.TH "hetf2" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
hetf2 \- {he,sy}tf2: triangular factor, level 2
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBchetf2\fP (uplo, n, a, lda, ipiv, info)"
.br
.RI "\fBCHETF2\fP computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm calling Level 2 BLAS)\&. "
.ti -1c
.RI "subroutine \fBcsytf2\fP (uplo, n, a, lda, ipiv, info)"
.br
.RI "\fBCSYTF2\fP computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm)\&. "
.ti -1c
.RI "subroutine \fBdsytf2\fP (uplo, n, a, lda, ipiv, info)"
.br
.RI "\fBDSYTF2\fP computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm)\&. "
.ti -1c
.RI "subroutine \fBssytf2\fP (uplo, n, a, lda, ipiv, info)"
.br
.RI "\fBSSYTF2\fP computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm)\&. "
.ti -1c
.RI "subroutine \fBzhetf2\fP (uplo, n, a, lda, ipiv, info)"
.br
.RI "\fBZHETF2\fP computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm, calling Level 2 BLAS)\&. "
.ti -1c
.RI "subroutine \fBzsytf2\fP (uplo, n, a, lda, ipiv, info)"
.br
.RI "\fBZSYTF2\fP computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm)\&. "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine chetf2 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)"

.PP
\fBCHETF2\fP computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm calling Level 2 BLAS)\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CHETF2 computes the factorization of a complex Hermitian matrix A
 using the Bunch-Kaufman diagonal pivoting method:

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

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

 This is the unblocked version of the algorithm, calling Level 2 BLAS\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX array, dimension (LDA,N)
          On entry, 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\&.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L (see below for further details)\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIIPIV\fP 
.PP
.nf
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D\&.

          If UPLO = 'U':
             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 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':
             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 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\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, D(k,k) is exactly zero\&.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it
               is used to solve a system of equations\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  09-29-06 - patch from
    Bobby Cheng, MathWorks

    Replace l\&.210 and l\&.392
         IF( MAX( ABSAKK, COLMAX )\&.EQ\&.ZERO ) THEN
    by
         IF( (MAX( ABSAKK, COLMAX )\&.EQ\&.ZERO) \&.OR\&. SISNAN(ABSAKK) ) THEN

  01-01-96 - Based on modifications by
    J\&. Lewis, Boeing Computer Services Company
    A\&. Petitet, Computer Science Dept\&., Univ\&. of Tenn\&., Knoxville, USA

  If UPLO = 'U', then A = U*D*U**H, where
     U = P(n)*U(n)* \&.\&.\&. *P(k)U(k)* \&.\&.\&.,
  i\&.e\&., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k)\&.  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k)\&.
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k)\&.

  If UPLO = 'L', then A = L*D*L**H, where
     L = P(1)*L(1)* \&.\&.\&. *P(k)*L(k)* \&.\&.\&.,
  i\&.e\&., L is a product of terms P(k)*L(k), where k increases from 1 to
  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k)\&.  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k)\&.
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1)\&.
.fi
.PP
 
.RE
.PP

.SS "subroutine csytf2 (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)"

.PP
\fBCSYTF2\fP computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm)\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CSYTF2 computes the factorization of a complex symmetric matrix A
 using the Bunch-Kaufman diagonal pivoting method:

    A = U*D*U**T  or  A = L*D*L**T

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

 This is the unblocked version of the algorithm, calling Level 2 BLAS\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX array, dimension (LDA,N)
          On entry, 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\&.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L (see below for further details)\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIIPIV\fP 
.PP
.nf
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D\&.

          If UPLO = 'U':
             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 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':
             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 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\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, D(k,k) is exactly zero\&.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it
               is used to solve a system of equations\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  If UPLO = 'U', then A = U*D*U**T, where
     U = P(n)*U(n)* \&.\&.\&. *P(k)U(k)* \&.\&.\&.,
  i\&.e\&., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k)\&.  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k)\&.
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k)\&.

  If UPLO = 'L', then A = L*D*L**T, where
     L = P(1)*L(1)* \&.\&.\&. *P(k)*L(k)* \&.\&.\&.,
  i\&.e\&., L is a product of terms P(k)*L(k), where k increases from 1 to
  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k)\&.  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k)\&.
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1)\&.
.fi
.PP
 
.RE
.PP
\fBContributors:\fP
.RS 4

.PP
.nf
  09-29-06 - patch from
    Bobby Cheng, MathWorks

    Replace l\&.209 and l\&.377
         IF( MAX( ABSAKK, COLMAX )\&.EQ\&.ZERO ) THEN
    by
         IF( (MAX( ABSAKK, COLMAX )\&.EQ\&.ZERO) \&.OR\&. SISNAN(ABSAKK) ) THEN

  1-96 - Based on modifications by J\&. Lewis, Boeing Computer Services
         Company
.fi
.PP
 
.RE
.PP

.SS "subroutine dsytf2 (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)"

.PP
\fBDSYTF2\fP computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm)\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DSYTF2 computes the factorization of a real symmetric matrix A using
 the Bunch-Kaufman diagonal pivoting method:

    A = U*D*U**T  or  A = L*D*L**T

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

 This is the unblocked version of the algorithm, calling Level 2 BLAS\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, 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\&.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L (see below for further details)\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIIPIV\fP 
.PP
.nf
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D\&.

          If UPLO = 'U':
             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 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':
             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 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\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, D(k,k) is exactly zero\&.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it
               is used to solve a system of equations\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  If UPLO = 'U', then A = U*D*U**T, where
     U = P(n)*U(n)* \&.\&.\&. *P(k)U(k)* \&.\&.\&.,
  i\&.e\&., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k)\&.  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k)\&.
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k)\&.

  If UPLO = 'L', then A = L*D*L**T, where
     L = P(1)*L(1)* \&.\&.\&. *P(k)*L(k)* \&.\&.\&.,
  i\&.e\&., L is a product of terms P(k)*L(k), where k increases from 1 to
  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k)\&.  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k)\&.
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1)\&.
.fi
.PP
 
.RE
.PP
\fBContributors:\fP
.RS 4

.PP
.nf
  09-29-06 - patch from
    Bobby Cheng, MathWorks

    Replace l\&.204 and l\&.372
         IF( MAX( ABSAKK, COLMAX )\&.EQ\&.ZERO ) THEN
    by
         IF( (MAX( ABSAKK, COLMAX )\&.EQ\&.ZERO) \&.OR\&. DISNAN(ABSAKK) ) THEN

  01-01-96 - Based on modifications by
    J\&. Lewis, Boeing Computer Services Company
    A\&. Petitet, Computer Science Dept\&., Univ\&. of Tenn\&., Knoxville, USA
  1-96 - Based on modifications by J\&. Lewis, Boeing Computer Services
         Company
.fi
.PP
 
.RE
.PP

.SS "subroutine ssytf2 (character uplo, integer n, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)"

.PP
\fBSSYTF2\fP computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm)\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SSYTF2 computes the factorization of a real symmetric matrix A using
 the Bunch-Kaufman diagonal pivoting method:

    A = U*D*U**T  or  A = L*D*L**T

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

 This is the unblocked version of the algorithm, calling Level 2 BLAS\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is REAL array, dimension (LDA,N)
          On entry, 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\&.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L (see below for further details)\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIIPIV\fP 
.PP
.nf
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D\&.

          If UPLO = 'U':
             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 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':
             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 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\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, D(k,k) is exactly zero\&.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it
               is used to solve a system of equations\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  If UPLO = 'U', then A = U*D*U**T, where
     U = P(n)*U(n)* \&.\&.\&. *P(k)U(k)* \&.\&.\&.,
  i\&.e\&., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k)\&.  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k)\&.
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k)\&.

  If UPLO = 'L', then A = L*D*L**T, where
     L = P(1)*L(1)* \&.\&.\&. *P(k)*L(k)* \&.\&.\&.,
  i\&.e\&., L is a product of terms P(k)*L(k), where k increases from 1 to
  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k)\&.  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k)\&.
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1)\&.
.fi
.PP
 
.RE
.PP
\fBContributors:\fP
.RS 4

.PP
.nf
  09-29-06 - patch from
    Bobby Cheng, MathWorks

    Replace l\&.204 and l\&.372
         IF( MAX( ABSAKK, COLMAX )\&.EQ\&.ZERO ) THEN
    by
         IF( (MAX( ABSAKK, COLMAX )\&.EQ\&.ZERO) \&.OR\&. SISNAN(ABSAKK) ) THEN

  01-01-96 - Based on modifications by
    J\&. Lewis, Boeing Computer Services Company
    A\&. Petitet, Computer Science Dept\&., Univ\&. of Tenn\&., Knoxville, USA
  1-96 - Based on modifications by J\&. Lewis, Boeing Computer Services
         Company
.fi
.PP
 
.RE
.PP

.SS "subroutine zhetf2 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)"

.PP
\fBZHETF2\fP computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm, calling Level 2 BLAS)\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZHETF2 computes the factorization of a complex Hermitian matrix A
 using the Bunch-Kaufman diagonal pivoting method:

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

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

 This is the unblocked version of the algorithm, calling Level 2 BLAS\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX*16 array, dimension (LDA,N)
          On entry, 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\&.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L (see below for further details)\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIIPIV\fP 
.PP
.nf
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D\&.

          If UPLO = 'U':
             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 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':
             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 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\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, D(k,k) is exactly zero\&.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it
               is used to solve a system of equations\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  If UPLO = 'U', then A = U*D*U**H, where
     U = P(n)*U(n)* \&.\&.\&. *P(k)U(k)* \&.\&.\&.,
  i\&.e\&., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k)\&.  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k)\&.
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k)\&.

  If UPLO = 'L', then A = L*D*L**H, where
     L = P(1)*L(1)* \&.\&.\&. *P(k)*L(k)* \&.\&.\&.,
  i\&.e\&., L is a product of terms P(k)*L(k), where k increases from 1 to
  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k)\&.  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k)\&.
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1)\&.
.fi
.PP
 
.RE
.PP
\fBContributors:\fP
.RS 4

.PP
.nf
  09-29-06 - patch from
    Bobby Cheng, MathWorks

    Replace l\&.210 and l\&.393
         IF( MAX( ABSAKK, COLMAX )\&.EQ\&.ZERO ) THEN
    by
         IF( (MAX( ABSAKK, COLMAX )\&.EQ\&.ZERO) \&.OR\&. DISNAN(ABSAKK) ) THEN

  01-01-96 - Based on modifications by
    J\&. Lewis, Boeing Computer Services Company
    A\&. Petitet, Computer Science Dept\&., Univ\&. of Tenn\&., Knoxville, USA
.fi
.PP
 
.RE
.PP

.SS "subroutine zsytf2 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, integer, dimension( * ) ipiv, integer info)"

.PP
\fBZSYTF2\fP computes the factorization of a real symmetric indefinite matrix, using the diagonal pivoting method (unblocked algorithm)\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZSYTF2 computes the factorization of a complex symmetric matrix A
 using the Bunch-Kaufman diagonal pivoting method:

    A = U*D*U**T  or  A = L*D*L**T

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

 This is the unblocked version of the algorithm, calling Level 2 BLAS\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored:
          = 'U':  Upper triangular
          = 'L':  Lower triangular
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrix A\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX*16 array, dimension (LDA,N)
          On entry, 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\&.

          On exit, the block diagonal matrix D and the multipliers used
          to obtain the factor U or L (see below for further details)\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
.fi
.PP
.br
\fIIPIV\fP 
.PP
.nf
          IPIV is INTEGER array, dimension (N)
          Details of the interchanges and the block structure of D\&.

          If UPLO = 'U':
             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 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':
             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 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\&.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
          > 0: if INFO = k, D(k,k) is exactly zero\&.  The factorization
               has been completed, but the block diagonal matrix D is
               exactly singular, and division by zero will occur if it
               is used to solve a system of equations\&.
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
  If UPLO = 'U', then A = U*D*U**T, where
     U = P(n)*U(n)* \&.\&.\&. *P(k)U(k)* \&.\&.\&.,
  i\&.e\&., U is a product of terms P(k)*U(k), where k decreases from n to
  1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k)\&.  P(k) is a permutation matrix as
  defined by IPIV(k), and U(k) is a unit upper triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    v    0   )   k-s
     U(k) =  (   0    I    0   )   s
             (   0    0    I   )   n-k
                k-s   s   n-k

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k)\&.
  If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
  and A(k,k), and v overwrites A(1:k-2,k-1:k)\&.

  If UPLO = 'L', then A = L*D*L**T, where
     L = P(1)*L(1)* \&.\&.\&. *P(k)*L(k)* \&.\&.\&.,
  i\&.e\&., L is a product of terms P(k)*L(k), where k increases from 1 to
  n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
  and 2-by-2 diagonal blocks D(k)\&.  P(k) is a permutation matrix as
  defined by IPIV(k), and L(k) is a unit lower triangular matrix, such
  that if the diagonal block D(k) is of order s (s = 1 or 2), then

             (   I    0     0   )  k-1
     L(k) =  (   0    I     0   )  s
             (   0    v     I   )  n-k-s+1
                k-1   s  n-k-s+1

  If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k)\&.
  If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
  and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1)\&.
.fi
.PP
 
.RE
.PP
\fBContributors:\fP
.RS 4

.PP
.nf
  09-29-06 - patch from
    Bobby Cheng, MathWorks

    Replace l\&.209 and l\&.377
         IF( MAX( ABSAKK, COLMAX )\&.EQ\&.ZERO ) THEN
    by
         IF( (MAX( ABSAKK, COLMAX )\&.EQ\&.ZERO) \&.OR\&. DISNAN(ABSAKK) ) THEN

  1-96 - Based on modifications by J\&. Lewis, Boeing Computer Services
         Company
.fi
.PP
 
.RE
.PP

.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.