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 is INTEGER


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



          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 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 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.

Definition at line 129 of file cpbequ.f.

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 is INTEGER


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



          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 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 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.

Definition at line 128 of file dpbequ.f.

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 is INTEGER


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



          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 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 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.

Definition at line 128 of file spbequ.f.

subroutine zpbequ (character uplo, integer n, integer kd, complex16, 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 is INTEGER


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



          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 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 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.

Definition at line 129 of file zpbequ.f.

Author

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