sytf2_rk#

Functions

void ssytf2_rk(
    const char*          uplo,
    const INT            n,
          f32*  restrict A,
    const INT            lda,
          f32*  restrict E,
          INT*  restrict ipiv,
          INT*           info
);
void ssytf2_rk(const char *uplo, const INT n, f32 *restrict A, const INT lda, f32 *restrict E, INT *restrict ipiv, INT *info)#

SSYTF2_RK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: 

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.

This is the unblocked version of the algorithm, calling Level 2 BLAS.

  • 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 are stored on exit in array E), and

  • If uplo='U': factor U in the superdiagonal part of A. If uplo='L': factor L in the subdiagonal part of A.

  • 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]<0 and ipiv[k-1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k-1 and -ipiv[k-1]-1 were interchanged, 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]<0 and ipiv[k+1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k+1 and -ipiv[k+1]-1 were interchanged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

Parameters

in
lda

The leading dimension of the array A. lda>=max(1,n).

out
E

Array of dimension n. On exit, 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=1:n-1, E[0] is set to 0; if uplo='L': E[i]=D(i+1,i), i=0:n-2, E[n-1] is set to 0. For a 1-by-1 diagonal block D(k), the element E[k] is set to 0 in both uplo='U' and uplo='L' cases.

out
ipiv

Array of dimension n. Pivot indices (0-based). If uplo='U':

in
uplo

  • 'U': Upper triangular

  • 'L': Lower triangular

in
n

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

inout
A

Array of dimension (lda,n). On entry, the symmetric matrix A. On exit, contains:

out
info

  • info=0: successful exit

  • info<0: if info=-k, the k-th argument had an illegal value

  • info>0: if info=k, the matrix A is singular, because column k of the triangular part of A (upper or lower, per uplo) contains all zeros. D(k,k) is exactly zero, and superdiagonal (or subdiagonal) elements of column k of U (or L) are all zeros. 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. info only stores the first occurrence of a singularity; the factorization always completes.

Functions

void dsytf2_rk(
    const char*          uplo,
    const INT            n,
          f64*  restrict A,
    const INT            lda,
          f64*  restrict E,
          INT*  restrict ipiv,
          INT*           info
);
void dsytf2_rk(const char *uplo, const INT n, f64 *restrict A, const INT lda, f64 *restrict E, INT *restrict ipiv, INT *info)#

DSYTF2_RK computes the factorization of a real symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: 

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.

This is the unblocked version of the algorithm, calling Level 2 BLAS.

  • 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 are stored on exit in array E), and

  • If uplo='U': factor U in the superdiagonal part of A. If uplo='L': factor L in the subdiagonal part of A.

  • 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]<0 and ipiv[k-1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k-1 and -ipiv[k-1]-1 were interchanged, 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]<0 and ipiv[k+1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k+1 and -ipiv[k+1]-1 were interchanged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

Parameters

in
lda

The leading dimension of the array A. lda>=max(1,n).

out
E

Array of dimension n. On exit, 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=1:n-1, E[0] is set to 0; if uplo='L': E[i]=D(i+1,i), i=0:n-2, E[n-1] is set to 0. For a 1-by-1 diagonal block D(k), the element E[k] is set to 0 in both uplo='U' and uplo='L' cases.

out
ipiv

Array of dimension n. Pivot indices (0-based). If uplo='U':

in
uplo

  • 'U': Upper triangular

  • 'L': Lower triangular

in
n

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

inout
A

Array of dimension (lda,n). On entry, the symmetric matrix A. On exit, contains:

out
info

  • info=0: successful exit

  • info<0: if info=-k, the k-th argument had an illegal value

  • info>0: if info=k, the matrix A is singular, because column k of the triangular part of A (upper or lower, per uplo) contains all zeros. D(k,k) is exactly zero, and superdiagonal (or subdiagonal) elements of column k of U (or L) are all zeros. 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. info only stores the first occurrence of a singularity; the factorization always completes.

Functions

void csytf2_rk(
    const char*          uplo,
    const INT            n,
          c64*  restrict A,
    const INT            lda,
          c64*  restrict E,
          INT*  restrict ipiv,
          INT*           info
);
void csytf2_rk(const char *uplo, const INT n, c64 *restrict A, const INT lda, c64 *restrict E, INT *restrict ipiv, INT *info)#

CSYTF2_RK computes the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: 

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.

This is the unblocked version of the algorithm, calling Level 2 BLAS.

  • 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 are stored on exit in array E), and

  • If uplo='U': factor U in the superdiagonal part of A. If uplo='L': factor L in the subdiagonal part of A.

  • 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]<0 and ipiv[k-1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k-1 and -ipiv[k-1]-1 were interchanged, 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]<0 and ipiv[k+1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k+1 and -ipiv[k+1]-1 were interchanged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

Parameters

in
lda

The leading dimension of the array A. lda>=max(1,n).

out
E

Complex array of dimension n. On exit, 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=1:n-1, E[0] is set to 0; if uplo='L': E[i]=D(i+1,i), i=0:n-2, E[n-1] is set to 0. For a 1-by-1 diagonal block D(k), the element E[k] is set to 0 in both uplo='U' and uplo='L' cases.

out
ipiv

Array of dimension n. Pivot indices (0-based). If uplo='U':

in
uplo

  • 'U': Upper triangular

  • 'L': Lower triangular

in
n

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

inout
A

Complex array of dimension (lda,n). On entry, the symmetric matrix A. On exit, contains:

out
info

  • info=0: successful exit

  • info<0: if info=-k, the k-th argument had an illegal value

  • info>0: if info=k, the matrix A is singular, because column k of the triangular part of A (upper or lower, per uplo) contains all zeros. D(k,k) is exactly zero, and superdiagonal (or subdiagonal) elements of column k of U (or L) are all zeros. 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. info only stores the first occurrence of a singularity; the factorization always completes.

Functions

void zsytf2_rk(
    const char*          uplo,
    const INT            n,
          c128* restrict A,
    const INT            lda,
          c128* restrict E,
          INT*  restrict ipiv,
          INT*           info
);
void zsytf2_rk(const char *uplo, const INT n, c128 *restrict A, const INT lda, c128 *restrict E, INT *restrict ipiv, INT *info)#

ZSYTF2_RK computes the factorization of a complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: 

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.

This is the unblocked version of the algorithm, calling Level 2 BLAS.

  • 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 are stored on exit in array E), and

  • If uplo='U': factor U in the superdiagonal part of A. If uplo='L': factor L in the subdiagonal part of A.

  • 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]<0 and ipiv[k-1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k-1 and -ipiv[k-1]-1 were interchanged, 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]<0 and ipiv[k+1]<0, then rows and columns k and -ipiv[k]-1 were interchanged and rows and columns k+1 and -ipiv[k+1]-1 were interchanged, D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

Parameters

in
lda

The leading dimension of the array A. lda>=max(1,n).

out
E

Complex array of dimension n. On exit, 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=1:n-1, E[0] is set to 0; if uplo='L': E[i]=D(i+1,i), i=0:n-2, E[n-1] is set to 0. For a 1-by-1 diagonal block D(k), the element E[k] is set to 0 in both uplo='U' and uplo='L' cases.

out
ipiv

Array of dimension n. Pivot indices (0-based). If uplo='U':

in
uplo

  • 'U': Upper triangular

  • 'L': Lower triangular

in
n

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

inout
A

Complex array of dimension (lda,n). On entry, the symmetric matrix A. On exit, contains:

out
info

  • info=0: successful exit

  • info<0: if info=-k, the k-th argument had an illegal value

  • info>0: if info=k, the matrix A is singular, because column k of the triangular part of A (upper or lower, per uplo) contains all zeros. D(k,k) is exactly zero, and superdiagonal (or subdiagonal) elements of column k of U (or L) are all zeros. 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. info only stores the first occurrence of a singularity; the factorization always completes.