.TH "gelsd" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME gelsd \- gelsd: least squares using SVD, divide and conquer .SH SYNOPSIS .br .PP .SS "Functions" .in +1c .ti -1c .RI "subroutine \fBcgelsd\fP (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info)" .br .RI "\fB CGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices\fP " .ti -1c .RI "subroutine \fBdgelsd\fP (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info)" .br .RI "\fB DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices\fP " .ti -1c .RI "subroutine \fBsgelsd\fP (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info)" .br .RI "\fB SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices\fP " .ti -1c .RI "subroutine \fBzgelsd\fP (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, rwork, iwork, info)" .br .RI "\fB ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices\fP " .in -1c .SH "Detailed Description" .PP .SH "Function Documentation" .PP .SS "subroutine cgelsd (integer m, integer n, integer nrhs, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, real, dimension( * ) s, real rcond, integer rank, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)" .PP \fB CGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices\fP .PP \fBPurpose:\fP .RS 4 .PP .nf CGELSD computes the minimum-norm solution to a real linear least squares problem: minimize 2-norm(| b - A*x |) using the singular value decomposition (SVD) of A\&. A is an M-by-N matrix which may be rank-deficient\&. Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X\&. The problem is solved in three steps: (1) Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a 'bidiagonal least squares problem' (BLS) (2) Solve the BLS using a divide and conquer approach\&. (3) Apply back all the Householder transformations to solve the original least squares problem\&. The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The number of rows of the matrix A\&. M >= 0\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix A\&. N >= 0\&. .fi .PP .br \fINRHS\fP .PP .nf NRHS is INTEGER The number of right hand sides, i\&.e\&., the number of columns of the matrices B and X\&. NRHS >= 0\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX array, dimension (LDA,N) On entry, the M-by-N matrix A\&. On exit, A has been destroyed\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .br \fIB\fP .PP .nf B is COMPLEX array, dimension (LDB,NRHS) On entry, the M-by-NRHS right hand side matrix B\&. On exit, B is overwritten by the N-by-NRHS solution matrix X\&. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of the modulus of elements n+1:m in that column\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,M,N)\&. .fi .PP .br \fIS\fP .PP .nf S is REAL array, dimension (min(M,N)) The singular values of A in decreasing order\&. The condition number of A in the 2-norm = S(1)/S(min(m,n))\&. .fi .PP .br \fIRCOND\fP .PP .nf RCOND is REAL RCOND is used to determine the effective rank of A\&. Singular values S(i) <= RCOND*S(1) are treated as zero\&. If RCOND < 0, machine precision is used instead\&. .fi .PP .br \fIRANK\fP .PP .nf RANK is INTEGER The effective rank of A, i\&.e\&., the number of singular values which are greater than RCOND*S(1)\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is COMPLEX array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&. .fi .PP .br \fILWORK\fP .PP .nf LWORK is INTEGER The dimension of the array WORK\&. LWORK must be at least 1\&. The exact minimum amount of workspace needed depends on M, N and NRHS\&. As long as LWORK is at least 2 * N + N * NRHS if M is greater than or equal to N or 2 * M + M * NRHS if M is less than N, the code will execute correctly\&. For good performance, LWORK should generally be larger\&. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the array WORK and the minimum sizes of the arrays RWORK and IWORK, and returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK is issued by XERBLA\&. .fi .PP .br \fIRWORK\fP .PP .nf RWORK is REAL array, dimension (MAX(1,LRWORK)) LRWORK >= 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ) if M is greater than or equal to N or 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS + MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ) if M is less than N, the code will execute correctly\&. SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK\&. .fi .PP .br \fIIWORK\fP .PP .nf IWORK is INTEGER array, dimension (MAX(1,LIWORK)) LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN), where MINMN = MIN( M,N )\&. On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK\&. .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: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero\&. .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 \fBContributors:\fP .RS 4 Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA .br Osni Marques, LBNL/NERSC, USA .br .RE .PP .SS "subroutine dgelsd (integer m, integer n, integer nrhs, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) s, double precision rcond, integer rank, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)" .PP \fB DGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices\fP .PP \fBPurpose:\fP .RS 4 .PP .nf DGELSD computes the minimum-norm solution to a real linear least squares problem: minimize 2-norm(| b - A*x |) using the singular value decomposition (SVD) of A\&. A is an M-by-N matrix which may be rank-deficient\&. Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X\&. The problem is solved in three steps: (1) Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a 'bidiagonal least squares problem' (BLS) (2) Solve the BLS using a divide and conquer approach\&. (3) Apply back all the Householder transformations to solve the original least squares problem\&. The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The number of rows of A\&. M >= 0\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of A\&. N >= 0\&. .fi .PP .br \fINRHS\fP .PP .nf NRHS is INTEGER The number of right hand sides, i\&.e\&., the number of columns of the matrices B and X\&. NRHS >= 0\&. .fi .PP .br \fIA\fP .PP .nf A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the M-by-N matrix A\&. On exit, A has been destroyed\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .br \fIB\fP .PP .nf B is DOUBLE PRECISION array, dimension (LDB,NRHS) On entry, the M-by-NRHS right hand side matrix B\&. On exit, B is overwritten by the N-by-NRHS solution matrix X\&. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that column\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,max(M,N))\&. .fi .PP .br \fIS\fP .PP .nf S is DOUBLE PRECISION array, dimension (min(M,N)) The singular values of A in decreasing order\&. The condition number of A in the 2-norm = S(1)/S(min(m,n))\&. .fi .PP .br \fIRCOND\fP .PP .nf RCOND is DOUBLE PRECISION RCOND is used to determine the effective rank of A\&. Singular values S(i) <= RCOND*S(1) are treated as zero\&. If RCOND < 0, machine precision is used instead\&. .fi .PP .br \fIRANK\fP .PP .nf RANK is INTEGER The effective rank of A, i\&.e\&., the number of singular values which are greater than RCOND*S(1)\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&. .fi .PP .br \fILWORK\fP .PP .nf LWORK is INTEGER The dimension of the array WORK\&. LWORK must be at least 1\&. The exact minimum amount of workspace needed depends on M, N and NRHS\&. As long as LWORK is at least 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N or 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, if M is less than N, the code will execute correctly\&. SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) For good performance, LWORK should generally be larger\&. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA\&. .fi .PP .br \fIIWORK\fP .PP .nf IWORK is INTEGER array, dimension (MAX(1,LIWORK)) LIWORK >= max(1, 3 * MINMN * NLVL + 11 * MINMN), where MINMN = MIN( M,N )\&. On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK\&. .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: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero\&. .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 \fBContributors:\fP .RS 4 Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA .br Osni Marques, LBNL/NERSC, USA .br .RE .PP .SS "subroutine sgelsd (integer m, integer n, integer nrhs, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) s, real rcond, integer rank, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)" .PP \fB SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices\fP .PP \fBPurpose:\fP .RS 4 .PP .nf SGELSD computes the minimum-norm solution to a real linear least squares problem: minimize 2-norm(| b - A*x |) using the singular value decomposition (SVD) of A\&. A is an M-by-N matrix which may be rank-deficient\&. Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X\&. The problem is solved in three steps: (1) Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a 'bidiagonal least squares problem' (BLS) (2) Solve the BLS using a divide and conquer approach\&. (3) Apply back all the Householder transformations to solve the original least squares problem\&. The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The number of rows of A\&. M >= 0\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of A\&. N >= 0\&. .fi .PP .br \fINRHS\fP .PP .nf NRHS is INTEGER The number of right hand sides, i\&.e\&., the number of columns of the matrices B and X\&. NRHS >= 0\&. .fi .PP .br \fIA\fP .PP .nf A is REAL array, dimension (LDA,N) On entry, the M-by-N matrix A\&. On exit, A has been destroyed\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .br \fIB\fP .PP .nf B is REAL array, dimension (LDB,NRHS) On entry, the M-by-NRHS right hand side matrix B\&. On exit, B is overwritten by the N-by-NRHS solution matrix X\&. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of elements n+1:m in that column\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,max(M,N))\&. .fi .PP .br \fIS\fP .PP .nf S is REAL array, dimension (min(M,N)) The singular values of A in decreasing order\&. The condition number of A in the 2-norm = S(1)/S(min(m,n))\&. .fi .PP .br \fIRCOND\fP .PP .nf RCOND is REAL RCOND is used to determine the effective rank of A\&. Singular values S(i) <= RCOND*S(1) are treated as zero\&. If RCOND < 0, machine precision is used instead\&. .fi .PP .br \fIRANK\fP .PP .nf RANK is INTEGER The effective rank of A, i\&.e\&., the number of singular values which are greater than RCOND*S(1)\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&. .fi .PP .br \fILWORK\fP .PP .nf LWORK is INTEGER The dimension of the array WORK\&. LWORK must be at least 1\&. The exact minimum amount of workspace needed depends on M, N and NRHS\&. As long as LWORK is at least 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N or 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, if M is less than N, the code will execute correctly\&. SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) For good performance, LWORK should generally be larger\&. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the array WORK and the minimum size of the array IWORK, and returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK is issued by XERBLA\&. .fi .PP .br \fIIWORK\fP .PP .nf IWORK is INTEGER array, dimension (MAX(1,LIWORK)) LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN), where MINMN = MIN( M,N )\&. On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK\&. .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: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero\&. .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 \fBContributors:\fP .RS 4 Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA .br Osni Marques, LBNL/NERSC, USA .br .RE .PP .SS "subroutine zgelsd (integer m, integer n, integer nrhs, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) s, double precision rcond, integer rank, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)" .PP \fB ZGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices\fP .PP \fBPurpose:\fP .RS 4 .PP .nf ZGELSD computes the minimum-norm solution to a real linear least squares problem: minimize 2-norm(| b - A*x |) using the singular value decomposition (SVD) of A\&. A is an M-by-N matrix which may be rank-deficient\&. Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X\&. The problem is solved in three steps: (1) Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a 'bidiagonal least squares problem' (BLS) (2) Solve the BLS using a divide and conquer approach\&. (3) Apply back all the Householder transformations to solve the original least squares problem\&. The effective rank of A is determined by treating as zero those singular values which are less than RCOND times the largest singular value\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIM\fP .PP .nf M is INTEGER The number of rows of the matrix A\&. M >= 0\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The number of columns of the matrix A\&. N >= 0\&. .fi .PP .br \fINRHS\fP .PP .nf NRHS is INTEGER The number of right hand sides, i\&.e\&., the number of columns of the matrices B and X\&. NRHS >= 0\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX*16 array, dimension (LDA,N) On entry, the M-by-N matrix A\&. On exit, A has been destroyed\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,M)\&. .fi .PP .br \fIB\fP .PP .nf B is COMPLEX*16 array, dimension (LDB,NRHS) On entry, the M-by-NRHS right hand side matrix B\&. On exit, B is overwritten by the N-by-NRHS solution matrix X\&. If m >= n and RANK = n, the residual sum-of-squares for the solution in the i-th column is given by the sum of squares of the modulus of elements n+1:m in that column\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,M,N)\&. .fi .PP .br \fIS\fP .PP .nf S is DOUBLE PRECISION array, dimension (min(M,N)) The singular values of A in decreasing order\&. The condition number of A in the 2-norm = S(1)/S(min(m,n))\&. .fi .PP .br \fIRCOND\fP .PP .nf RCOND is DOUBLE PRECISION RCOND is used to determine the effective rank of A\&. Singular values S(i) <= RCOND*S(1) are treated as zero\&. If RCOND < 0, machine precision is used instead\&. .fi .PP .br \fIRANK\fP .PP .nf RANK is INTEGER The effective rank of A, i\&.e\&., the number of singular values which are greater than RCOND*S(1)\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&. .fi .PP .br \fILWORK\fP .PP .nf LWORK is INTEGER The dimension of the array WORK\&. LWORK must be at least 1\&. The exact minimum amount of workspace needed depends on M, N and NRHS\&. As long as LWORK is at least 2*N + N*NRHS if M is greater than or equal to N or 2*M + M*NRHS if M is less than N, the code will execute correctly\&. For good performance, LWORK should generally be larger\&. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the array WORK and the minimum sizes of the arrays RWORK and IWORK, and returns these values as the first entries of the WORK, RWORK and IWORK arrays, and no error message related to LWORK is issued by XERBLA\&. .fi .PP .br \fIRWORK\fP .PP .nf RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK)) LRWORK >= 10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ) if M is greater than or equal to N or 10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS + MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS ) if M is less than N, the code will execute correctly\&. SMLSIZ is returned by ILAENV and is equal to the maximum size of the subproblems at the bottom of the computation tree (usually about 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 ) On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK\&. .fi .PP .br \fIIWORK\fP .PP .nf IWORK is INTEGER array, dimension (MAX(1,LIWORK)) LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN), where MINMN = MIN( M,N )\&. On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK\&. .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: the algorithm for computing the SVD failed to converge; if INFO = i, i off-diagonal elements of an intermediate bidiagonal form did not converge to zero\&. .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 \fBContributors:\fP .RS 4 Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA .br Osni Marques, LBNL/NERSC, USA .br .RE .PP .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.