CHESVX uses the diagonal pivoting factorization to compute the


 solution to a complex system of linear equations A * X = B,


 where A is an N-by-N Hermitian matrix 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.


    The form of the factorization is


       A = U * D * U**H,  if UPLO = 'U', or


       A = L * D * L**H,  if UPLO = 'L',


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


    triangular matrices, and D is Hermitian 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, AF and IPIV contain the factored form


                  of A.  A, AF and IPIV will not be modified.


          = 'N':  The matrix A will be copied to AF 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.

A

          A is COMPLEX array, dimension (LDA,N)


          The Hermitian matrix A.  If UPLO = 'U', the leading N-by-N


          upper triangular part of A contains the upper triangular part


          of the matrix A, and the strictly lower triangular part of A


          is not referenced.  If UPLO = 'L', the leading N-by-N lower


          triangular part of A contains the lower triangular part of


          the matrix A, and the strictly upper triangular part of A is


          not referenced.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

AF

          AF is COMPLEX array, dimension (LDAF,N)


          If FACT = 'F', then AF 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**H or A = L*D*L**H as computed by CHETRF.


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


          returns the block diagonal matrix D and the multipliers used


          to obtain the factor U or L from the factorization


          A = U*D*U**H or A = L*D*L**H.

LDAF

          LDAF is INTEGER


          The leading dimension of the array AF.  LDAF >= max(1,N).

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 CHETRF.


          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 CHETRF.

B

          B is COMPLEX 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 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 REAL


          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 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

          BERR is 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

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The length of WORK.  LWORK >= max(1,2*N), and for best


          performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where


          NB is the optimal blocksize for CHETRF.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL 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.

 CSYSVX uses the diagonal pivoting factorization to compute the


 solution to a complex system of linear equations A * X = B,


 where A is an N-by-N symmetric matrix 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.


    The form of the factorization is


       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, AF and IPIV contain the factored form


                  of A.  A, AF and IPIV will not be modified.


          = 'N':  The matrix A will be copied to AF 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.

A

          A is COMPLEX array, dimension (LDA,N)


          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N


          upper triangular part of A contains the upper triangular part


          of the matrix A, and the strictly lower triangular part of A


          is not referenced.  If UPLO = 'L', the leading N-by-N lower


          triangular part of A contains the lower triangular part of


          the matrix A, and the strictly upper triangular part of A is


          not referenced.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

AF

          AF is COMPLEX array, dimension (LDAF,N)


          If FACT = 'F', then AF 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 CSYTRF.


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


          returns 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.

LDAF

          LDAF is INTEGER


          The leading dimension of the array AF.  LDAF >= max(1,N).

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 CSYTRF.


          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 CSYTRF.

B

          B is COMPLEX 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 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 REAL


          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 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

          BERR is 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

          WORK is COMPLEX array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The length of WORK.  LWORK >= max(1,2*N), and for best


          performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where


          NB is the optimal blocksize for CSYTRF.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

RWORK

          RWORK is REAL 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.

 DSYSVX uses the diagonal pivoting factorization to compute the


 solution to a real system of linear equations A * X = B,


 where A is an N-by-N symmetric matrix 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.


    The form of the factorization is


       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, AF and IPIV contain the factored form of


                  A.  AF and IPIV will not be modified.


          = 'N':  The matrix A will be copied to AF 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.

A

          A is DOUBLE PRECISION array, dimension (LDA,N)


          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N


          upper triangular part of A contains the upper triangular part


          of the matrix A, and the strictly lower triangular part of A


          is not referenced.  If UPLO = 'L', the leading N-by-N lower


          triangular part of A contains the lower triangular part of


          the matrix A, and the strictly upper triangular part of A is


          not referenced.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

AF

          AF is DOUBLE PRECISION array, dimension (LDAF,N)


          If FACT = 'F', then AF 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 DSYTRF.


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


          returns 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.

LDAF

          LDAF is INTEGER


          The leading dimension of the array AF.  LDAF >= max(1,N).

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 DSYTRF.


          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 DSYTRF.

B

          B is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The length of WORK.  LWORK >= max(1,3*N), and for best


          performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where


          NB is the optimal blocksize for DSYTRF.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER 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.

 SSYSVX uses the diagonal pivoting factorization to compute the


 solution to a real system of linear equations A * X = B,


 where A is an N-by-N symmetric matrix 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.


    The form of the factorization is


       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, AF and IPIV contain the factored form of


                  A.  AF and IPIV will not be modified.


          = 'N':  The matrix A will be copied to AF 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.

A

          A is REAL array, dimension (LDA,N)


          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N


          upper triangular part of A contains the upper triangular part


          of the matrix A, and the strictly lower triangular part of A


          is not referenced.  If UPLO = 'L', the leading N-by-N lower


          triangular part of A contains the lower triangular part of


          the matrix A, and the strictly upper triangular part of A is


          not referenced.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

AF

          AF is REAL array, dimension (LDAF,N)


          If FACT = 'F', then AF 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 SSYTRF.


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


          returns 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.

LDAF

          LDAF is INTEGER


          The leading dimension of the array AF.  LDAF >= max(1,N).

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 SSYTRF.


          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 SSYTRF.

B

          B is REAL 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 REAL 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 REAL


          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 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

          BERR is 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

          WORK is REAL array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The length of WORK.  LWORK >= max(1,3*N), and for best


          performance, when FACT = 'N', LWORK >= max(1,3*N,N*NB), where


          NB is the optimal blocksize for SSYTRF.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER 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.

 ZHESVX uses the diagonal pivoting factorization to compute the


 solution to a complex system of linear equations A * X = B,


 where A is an N-by-N Hermitian matrix 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.


    The form of the factorization is


       A = U * D * U**H,  if UPLO = 'U', or


       A = L * D * L**H,  if UPLO = 'L',


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


    triangular matrices, and D is Hermitian 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, AF and IPIV contain the factored form


                  of A.  A, AF and IPIV will not be modified.


          = 'N':  The matrix A will be copied to AF 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.

A

          A is COMPLEX*16 array, dimension (LDA,N)


          The Hermitian matrix A.  If UPLO = 'U', the leading N-by-N


          upper triangular part of A contains the upper triangular part


          of the matrix A, and the strictly lower triangular part of A


          is not referenced.  If UPLO = 'L', the leading N-by-N lower


          triangular part of A contains the lower triangular part of


          the matrix A, and the strictly upper triangular part of A is


          not referenced.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

AF

          AF is COMPLEX*16 array, dimension (LDAF,N)


          If FACT = 'F', then AF 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**H or A = L*D*L**H as computed by ZHETRF.


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


          returns the block diagonal matrix D and the multipliers used


          to obtain the factor U or L from the factorization


          A = U*D*U**H or A = L*D*L**H.

LDAF

          LDAF is INTEGER


          The leading dimension of the array AF.  LDAF >= max(1,N).

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 ZHETRF.


          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 ZHETRF.

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 (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The length of WORK.  LWORK >= max(1,2*N), and for best


          performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where


          NB is the optimal blocksize for ZHETRF.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

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.

 ZSYSVX uses the diagonal pivoting factorization to compute the


 solution to a complex system of linear equations A * X = B,


 where A is an N-by-N symmetric matrix 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.


    The form of the factorization is


       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, AF and IPIV contain the factored form


                  of A.  A, AF and IPIV will not be modified.


          = 'N':  The matrix A will be copied to AF 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.

A

          A is COMPLEX*16 array, dimension (LDA,N)


          The symmetric matrix A.  If UPLO = 'U', the leading N-by-N


          upper triangular part of A contains the upper triangular part


          of the matrix A, and the strictly lower triangular part of A


          is not referenced.  If UPLO = 'L', the leading N-by-N lower


          triangular part of A contains the lower triangular part of


          the matrix A, and the strictly upper triangular part of A is


          not referenced.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

AF

          AF is COMPLEX*16 array, dimension (LDAF,N)


          If FACT = 'F', then AF 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 ZSYTRF.


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


          returns 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.

LDAF

          LDAF is INTEGER


          The leading dimension of the array AF.  LDAF >= max(1,N).

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 ZSYTRF.


          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 ZSYTRF.

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 (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The length of WORK.  LWORK >= max(1,2*N), and for best


          performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where


          NB is the optimal blocksize for ZSYTRF.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

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.