dlatme dlatmr dlatms dlatmt

dlatme() void dlatme ( int  n, char  dist, int  iseed[], double  d[], int  mode, double  cond, double  dmax, char  ei[], char  rsign, char  upper, char  sim, double  ds[], int  modes, double  conds, int  kl, int  ku, double  anorm, int  lda, double  a[], double  work[], int *  info  ) Generates random non-symmetric square matrices with specified eigenvalues Purpose This routine generates random non-symmetric square matrices with specified eigenvalues. dlatme operates by applying the following sequence of operations: Set the diagonal to d, where d may be input or computed according to mode, cond, dmax, and rsign as described below. If complex conjugate pairs are desired (mode = 0 and ei[0] = 'R', or mode = 5), certain pairs of adjacent elements of d are interpreted as the real and complex parts of a complex conjugate pair; A thus becomes block diagonal, with 1 x 1 and 2 x 2 blocks. If upper = 'T', the upper triangle of A is set to random values out of distribution dist. If sim = 'T', A is multiplied on the left by a random matrix X, whose singular values are specified by ds, modes, and conds, and on the right by X inverse. If kl < n-1, the lower bandwidth is reduced to kl using Householder transformations. If ku < n-1, the upper bandwidth is reduced to ku. If anorm is not negative, the matrix is scaled to have maximum-element-norm anorm. (Note: Since the matrix cannot be reduced beyond Hessenberg form, no packing options are available.) Parameters [in] n Number of columns (or rows) of A. (n >= 0) [in] dist The type of distribution to be used to generate the random eigenvalues/singular values. = 'U': Uniform distribution on (0, 1). = 'S': Uniform distribution on (-1, 1). = 'N': Normal distribution on (0, 1). [in,out] iseed[] Array iseed[liseed] (liseed >= 4) [in] The seed of the random number generator. (0 <= iseed[i] <= 4095, and iseed[3] should be odd) [out] The values of iseed[] are changed on exit, and can be used in the next call to continue the same random number sequence. [in,out] d[] Array d[ld] (ld >= n) This array is used to specify the eigenvalues of A. [in] If mode = 0, d[] is assumed to contain the input eigenvalues (but see the description of ei[]). [out] If mode != 0, the eigenvalues will be computed according to mode, cond, dmax and rsign, and placed in d[]. [in] mode Describes how the eigenvalues are to be specified. = 0: Use d[] (with ei[]) as input. = 1: d[0] = 1 and d[1] to d[n-1] = 1/cond. = 2: d[0] to d[n-2] = 1 and d[n-1] = 1/cond. = 3: d[i] = cond^(-i/(n-1)) (i = 0 to n-1). = 4: d[i] = 1-i/(n-1)*(1-1/cond) (i = 0 to n-1). = 5: Random numbers in the range (1/cond, 1) such that their logarithms are uniformly distributed. Each odd-even pair of elements will be either used as two real eigenvalues or as the real and imaginary part of a complex conjugate pair of eigenvalues; the choice of which is done is random, with 50-50 probability, for each pair. = 6: Random numbers from same distribution as the rest of the matrix. < 0: Same as |mode| but the order of the elements of d[] is reversed. Thus if mode is between 1 and 4, D has entries ranging from 1 to 1/cond, if between -1 and -4, d has entries ranging from 1/cond to 1. [in] cond This is used as described under mode above. (cond >= 1) [in] dmax If mode != -6, 0 nor 6, the contents of d[], as computed according to mode and cond, will be scaled by dmax/max(abs(d[i])). Note that dmax need not be positive: if dmax is negative (or zero), d[] will be scaled by a negative number (or zero). [in] ei[] Array ei[lei] (lei >= n) To be referenced only when mode = 0. ei[0] = ' ':   All input eigenvalues in d[] will be real. ei[0] = 'R':   ei[] specifies which elements of d[] (on input) are real eigenvalues and which are the real and imaginary parts of a complex conjugate pair of eigenvalues.   If ei[j-1] = 'R' and ei[j] = 'I', the j-th eigenvalue is d[j-1] + d[j]i, and the (j+1)-th eigenvalue is the conjugate complex, d[j-1] - d[j]i.   If ei[j-1] = ei[j] = 'R', the j-th eigenvalue is d[j-1] (real).   Two adjacent elements of ei[] both should not have the value 'I'. ei[0] should not be 'I'. [in] rsign To be used except for mode = -6, 0 and 6. = 'T': The elements of d[], as computed according to mode and cond, will be multiplied by a random sign (+1 or -1). = 'F': They will not be multiplied. [in] upper = 'T': The elements of A above the diagonal (and above the 2 x 2 diagonal blocks, if A has complex eigenvalues) will be set to random numbers out of dist. = 'F': They will not be set. [in] sim = 'T': A will be operated on by a "similarity transform", i.e. multiplied on the left by a matrix X and on the right by X inverse. X = U*S*V, where U and V are random unitary matrices and S is a (diagonal) matrix of singular values specified by ds[], modes and conds. = 'F': A will not be trandformed. [in,out] ds[] Array ds[lds] (lds >= n) This array is used to specify the singular values of X, in the same way that d[] specifies the eigenvalues of A. If modes = 0, ds[] contains the singular values, which may not be zero. [in] modes [in] conds Same as mode and cond, but for specifying the diagonal of S. modes = -6 and 6 are not allowed (since they would result in randomly ill-conditioned eigenvalues.) [in] kl The lower bandwidth of the matrix. (kl >= 1) kl = 1 specifies upper Hessenberg form. If kl >= n-1, then A will have full lower bandwidth. [in] ku The upper bandwidth of the matrix. (ku >= 1) If ku >= n-1, then A will have full upper bandwidth; if ku >= n-1 and kl >= n-1, then A will be dense. Only one of ku and kl may be less than n-1. [in] anorm If anorm >= 0, then A will be scaled by a non-negative real number to make the maximum-element-norm of A to be anorm. [in] lda Leading dimension of two dimensional array a[][]. (lda >= n) [out] a[][] Array a[la][lda] (la >= n) The desired test matrix. [out] work[] Array work[lwork] (lwork >= 3*n) Work array. [out] info = 0: Successful exit = -1: The argument n had an illegal value (n < 0) = -2: The argument dist had an illegal value (dist != 'U', 'S' nor 'N') = -5: The argument mode had an illegal value (mode < -6 or mode > 6) = -6: The argument cond had an illegal value (cond < 1) = -8: The argument ei had an illegal value (invalid ei[] value(s)) = -9: The argument rsign had an illegal value (rsign != 'T' nor 'F') = -10: The argument upper had an illegal value (upper != 'T' nor 'F') = -11: The argument sim had an illegal value (sim != 'T' nor 'F') = -12: The argument ds had an illegal value (invalid ds[] value(s)) = -13: The argument modes had an illegal value (modes < -5 or modes > 5) = -14: The argument conds had an illegal value (conds < 1) = -15: The argument kl had an illegal value (kl < 1) = -16: The argument ku had an illegal value (ku > 1 or both ku and kl < n-1) = -18: The argument lda had an illegal value (lda < n) = 1: Error return from dlatm1 (computing d[]) = 2: Cannot scale to dmax (max. singular value is 0) = 3: Error return from dlatm1 (computing ds[]) = 4: Error return from dlarge = 5: Zero singular value from dlatm1 Reference LAPACK