ZSPSVX uses the diagonal pivoting factorization A = U*D*U**T or


 A = L*D*L**T to compute the solution to a complex system of linear


 equations A * X = B, where A is an N-by-N symmetric matrix stored


 in packed format and X and B are N-by-NRHS matrices.


 Error bounds on the solution and a condition estimate are also


 provided.

 The following steps are performed:


 1. If FACT = 'N', the diagonal pivoting method is used to factor A as


       A = U * D * U**T,  if UPLO = 'U', or


       A = L * D * L**T,  if UPLO = 'L',


    where U (or L) is a product of permutation and unit upper (lower)


    triangular matrices and D is symmetric and block diagonal with


    1-by-1 and 2-by-2 diagonal blocks.


 2. If some D(i,i)=0, so that D is exactly singular, then the routine


    returns with INFO = i. Otherwise, the factored form of A is used


    to estimate the condition number of the matrix A.  If the


    reciprocal of the condition number is less than machine precision,


    INFO = N+1 is returned as a warning, but the routine still goes on


    to solve for X and compute error bounds as described below.


 3. The system of equations is solved for X using the factored form


    of A.


 4. Iterative refinement is applied to improve the computed solution


    matrix and calculate error bounds and backward error estimates


    for it.

FACT

          FACT is CHARACTER*1


          Specifies whether or not the factored form of A has been


          supplied on entry.


          = 'F':  On entry, AFP and IPIV contain the factored form


                  of A.  AP, AFP and IPIV will not be modified.


          = 'N':  The matrix A will be copied to AFP and factored.

UPLO

          UPLO is CHARACTER*1


          = 'U':  Upper triangle of A is stored;


          = 'L':  Lower triangle of A is stored.

N

          N is INTEGER


          The number of linear equations, i.e., the order of the


          matrix A.  N >= 0.

NRHS

          NRHS is INTEGER


          The number of right hand sides, i.e., the number of columns


          of the matrices B and X.  NRHS >= 0.

AP

          AP is COMPLEX*16 array, dimension (N*(N+1)/2)


          The upper or lower triangle of the symmetric matrix A, packed


          columnwise in a linear array.  The j-th column of A is stored


          in the array AP as follows:


          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;


          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.


          See below for further details.

AFP

          AFP is COMPLEX*16 array, dimension (N*(N+1)/2)


          If FACT = 'F', then AFP is an input argument and on entry


          contains the block diagonal matrix D and the multipliers used


          to obtain the factor U or L from the factorization


          A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as


          a packed triangular matrix in the same storage format as A.


          If FACT = 'N', then AFP is an output argument and on exit


          contains the block diagonal matrix D and the multipliers used


          to obtain the factor U or L from the factorization


          A = U*D*U**T or A = L*D*L**T as computed by ZSPTRF, stored as


          a packed triangular matrix in the same storage format as A.

IPIV

          IPIV is INTEGER array, dimension (N)


          If FACT = 'F', then IPIV is an input argument and on entry


          contains details of the interchanges and the block structure


          of D, as determined by ZSPTRF.


          If IPIV(k) > 0, then rows and columns k and IPIV(k) were


          interchanged and D(k,k) is a 1-by-1 diagonal block.


          If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and


          columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)


          is a 2-by-2 diagonal block.  If UPLO = 'L' and IPIV(k) =


          IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were


          interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.


          If FACT = 'N', then IPIV is an output argument and on exit


          contains details of the interchanges and the block structure


          of D, as determined by ZSPTRF.

B

          B is COMPLEX*16 array, dimension (LDB,NRHS)


          The N-by-NRHS right hand side matrix B.

LDB

          LDB is INTEGER


          The leading dimension of the array B.  LDB >= max(1,N).

X

          X is COMPLEX*16 array, dimension (LDX,NRHS)


          If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.

LDX

          LDX is INTEGER


          The leading dimension of the array X.  LDX >= max(1,N).

RCOND

          RCOND is DOUBLE PRECISION


          The estimate of the reciprocal condition number of the matrix


          A.  If RCOND is less than the machine precision (in


          particular, if RCOND = 0), the matrix is singular to working


          precision.  This condition is indicated by a return code of


          INFO > 0.

FERR

          FERR is DOUBLE PRECISION 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

          BERR is DOUBLE PRECISION 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

          WORK is COMPLEX*16 array, dimension (2*N)

RWORK

          RWORK is DOUBLE PRECISION array, dimension (N)

INFO

          INFO is INTEGER


          = 0: successful exit


          < 0: if INFO = -i, the i-th argument had an illegal value


          > 0:  if INFO = i, and i is


                <= N:  D(i,i) is exactly zero.  The factorization


                       has been completed but the factor D is exactly


                       singular, so the solution and error bounds could


                       not be computed. RCOND = 0 is returned.


                = N+1: D is nonsingular, but RCOND is less than machine


                       precision, meaning that the matrix is singular


                       to working precision.  Nevertheless, the


                       solution and error bounds are computed because


                       there are a number of situations where the


                       computed solution can be more accurate than the


                       value of RCOND would suggest.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

April 2012

  The packed storage scheme is illustrated by the following example


  when N = 4, UPLO = 'U':


  Two-dimensional storage of the symmetric matrix A:


     a11 a12 a13 a14


         a22 a23 a24


             a33 a34     (aij = aji)


                 a44


  Packed storage of the upper triangle of A:


  AP = [ a11, a12, a22, a13, a23, a33, a14, a24, a34, a44 ]