pbequ(3)

NAME
pbequ - pbequ: equilibration

SYNOPSIS

  Functions

subroutine cpbequ (uplo, n, kd, ab, ldab, s, scond, amax, info)
CPBEQU
subroutine dpbequ (uplo, n, kd, ab, ldab, s, scond, amax, info)
DPBEQU
subroutine spbequ (uplo, n, kd, ab, ldab, s, scond, amax, info)
SPBEQU
subroutine zpbequ (uplo, n, kd, ab, ldab, s, scond, amax, info)
ZPBEQU

Detailed Description

Function Documentation

  subroutine cpbequ (character uplo, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) s, real scond, real amax, integer info)

CPBEQU

Purpose:

    !>
    !> 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
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangular of A is stored;
    !>          = 'L':  Lower triangular of A is stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    KD

    !>          KD is INTEGER
    !>          The number of superdiagonals of the matrix A if UPLO = 'U',
    !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    !>

    AB

    !>          AB is COMPLEX array, 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+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    !>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
    !>

    LDAB

    !>          LDAB is INTEGER
    !>          The leading dimension of the array A.  LDAB >= KD+1.
    !>

    S

    !>          S is REAL array, dimension (N)
    !>          If INFO = 0, S contains the scale factors for A.
    !>

    SCOND

    !>          SCOND is REAL
    !>          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.
    !>

    AMAX

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

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine dpbequ (character uplo, integer n, integer kd, double precision, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) s, double precision scond, double precision amax, integer info)

DPBEQU

Purpose:

    !>
    !> 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
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangular of A is stored;
    !>          = 'L':  Lower triangular of A is stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    KD

    !>          KD is INTEGER
    !>          The number of superdiagonals of the matrix A if UPLO = 'U',
    !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    !>

    AB

    !>          AB is DOUBLE PRECISION array, 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+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    !>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
    !>

    LDAB

    !>          LDAB is INTEGER
    !>          The leading dimension of the array A.  LDAB >= KD+1.
    !>

    S

    !>          S is DOUBLE PRECISION array, dimension (N)
    !>          If INFO = 0, S contains the scale factors for A.
    !>

    SCOND

    !>          SCOND is DOUBLE PRECISION
    !>          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.
    !>

    AMAX

    !>          AMAX is DOUBLE PRECISION
    !>          Absolute value of largest matrix element.  If AMAX is very
    !>          close to overflow or very close to underflow, the matrix
    !>          should be scaled.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine spbequ (character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) s, real scond, real amax, integer info)

SPBEQU

Purpose:

    !>
    !> 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
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangular of A is stored;
    !>          = 'L':  Lower triangular of A is stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    KD

    !>          KD is INTEGER
    !>          The number of superdiagonals of the matrix A if UPLO = 'U',
    !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    !>

    AB

    !>          AB is REAL array, 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+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    !>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
    !>

    LDAB

    !>          LDAB is INTEGER
    !>          The leading dimension of the array A.  LDAB >= KD+1.
    !>

    S

    !>          S is REAL array, dimension (N)
    !>          If INFO = 0, S contains the scale factors for A.
    !>

    SCOND

    !>          SCOND is REAL
    !>          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.
    !>

    AMAX

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

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

  subroutine zpbequ (character uplo, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, double precision, dimension( * ) s, double precision scond, double precision amax, integer info)

ZPBEQU

Purpose:

    !>
    !> 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
    UPLO

    !>          UPLO is CHARACTER*1
    !>          = 'U':  Upper triangular of A is stored;
    !>          = 'L':  Lower triangular of A is stored.
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    KD

    !>          KD is INTEGER
    !>          The number of superdiagonals of the matrix A if UPLO = 'U',
    !>          or the number of subdiagonals if UPLO = 'L'.  KD >= 0.
    !>

    AB

    !>          AB is COMPLEX*16 array, 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+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
    !>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
    !>

    LDAB

    !>          LDAB is INTEGER
    !>          The leading dimension of the array A.  LDAB >= KD+1.
    !>

    S

    !>          S is DOUBLE PRECISION array, dimension (N)
    !>          If INFO = 0, S contains the scale factors for A.
    !>

    SCOND

    !>          SCOND is DOUBLE PRECISION
    !>          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.
    !>

    AMAX

    !>          AMAX is DOUBLE PRECISION
    !>          Absolute value of largest matrix element.  If AMAX is very
    !>          close to overflow or very close to underflow, the matrix
    !>          should be scaled.
    !>

    INFO

    !>          INFO is INTEGER
    !>          = 0:  successful exit
    !>          < 0:  if INFO = -i, the i-th argument had an illegal value.
    !>          > 0:  if INFO = i, the i-th diagonal element is nonpositive.
    !>

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Author

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