pbequ

spbequ

Functions

void spbequ(
    const char*          uplo,
    const INT            n,
    const INT            kd,
    const f32*  restrict AB,
    const INT            ldab,
          f32*  restrict S,
          f32*           scond,
          f32*           amax,
          INT*           info
);

void spbequ(const char *uplo, const INT n, const INT kd, const f32 *restrict AB, const INT ldab, f32 *restrict S, f32 *scond, f32 *amax, INT *info)

SPBEQU computes row and column scalings intended to equilibrate a symmetric positive definite band matrix A and reduce its condition number (with respect to the two-norm).

S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.

Parameters

in

uplo

• 'U': Upper triangular of A is stored

• 'L': Lower triangular of A is stored

in

n

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

in

kd

The number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

in

AB

Array of dimension (ldab, n). The upper or lower triangle of the symmetric band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if uplo='U', AB[kd+i-j + j*ldab] = A(i,j) for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j + j*ldab] = A(i,j) for j<=i<=min(n-1,j+kd).

in

ldab

The leading dimension of the array AB. ldab>=kd+1.

out

S

Array of dimension n. If info=0, S contains the scale factors for A.

out

scond

If info=0, S contains the ratio of the smallest S(i) to the largest S(i). If scond>=0.1 and amax is neither too large nor too small, it is not worth scaling by S.

out

amax

Absolute value of largest matrix element. If amax is very close to overflow or very close to underflow, the matrix should be scaled.

out

info

• info=0: successful exit

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

• info>0: if info=i, the i-th diagonal element is nonpositive.

dpbequ

Functions

void dpbequ(
    const char*          uplo,
    const INT            n,
    const INT            kd,
    const f64*  restrict AB,
    const INT            ldab,
          f64*  restrict S,
          f64*           scond,
          f64*           amax,
          INT*           info
);

void dpbequ(const char *uplo, const INT n, const INT kd, const f64 *restrict AB, const INT ldab, f64 *restrict S, f64 *scond, f64 *amax, INT *info)

DPBEQU computes row and column scalings intended to equilibrate a symmetric positive definite band matrix A and reduce its condition number (with respect to the two-norm).

S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.

Parameters

in

uplo

• 'U': Upper triangular of A is stored

• 'L': Lower triangular of A is stored

in

n

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

in

kd

The number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

in

AB

Array of dimension (ldab, n). The upper or lower triangle of the symmetric band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if uplo='U', AB[kd+i-j + j*ldab] = A(i,j) for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j + j*ldab] = A(i,j) for j<=i<=min(n-1,j+kd).

in

ldab

The leading dimension of the array AB. ldab>=kd+1.

out

S

Array of dimension n. If info=0, S contains the scale factors for A.

out

scond

If info=0, S contains the ratio of the smallest S(i) to the largest S(i). If scond>=0.1 and amax is neither too large nor too small, it is not worth scaling by S.

out

amax

Absolute value of largest matrix element. If amax is very close to overflow or very close to underflow, the matrix should be scaled.

out

info

• info=0: successful exit

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

• info>0: if info=i, the i-th diagonal element is nonpositive.

cpbequ

Functions

void cpbequ(
    const char*          uplo,
    const INT            n,
    const INT            kd,
    const c64*  restrict AB,
    const INT            ldab,
          f32*  restrict S,
          f32*           scond,
          f32*           amax,
          INT*           info
);

void cpbequ(const char *uplo, const INT n, const INT kd, const c64 *restrict AB, const INT ldab, f32 *restrict S, f32 *scond, f32 *amax, INT *info)

CPBEQU computes row and column scalings intended to equilibrate a Hermitian positive definite band matrix A and reduce its condition number (with respect to the two-norm).

S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.

Parameters

in

uplo

• 'U': Upper triangular of A is stored

• 'L': Lower triangular of A is stored

in

n

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

in

kd

The number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

in

AB

Array of dimension (ldab, n). The upper or lower triangle of the Hermitian band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if uplo='U', AB[kd+i-j + j*ldab] = A(i,j) for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j + j*ldab] = A(i,j) for j<=i<=min(n-1,j+kd).

in

ldab

The leading dimension of the array AB. ldab>=kd+1.

out

S

Array of dimension n. If info=0, S contains the scale factors for A.

out

scond

If info=0, S contains the ratio of the smallest S(i) to the largest S(i). If scond>=0.1 and amax is neither too large nor too small, it is not worth scaling by S.

out

amax

Absolute value of largest matrix element. If amax is very close to overflow or very close to underflow, the matrix should be scaled.

out

info

• info=0: successful exit

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

• info>0: if info=i, the i-th diagonal element is nonpositive.

zpbequ

Functions

void zpbequ(
    const char*          uplo,
    const INT            n,
    const INT            kd,
    const c128* restrict AB,
    const INT            ldab,
          f64*  restrict S,
          f64*           scond,
          f64*           amax,
          INT*           info
);

void zpbequ(const char *uplo, const INT n, const INT kd, const c128 *restrict AB, const INT ldab, f64 *restrict S, f64 *scond, f64 *amax, INT *info)

ZPBEQU computes row and column scalings intended to equilibrate a Hermitian positive definite band matrix A and reduce its condition number (with respect to the two-norm).

S contains the scale factors, S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal. This choice of S puts the condition number of B within a factor N of the smallest possible condition number over all possible diagonal scalings.

Parameters

in

uplo

• 'U': Upper triangular of A is stored

• 'L': Lower triangular of A is stored

in

n

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

in

kd

The number of superdiagonals of the matrix A if uplo='U', or the number of subdiagonals if uplo='L'. kd>=0.

in

AB

Array of dimension (ldab, n). The upper or lower triangle of the Hermitian band matrix A, stored in the first kd+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if uplo='U', AB[kd+i-j + j*ldab] = A(i,j) for max(0,j-kd)<=i<=j; if uplo='L', AB[i-j + j*ldab] = A(i,j) for j<=i<=min(n-1,j+kd).

in

ldab

The leading dimension of the array AB. ldab>=kd+1.

out

S

Array of dimension n. If info=0, S contains the scale factors for A.

out

scond

If info=0, S contains the ratio of the smallest S(i) to the largest S(i). If scond>=0.1 and amax is neither too large nor too small, it is not worth scaling by S.

out

amax

Absolute value of largest matrix element. If amax is very close to overflow or very close to underflow, the matrix should be scaled.

out

info

• info=0: successful exit

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

• info>0: if info=i, the i-th diagonal element is nonpositive.