sptrf#

ssptrf

Functions

void ssptrf(
    const char*          uplo,
    const INT            n,
          f32*  restrict AP,
          INT*  restrict ipiv,
          INT*           info
);

void ssptrf(const char *uplo, const INT n, f32 *restrict AP, INT *restrict ipiv, INT *info)#

SSPTRF computes the factorization of a real symmetric matrix A stored in packed format 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, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

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

AP

Array of dimension n*(n+1)/2. On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if uplo='U', AP[i + j*(j+1)/2] = A(i,j) for 0<=i<=j; if uplo='L', AP[i + j*(2*n-j-1)/2] = A(i,j) for j<=i<=n-1. On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L, stored as a packed triangular matrix overwriting A (see below for further details).

out

ipiv

Array of dimension n. Pivot indices (0-based). 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]-1 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]-1 were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

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.

dsptrf

Functions

void dsptrf(
    const char*          uplo,
    const INT            n,
          f64*  restrict AP,
          INT*  restrict ipiv,
          INT*           info
);

void dsptrf(const char *uplo, const INT n, f64 *restrict AP, INT *restrict ipiv, INT *info)#

DSPTRF computes the factorization of a real symmetric matrix A stored in packed format 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, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

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

AP

Array of dimension n*(n+1)/2. On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if uplo='U', AP[i + j*(j+1)/2] = A(i,j) for 0<=i<=j; if uplo='L', AP[i + j*(2*n-j-1)/2] = A(i,j) for j<=i<=n-1. On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L, stored as a packed triangular matrix overwriting A (see below for further details).

out

ipiv

Array of dimension n. Pivot indices (0-based). 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]-1 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]-1 were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

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.

csptrf

Functions

void csptrf(
    const char*          uplo,
    const INT            n,
          c64*  restrict AP,
          INT*  restrict ipiv,
          INT*           info
);

void csptrf(const char *uplo, const INT n, c64 *restrict AP, INT *restrict ipiv, INT *info)#

CSPTRF computes the factorization of a complex symmetric matrix A stored in packed format 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, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

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

AP

Complex array of dimension n*(n+1)/2. On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if uplo='U', AP[i + j*(j+1)/2] = A(i,j) for 0<=i<=j; if uplo='L', AP[i + j*(2*n-j-1)/2] = A(i,j) for j<=i<=n-1. On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L, stored as a packed triangular matrix overwriting A (see below for further details).

out

ipiv

Array of dimension n. Pivot indices (0-based). 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]-1 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]-1 were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

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.

zsptrf

Functions

void zsptrf(
    const char*          uplo,
    const INT            n,
          c128* restrict AP,
          INT*  restrict ipiv,
          INT*           info
);

void zsptrf(const char *uplo, const INT n, c128 *restrict AP, INT *restrict ipiv, INT *info)#

ZSPTRF computes the factorization of a complex symmetric matrix A stored in packed format 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, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

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

AP

Complex array of dimension n*(n+1)/2. On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if uplo='U', AP[i + j*(j+1)/2] = A(i,j) for 0<=i<=j; if uplo='L', AP[i + j*(2*n-j-1)/2] = A(i,j) for j<=i<=n-1. On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L, stored as a packed triangular matrix overwriting A (see below for further details).

out

ipiv

Array of dimension n. Pivot indices (0-based). 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]-1 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]-1 were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

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.