.TH "pbequ" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME pbequ \- pbequ: equilibration .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBcpbequ\fP (uplo, n, kd, ab, ldab, s, scond, amax, info)" .br .RI "\fBCPBEQU\fP " .ti -1c .RI "subroutine \fBdpbequ\fP (uplo, n, kd, ab, ldab, s, scond, amax, info)" .br .RI "\fBDPBEQU\fP " .ti -1c .RI "subroutine \fBspbequ\fP (uplo, n, kd, ab, ldab, s, scond, amax, info)" .br .RI "\fBSPBEQU\fP " .ti -1c .RI "subroutine \fBzpbequ\fP (uplo, n, kd, ab, ldab, s, scond, amax, info)" .br .RI "\fBZPBEQU\fP " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine cpbequ (character uplo, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) s, real scond, real amax, integer info)" .PP \fBCPBEQU\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': Upper triangular of A is stored; = 'L': Lower triangular of A is stored\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIKD\fP .PP .nf KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'\&. KD >= 0\&. .fi .PP .br \fIAB\fP .PP .nf 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)\&. .fi .PP .br \fILDAB\fP .PP .nf LDAB is INTEGER The leading dimension of the array A\&. LDAB >= KD+1\&. .fi .PP .br \fIS\fP .PP .nf S is REAL array, dimension (N) If INFO = 0, S contains the scale factors for A\&. .fi .PP .br \fISCOND\fP .PP .nf 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\&. .fi .PP .br \fIAMAX\fP .PP .nf 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\&. .fi .PP .br \fIINFO\fP .PP .nf 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\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "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)" .PP \fBDPBEQU\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': Upper triangular of A is stored; = 'L': Lower triangular of A is stored\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIKD\fP .PP .nf KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'\&. KD >= 0\&. .fi .PP .br \fIAB\fP .PP .nf 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)\&. .fi .PP .br \fILDAB\fP .PP .nf LDAB is INTEGER The leading dimension of the array A\&. LDAB >= KD+1\&. .fi .PP .br \fIS\fP .PP .nf S is DOUBLE PRECISION array, dimension (N) If INFO = 0, S contains the scale factors for A\&. .fi .PP .br \fISCOND\fP .PP .nf 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\&. .fi .PP .br \fIAMAX\fP .PP .nf 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\&. .fi .PP .br \fIINFO\fP .PP .nf 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\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "subroutine spbequ (character uplo, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) s, real scond, real amax, integer info)" .PP \fBSPBEQU\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': Upper triangular of A is stored; = 'L': Lower triangular of A is stored\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIKD\fP .PP .nf KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'\&. KD >= 0\&. .fi .PP .br \fIAB\fP .PP .nf 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)\&. .fi .PP .br \fILDAB\fP .PP .nf LDAB is INTEGER The leading dimension of the array A\&. LDAB >= KD+1\&. .fi .PP .br \fIS\fP .PP .nf S is REAL array, dimension (N) If INFO = 0, S contains the scale factors for A\&. .fi .PP .br \fISCOND\fP .PP .nf 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\&. .fi .PP .br \fIAMAX\fP .PP .nf 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\&. .fi .PP .br \fIINFO\fP .PP .nf 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\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SS "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)" .PP \fBZPBEQU\fP .PP \fBPurpose:\fP .RS 4 .PP .nf 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIUPLO\fP .PP .nf UPLO is CHARACTER*1 = 'U': Upper triangular of A is stored; = 'L': Lower triangular of A is stored\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrix A\&. N >= 0\&. .fi .PP .br \fIKD\fP .PP .nf KD is INTEGER The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'\&. KD >= 0\&. .fi .PP .br \fIAB\fP .PP .nf 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)\&. .fi .PP .br \fILDAB\fP .PP .nf LDAB is INTEGER The leading dimension of the array A\&. LDAB >= KD+1\&. .fi .PP .br \fIS\fP .PP .nf S is DOUBLE PRECISION array, dimension (N) If INFO = 0, S contains the scale factors for A\&. .fi .PP .br \fISCOND\fP .PP .nf 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\&. .fi .PP .br \fIAMAX\fP .PP .nf 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\&. .fi .PP .br \fIINFO\fP .PP .nf 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\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.