it.
</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">  FACT    (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          Specifies whether or not the factored form of A has been
</span><span class="comment">*</span><span class="comment">          supplied on entry.
</span><span class="comment">*</span><span class="comment">          = 'F':  On entry, AF and IPIV contain the factored form of
</span><span class="comment">*</span><span class="comment">                  A.  AF and IPIV will not be modified.
</span><span class="comment">*</span><span class="comment">          = 'N':  The matrix A will be copied to AF and factored.
</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">          = 'U':  Upper triangle of A is stored;
</span><span class="comment">*</span><span class="comment">          = 'L':  Lower triangle of A is stored.
</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 number of linear equations, i.e., the order of the
</span><span class="comment">*</span><span class="comment">          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 matrices B and X.  NRHS &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
</span><span class="comment">*</span><span class="comment">          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N
</span><span class="comment">*</span><span class="comment">          upper triangular part of A contains the upper triangular part
</span><span class="comment">*</span><span class="comment">          of the matrix A, and the strictly lower triangular part of A
</span><span class="comment">*</span><span class="comment">          is not referenced.  If UPLO = 'L', the leading N-by-N lower
</span><span class="comment">*</span><span class="comment">          triangular part of A contains the lower triangular part of
</span><span class="comment">*</span><span class="comment">          the matrix A, and the strictly upper triangular part of A is
</span><span class="comment">*</span><span class="comment">          not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDA     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array A.  LDA &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  AF      (input or output) DOUBLE PRECISION array, dimension (LDAF,N)
</span><span class="comment">*</span><span class="comment">          If FACT = 'F', then AF is an input argument and on entry
</span><span class="comment">*</span><span class="comment">          contains the block diagonal matrix D and the multipliers used
</span><span class="comment">*</span><span class="comment">          to obtain the factor U or L from the factorization
</span><span class="comment">*</span><span class="comment">          A = U*D*U**T or A = L*D*L**T as computed by <a name="DSYTRF.96"></a><a href="dsytrf.f.html#DSYTRF.1">DSYTRF</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">          If FACT = 'N', then AF is an output argument and on exit
</span><span class="comment">*</span><span class="comment">          returns the block diagonal matrix D and the multipliers used
</span><span class="comment">*</span><span class="comment">          to obtain the factor U or L from the factorization
</span><span class="comment">*</span><span class="comment">          A = U*D*U**T or A = L*D*L**T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDAF    (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array AF.  LDAF &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  IPIV    (input or output) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment">          If FACT = 'F', then IPIV is an input argument and on entry
</span><span class="comment">*</span><span class="comment">          contains details of the interchanges and the block structure
</span><span class="comment">*</span><span class="comment">          of D, as determined by <a name="DSYTRF.109"></a><a href="dsytrf.f.html#DSYTRF.1">DSYTRF</a>.
</span><span class="comment">*</span><span class="comment">          If IPIV(k) &gt; 0, then rows and columns k and IPIV(k) were
</span><span class="comment">*</span><span class="comment">          interchanged and D(k,k) is a 1-by-1 diagonal block.
</span><span class="comment">*</span><span class="comment">          If UPLO = 'U' and IPIV(k) = IPIV(k-1) &lt; 0, then rows and
</span><span class="comment">*</span><span class="comment">          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
</span><span class="comment">*</span><span class="comment">          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =
</span><span class="comment">*</span><span class="comment">          IPIV(k+1) &lt; 0, then rows and columns k+1 and -IPIV(k) were
</span><span class="comment">*</span><span class="comment">          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">          If FACT = 'N', then IPIV is an output argument and on exit
</span><span class="comment">*</span><span class="comment">          contains details of the interchanges and the block structure
</span><span class="comment">*</span><span class="comment">          of D, as determined by <a name="DSYTRF.120"></a><a href="dsytrf.f.html#DSYTRF.1">DSYTRF</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
</span><span class="comment">*</span><span class="comment">          The N-by-NRHS 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       (output) DOUBLE PRECISION array, dimension (LDX,NRHS)
</span><span class="comment">*</span><span class="comment">          If INFO = 0 or INFO = N+1, the N-by-NRHS 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">  RCOND   (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment">          The estimate of the reciprocal condition number of the matrix
</span><span class="comment">*</span><span class="comment">          A.  If RCOND is less than the machine precision (in
</span><span class="comment">*</span><span class="comment">          particular, if RCOND = 0), the matrix is singular to working
</span><span class="comment">*</span><span class="comment">          precision.  This condition is indicated by a return code of
</span><span class="comment">*</span><span class="comment">          INFO &gt; 0.
</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 estimated 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)
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