sytrf#

Functions

void ssytrf(
    const char*          uplo,
    const INT            n,
          f32*  restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          f32*  restrict work,
    const INT            lwork,
          INT*           info
);
void ssytrf(const char *uplo, const INT n, f32 *restrict A, const INT lda, INT *restrict ipiv, f32 *restrict work, const INT lwork, INT *info)#

SSYTRF computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method.

The form of the factorization is 

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, 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*U**T, where U = P(n-1)*U(n-1)* … P(k)*U(k) …, i.e., U is a product of terms P(k)*U(k), where k decreases from n-1 to 0 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+1
U(k) =  (   0    I    0   )   s
        (   0    0    I   )   n-1-k
           k-s+1  s   n-1-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(0: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(0:k-2,k-1:k).

If uplo='L', then A = L*D*L**T, where L = P(0)*L(0)* … P(k)*L(k) …, i.e., L is a product of terms P(k)*L(k), where k increases from 0 to n-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 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
L(k) =  (   0    I     0   )  s
        (   0    v     I   )  n-k-s
           k     s   n-k-s

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n-1,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-1,k:k+1).

Parameters

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).

in
lda

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

out
ipiv

Array of dimension n. Pivot indices (0-based). If ipiv[k]>=0: rows/columns k and ipiv[k] were interchanged, D(k,k) is a 1-by-1 block. If ipiv[k]<0 (upper): rows/columns k-1 and -ipiv[k]-1 were interchanged, D(k-1:k,k-1:k) is a 2-by-2 block, and ipiv[k-1]=ipiv[k]. If ipiv[k]<0 (lower): rows/columns k+1 and -ipiv[k]-1 were interchanged, D(k:k+1,k:k+1) is a 2-by-2 block, and ipiv[k+1]=ipiv[k].

out
work

Array of dimension max(1,lwork). On exit, if info=0, work[0] returns the optimal lwork.

in
lwork

The length of work. lwork>=1. For best performance lwork>=n*nb, where nb is the block size returned by ILAENV. 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.

out
info

  • info=0: successful exit

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

  • info>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.

Functions

void dsytrf(
    const char*          uplo,
    const INT            n,
          f64*  restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          f64*  restrict work,
    const INT            lwork,
          INT*           info
);
void dsytrf(const char *uplo, const INT n, f64 *restrict A, const INT lda, INT *restrict ipiv, f64 *restrict work, const INT lwork, INT *info)#

DSYTRF computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method.

The form of the factorization is 

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, 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*U**T, where U = P(n-1)*U(n-1)* … P(k)*U(k) …, i.e., U is a product of terms P(k)*U(k), where k decreases from n-1 to 0 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+1
U(k) =  (   0    I    0   )   s
        (   0    0    I   )   n-1-k
           k-s+1  s   n-1-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(0: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(0:k-2,k-1:k).

If uplo='L', then A = L*D*L**T, where L = P(0)*L(0)* … P(k)*L(k) …, i.e., L is a product of terms P(k)*L(k), where k increases from 0 to n-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 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
L(k) =  (   0    I     0   )  s
        (   0    v     I   )  n-k-s
           k     s   n-k-s

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n-1,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-1,k:k+1).

Parameters

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).

in
lda

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

out
ipiv

Array of dimension n. Pivot indices (0-based). If ipiv[k]>=0: rows/columns k and ipiv[k] were interchanged, D(k,k) is a 1-by-1 block. If ipiv[k]<0 (upper): rows/columns k-1 and -ipiv[k]-1 were interchanged, D(k-1:k,k-1:k) is a 2-by-2 block, and ipiv[k-1]=ipiv[k]. If ipiv[k]<0 (lower): rows/columns k+1 and -ipiv[k]-1 were interchanged, D(k:k+1,k:k+1) is a 2-by-2 block, and ipiv[k+1]=ipiv[k].

out
work

Array of dimension max(1,lwork). On exit, if info=0, work[0] returns the optimal lwork.

in
lwork

The length of work. lwork>=1. For best performance lwork>=n*nb, where nb is the block size returned by ILAENV. 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.

out
info

  • info=0: successful exit

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

  • info>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.

Functions

void csytrf(
    const char*          uplo,
    const INT            n,
          c64*  restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          c64*  restrict work,
    const INT            lwork,
          INT*           info
);
void csytrf(const char *uplo, const INT n, c64 *restrict A, const INT lda, INT *restrict ipiv, c64 *restrict work, const INT lwork, INT *info)#

CSYTRF computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method.

The form of the factorization is 

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, 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*U**T, where U = P(n-1)*U(n-1)* … P(k)*U(k) …, i.e., U is a product of terms P(k)*U(k), where k decreases from n-1 to 0 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+1
U(k) =  (   0    I    0   )   s
        (   0    0    I   )   n-1-k
           k-s+1  s   n-1-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(0: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(0:k-2,k-1:k).

If uplo='L', then A = L*D*L**T, where L = P(0)*L(0)* … P(k)*L(k) …, i.e., L is a product of terms P(k)*L(k), where k increases from 0 to n-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 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
L(k) =  (   0    I     0   )  s
        (   0    v     I   )  n-k-s
           k     s   n-k-s

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n-1,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-1,k:k+1).

Parameters

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).

in
lda

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

out
ipiv

Array of dimension n. Pivot indices (0-based). If ipiv[k]>=0: rows/columns k and ipiv[k] were interchanged, D(k,k) is a 1-by-1 block. If ipiv[k]<0 (upper): rows/columns k-1 and -ipiv[k]-1 were interchanged, D(k-1:k,k-1:k) is a 2-by-2 block, and ipiv[k-1]=ipiv[k]. If ipiv[k]<0 (lower): rows/columns k+1 and -ipiv[k]-1 were interchanged, D(k:k+1,k:k+1) is a 2-by-2 block, and ipiv[k+1]=ipiv[k].

out
work

Complex array of dimension max(1,lwork). On exit, if info=0, work[0] returns the optimal lwork.

in
lwork

The length of work. lwork>=1. For best performance lwork>=n*nb, where nb is the block size returned by ILAENV. 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.

out
info

  • info=0: successful exit

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

  • info>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.

Functions

void zsytrf(
    const char*          uplo,
    const INT            n,
          c128* restrict A,
    const INT            lda,
          INT*  restrict ipiv,
          c128* restrict work,
    const INT            lwork,
          INT*           info
);
void zsytrf(const char *uplo, const INT n, c128 *restrict A, const INT lda, INT *restrict ipiv, c128 *restrict work, const INT lwork, INT *info)#

ZSYTRF computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method.

The form of the factorization is 

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, 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*U**T, where U = P(n-1)*U(n-1)* … P(k)*U(k) …, i.e., U is a product of terms P(k)*U(k), where k decreases from n-1 to 0 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+1
U(k) =  (   0    I    0   )   s
        (   0    0    I   )   n-1-k
           k-s+1  s   n-1-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(0: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(0:k-2,k-1:k).

If uplo='L', then A = L*D*L**T, where L = P(0)*L(0)* … P(k)*L(k) …, i.e., L is a product of terms P(k)*L(k), where k increases from 0 to n-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 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
L(k) =  (   0    I     0   )  s
        (   0    v     I   )  n-k-s
           k     s   n-k-s

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n-1,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-1,k:k+1).

Parameters

in
uplo

  • 'U': Upper triangle of A is stored

  • 'L': Lower triangle of A is stored

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, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).

in
lda

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

out
ipiv

Array of dimension n. Pivot indices (0-based). If ipiv[k]>=0: rows/columns k and ipiv[k] were interchanged, D(k,k) is a 1-by-1 block. If ipiv[k]<0 (upper): rows/columns k-1 and -ipiv[k]-1 were interchanged, D(k-1:k,k-1:k) is a 2-by-2 block, and ipiv[k-1]=ipiv[k]. If ipiv[k]<0 (lower): rows/columns k+1 and -ipiv[k]-1 were interchanged, D(k:k+1,k:k+1) is a 2-by-2 block, and ipiv[k+1]=ipiv[k].

out
work

Complex array of dimension max(1,lwork). On exit, if info=0, work[0] returns the optimal lwork.

in
lwork

The length of work. lwork>=1. For best performance lwork>=n*nb, where nb is the block size returned by ILAENV. 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.

out
info

  • info=0: successful exit

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

  • info>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.