DecompositionOperations<TReal, TComplex>.SymmetricDecompose Method

public void SymmetricDecompose(
	MatrixTriangle storedTriangle,
	int n,
	Array2D<TReal> a,
	Array1D<int> ipiv,
	out int info
)

Public Sub SymmetricDecompose ( 
	storedTriangle As MatrixTriangle,
	n As Integer,
	a As Array2D(Of TReal),
	ipiv As Array1D(Of Integer),
	<OutAttribute> ByRef info As Integer
)

public:
void SymmetricDecompose(
	MatrixTriangle storedTriangle, 
	int n, 
	Array2D<TReal> a, 
	Array1D<int> ipiv, 
	[OutAttribute] int% info
)

member SymmetricDecompose : 
        storedTriangle : MatrixTriangle * 
        n : int * 
        a : Array2D<'TReal> * 
        ipiv : Array1D<int> * 
        info : int byref -> unit 

Parameters

storedTriangle  MatrixTriangle

            = 'U':  Upper triangle of A is stored;
            = 'L':  Lower triangle of A is stored.
            

n  Int32

            The order of the matrix A.  N >= 0.
            

a  Array2D<TReal>

            A is TReal 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).
            

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

ipiv  Array1D<Int32>

            Dimension (N)
            Details of the interchanges and the block structure of D.
            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.
            

info  Int32

            = 0:  successful exit
            < 0:  if INFO = -i, the i-th argument had an illegal value
            > 0:  if INFO = i, D(i,i) 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.
            

            The form of the
            factorization is
               A = U*D*UT  or  A = L*D*LT
            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.
            This is the blocked version of the algorithm, calling Level 3 BLAS.
            

Further Details:

            If UPLO = 'U', then A = U*D*UT, 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*LT, 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).
            

This method corresponds to the LAPACK routine ?SYTRF.

public void SymmetricDecompose(
	MatrixTriangle storedTriangle,
	int n,
	Span2D<TReal> a,
	Span<int> ipiv,
	out int info
)

Public Sub SymmetricDecompose ( 
	storedTriangle As MatrixTriangle,
	n As Integer,
	a As Span2D(Of TReal),
	ipiv As Span(Of Integer),
	<OutAttribute> ByRef info As Integer
)

public:
void SymmetricDecompose(
	MatrixTriangle storedTriangle, 
	int n, 
	Span2D<TReal> a, 
	Span<int> ipiv, 
	[OutAttribute] int% info
)

member SymmetricDecompose : 
        storedTriangle : MatrixTriangle * 
        n : int * 
        a : Span2D<'TReal> * 
        ipiv : Span<int> * 
        info : int byref -> unit 

Parameters

storedTriangle  MatrixTriangle

            = 'U':  Upper triangle of A is stored;
            = 'L':  Lower triangle of A is stored.
            

n  Int32

            The order of the matrix A.  N >= 0.
            

a  Span2D<TReal>

            A is TReal 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).
            

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

ipiv  Span<Int32>

            Dimension (N)
            Details of the interchanges and the block structure of D.
            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.
            

info  Int32

            = 0:  successful exit
            < 0:  if INFO = -i, the i-th argument had an illegal value
            > 0:  if INFO = i, D(i,i) 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.
            

            The form of the
            factorization is
               A = U*D*UT  or  A = L*D*LT
            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.
            This is the blocked version of the algorithm, calling Level 3 BLAS.
            

Further Details:

            If UPLO = 'U', then A = U*D*UT, 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*LT, 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).
            

This method corresponds to the LAPACK routine ?SYTRF.

public abstract void SymmetricDecompose(
	MatrixTriangle storedTriangle,
	int n,
	Span<TReal> a,
	int lda,
	Span<int> ipiv,
	out int info
)

Public MustOverride Sub SymmetricDecompose ( 
	storedTriangle As MatrixTriangle,
	n As Integer,
	a As Span(Of TReal),
	lda As Integer,
	ipiv As Span(Of Integer),
	<OutAttribute> ByRef info As Integer
)

public:
virtual void SymmetricDecompose(
	MatrixTriangle storedTriangle, 
	int n, 
	Span<TReal> a, 
	int lda, 
	Span<int> ipiv, 
	[OutAttribute] int% info
) abstract

abstract SymmetricDecompose : 
        storedTriangle : MatrixTriangle * 
        n : int * 
        a : Span<'TReal> * 
        lda : int * 
        ipiv : Span<int> * 
        info : int byref -> unit 

Parameters

storedTriangle  MatrixTriangle

            = 'U':  Upper triangle of A is stored;
            = 'L':  Lower triangle of A is stored.
            

n  Int32

            The order of the matrix A.  N >= 0.
            

a  Span<TReal>

            A is TReal 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).
            

lda  Int32

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

ipiv  Span<Int32>

            Dimension (N)
            Details of the interchanges and the block structure of D.
            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.
            

info  Int32

            = 0:  successful exit
            < 0:  if INFO = -i, the i-th argument had an illegal value
            > 0:  if INFO = i, D(i,i) 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.
            

            The form of the
            factorization is
               A = U*D*UT  or  A = L*D*LT
            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.
            This is the blocked version of the algorithm, calling Level 3 BLAS.
            

Further Details:

            If UPLO = 'U', then A = U*D*UT, 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*LT, 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).
            

This method corresponds to the LAPACK routine ?SYTRF.

Reference