# hecon_3 - Man Page

{he,sy}con_3: condition number estimate

## Synopsis

### Functions

subroutine checon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, info)
CHECON_3
subroutine csycon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, info)
CSYCON_3
subroutine dsycon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, iwork, info)
DSYCON_3
subroutine ssycon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, iwork, info)
SSYCON_3
subroutine zhecon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, info)
ZHECON_3
subroutine zsycon_3 (uplo, n, a, lda, e, ipiv, anorm, rcond, work, info)
ZSYCON_3

## Function Documentation

### subroutine checon_3 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, real anorm, real rcond, complex, dimension( * ) work, integer info)

CHECON_3

Purpose:

``` CHECON_3 estimates the reciprocal of the condition number (in the
1-norm) of a complex Hermitian matrix A using the factorization
computed by CHETRF_RK or CHETRF_BK:

A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
This routine uses BLAS3 solver CHETRS_3.```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix:
= 'U':  Upper triangular, form is A = P*U*D*(U**H)*(P**T);
= 'L':  Lower triangular, form is A = P*L*D*(L**H)*(P**T).```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

A

```          A is COMPLEX array, dimension (LDA,N)
Diagonal of the block diagonal matrix D and factors U or L
as computed by CHETRF_RK and CHETRF_BK:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.```

LDA

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

E

```          E is COMPLEX array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.```

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CHETRF_RK or CHETRF_BK.```

ANORM

```          ANORM is REAL
The 1-norm of the original matrix A.```

RCOND

```          RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.```

WORK

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

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  June 2017,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester```

Definition at line 164 of file checon_3.f.

### subroutine csycon_3 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, real anorm, real rcond, complex, dimension( * ) work, integer info)

CSYCON_3

Purpose:

``` CSYCON_3 estimates the reciprocal of the condition number (in the
1-norm) of a complex symmetric matrix A using the factorization
computed by CSYTRF_RK or CSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
This routine uses BLAS3 solver CSYTRS_3.```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix:
= 'U':  Upper triangular, form is A = P*U*D*(U**T)*(P**T);
= 'L':  Lower triangular, form is A = P*L*D*(L**T)*(P**T).```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

A

```          A is COMPLEX array, dimension (LDA,N)
Diagonal of the block diagonal matrix D and factors U or L
as computed by CSYTRF_RK and CSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.```

LDA

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

E

```          E is COMPLEX array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.```

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by CSYTRF_RK or CSYTRF_BK.```

ANORM

```          ANORM is REAL
The 1-norm of the original matrix A.```

RCOND

```          RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.```

WORK

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

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  June 2017,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester```

Definition at line 164 of file csycon_3.f.

### subroutine dsycon_3 (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) e, integer, dimension( * ) ipiv, double precision anorm, double precision rcond, double precision, dimension( * ) work, integer, dimension( * ) iwork, integer info)

DSYCON_3

Purpose:

``` DSYCON_3 estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric matrix A using the factorization
computed by DSYTRF_RK or DSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
This routine uses BLAS3 solver DSYTRS_3.```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix:
= 'U':  Upper triangular, form is A = P*U*D*(U**T)*(P**T);
= 'L':  Lower triangular, form is A = P*L*D*(L**T)*(P**T).```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

A

```          A is DOUBLE PRECISION array, dimension (LDA,N)
Diagonal of the block diagonal matrix D and factors U or L
as computed by DSYTRF_RK and DSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.```

LDA

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

E

```          E is DOUBLE PRECISION array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.```

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by DSYTRF_RK or DSYTRF_BK.```

ANORM

```          ANORM is DOUBLE PRECISION
The 1-norm of the original matrix A.```

RCOND

```          RCOND is DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.```

WORK

`          WORK is DOUBLE PRECISION array, dimension (2*N)`

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```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  June 2017,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester```

Definition at line 169 of file dsycon_3.f.

### subroutine ssycon_3 (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) e, integer, dimension( * ) ipiv, real anorm, real rcond, real, dimension( * ) work, integer, dimension( * ) iwork, integer info)

SSYCON_3

Purpose:

``` SSYCON_3 estimates the reciprocal of the condition number (in the
1-norm) of a real symmetric matrix A using the factorization
computed by DSYTRF_RK or DSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
This routine uses BLAS3 solver SSYTRS_3.```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix:
= 'U':  Upper triangular, form is A = P*U*D*(U**T)*(P**T);
= 'L':  Lower triangular, form is A = P*L*D*(L**T)*(P**T).```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

A

```          A is REAL array, dimension (LDA,N)
Diagonal of the block diagonal matrix D and factors U or L
as computed by SSYTRF_RK and SSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.```

LDA

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

E

```          E is REAL array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.```

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by SSYTRF_RK or SSYTRF_BK.```

ANORM

```          ANORM is REAL
The 1-norm of the original matrix A.```

RCOND

```          RCOND is REAL
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.```

WORK

`          WORK is REAL array, dimension (2*N)`

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```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  June 2017,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester```

Definition at line 169 of file ssycon_3.f.

### subroutine zhecon_3 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, double precision anorm, double precision rcond, complex*16, dimension( * ) work, integer info)

ZHECON_3

Purpose:

``` ZHECON_3 estimates the reciprocal of the condition number (in the
1-norm) of a complex Hermitian matrix A using the factorization
computed by ZHETRF_RK or ZHETRF_BK:

A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
This routine uses BLAS3 solver ZHETRS_3.```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix:
= 'U':  Upper triangular, form is A = P*U*D*(U**H)*(P**T);
= 'L':  Lower triangular, form is A = P*L*D*(L**H)*(P**T).```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

A

```          A is COMPLEX*16 array, dimension (LDA,N)
Diagonal of the block diagonal matrix D and factors U or L
as computed by ZHETRF_RK and ZHETRF_BK:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.```

LDA

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

E

```          E is COMPLEX*16 array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.```

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by ZHETRF_RK or ZHETRF_BK.```

ANORM

```          ANORM is DOUBLE PRECISION
The 1-norm of the original matrix A.```

RCOND

```          RCOND is DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.```

WORK

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

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  June 2017,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester```

Definition at line 164 of file zhecon_3.f.

### subroutine zsycon_3 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, double precision anorm, double precision rcond, complex*16, dimension( * ) work, integer info)

ZSYCON_3

Purpose:

``` ZSYCON_3 estimates the reciprocal of the condition number (in the
1-norm) of a complex symmetric matrix A using the factorization
computed by ZSYTRF_RK or ZSYTRF_BK:

A = P*U*D*(U**T)*(P**T) or A = P*L*D*(L**T)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix,
U**T (or L**T) is the transpose of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is symmetric and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

An estimate is obtained for norm(inv(A)), and the reciprocal of the
condition number is computed as RCOND = 1 / (ANORM * norm(inv(A))).
This routine uses BLAS3 solver ZSYTRS_3.```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix:
= 'U':  Upper triangular, form is A = P*U*D*(U**T)*(P**T);
= 'L':  Lower triangular, form is A = P*L*D*(L**T)*(P**T).```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

A

```          A is COMPLEX*16 array, dimension (LDA,N)
Diagonal of the block diagonal matrix D and factors U or L
as computed by ZSYTRF_RK and ZSYTRF_BK:
a) ONLY diagonal elements of the symmetric block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.```

LDA

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

E

```          E is COMPLEX*16 array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.```

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by ZSYTRF_RK or ZSYTRF_BK.```

ANORM

```          ANORM is DOUBLE PRECISION
The 1-norm of the original matrix A.```

RCOND

```          RCOND is DOUBLE PRECISION
The reciprocal of the condition number of the matrix A,
computed as RCOND = 1/(ANORM * AINVNM), where AINVNM is an
estimate of the 1-norm of inv(A) computed in this routine.```

WORK

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

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  June 2017,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley

September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
School of Mathematics,
University of Manchester```

Definition at line 164 of file zsycon_3.f.

## Author

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## Info

Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK