Number of rows of

A

.

Number of columns of

A

.

On entry,

dist

specifies the type of distribution to be used to generate a random matrix .

If

dist

=

'U'

, real and imaginary parts are independent uniform( 0, 1 ).

If

dist

=

'S'

, real and imaginary parts are independent uniform( -1, 1 ).

If

dist

=

'N'

, real and imaginary parts are independent normal( 0, 1 ).

If

dist

=

'D'

, distribution is uniform on interior of unit disk.

Array, size 4.

On entry,

iseed

specifies the seed of the random number generator. They should lie between 0 and 4095 inclusive, and

iseed

[3]

should be odd. The random number generator uses a linear congruential sequence limited to small integers, and so should produce machine independent random numbers.

If

sym

=

'S'

, generated matrix is symmetric.

If

sym

=

'H'

, generated matrix is Hermitian.

If

sym

=

'N'

, generated matrix is nonsymmetric.

On entry this array specifies the diagonal entries of the diagonal of

A

.

d

may either be specified on entry, or set according to

mode

and

cond

as described below. If the matrix is Hermitian, the real part of

d

is taken. May be changed on exit if

mode

is nonzero.

On entry describes how

d

is to be used:

mode

= 0

means use

d

as input.

mode

= 1

sets

d

[0]

=1

and

d

[1:

n

- 1]

=1.0/

cond

.

mode

= 2

sets

d

[0:

n

- 2]

=1 and

d

[

n

- 1]

=1.0/

cond

.

mode

= 3

sets

d

[

i

- 1]

=

cond

**(-(

i

-1)/(

n

-1))

.

mode

= 4

sets

d

[

i

- 1]

=1 - (

i

-1)/(

n

-1)*(1 - 1/

cond

)

.

mode

= 5

sets

d

to random numbers in the range

( 1/

cond

, 1 )

such that their logarithms are uniformly distributed.

mode

= 6

sets

d

to random numbers from same distribution as the rest of the matrix.

mode

< 0

has the same meaning as

abs(

mode

)

, except that 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.

On entry, used as described under

mode

above. If used,

cond

must be

≥

1.

If

mode

is not -6, 0, or 6, the diagonal is scaled by

dmax

/ max(abs(

d

[

i

]

))

, so that maximum absolute entry of diagonal is

abs(

dmax

)

. If

dmax

is complex (or zero), the diagonal is scaled by a complex number (or zero).

If

mode

is not -6, 0, or 6, specifies the sign of the diagonal as follows:

For

slatmr

and

dlatmr

, if

rsign

=

'T'

, diagonal entries are multiplied 1 or -1 with a probability of 0.5.

For

clatmr

and

zlatmr

, if

rsign

=

'T'

, diagonal entries are multiplied by a random complex number uniformly distributed with absolute value 1.

If

r

sign

=

'F'

, diagonal entries are unchanged.

Specifies grading of matrix as follows:

If

grade

=

'N'

, there is no grading

If

grade

=

'L'

, matrix is premultiplied by diag(

dl

) (only if matrix is nonsymmetric)

If

grade

=

'R'

, matrix is postmultiplied by diag(

dr

) (only if matrix is nonsymmetric)

If

grade

=

'B'

, matrix is premultiplied by diag(

dl

) and postmultiplied by diag(

dr

) (only if matrix is nonsymmetric)

If

grade

=

'H'

, matrix is premultiplied by diag(

dl

) and postmultiplied by diag(

conjg(

dl

)

) (only if matrix is Hermitian or nonsymmetric)

If

grade

=

'S'

, matrix is premultiplied by diag(

dl

) and postmultiplied by diag(

dl

) (only if matrix is symmetric or nonsymmetric)

If

grade

=

'E'

, matrix is premultiplied by diag(

dl

) and postmultiplied by

inv( diag(

dl

) )

(only if matrix is nonsymmetric)

Array, size (

m

).

If

model

= 0

, then on entry this array specifies the diagonal entries of a diagonal matrix used as described under grade above.

If

model

is not zero, then

dl

is set according to

model

and

condl

, analogous to the way

D

is set according to

mode

and

cond

(except there is no

dmax

parameter for

dl

).

If

grade

=

'E'

, then

dl

cannot have zero entries.

Not referenced if

grade

=

'N'

or

'R'

. Changed on exit.

This specifies how the diagonal array

dl

is computed, just as

mode

specifies how

D

is computed.

When

model

is not zero, this specifies the condition number of the computed

dl

.

If

moder

= 0

, then on entry this array specifies the diagonal entries of a diagonal matrix used as described under

grade

above.

If

moder

is not zero, then

dr

is set according to

moder

and

condr

, analogous to the way

d

is set according to

mode

and

cond

(except there is no

dmax

parameter for

dr

).

Not referenced if

grade

=

'N'

,

'L'

,

'H'

'S'

or

'E'

.

This specifies how the diagonal array

dr

is to be computed, just as mode specifies how

d

is to be computed.

When

moder

is not zero, this specifies the condition number of the computed

dr

.

On entry specifies pivoting permutations as follows:

If

pivtng

=

'N'

or

' '

: no pivoting permutation.

If

pivtng

=

'L'

: left or row pivoting (matrix must be nonsymmetric).

If

pivtng

=

'R'

: right or column pivoting (matrix must be nonsymmetric).

If

pivtng

=

'B'

or

'F'

: both or full pivoting, i.e., on both sides. In this case,

m

must equal

n

.

If two calls to

?latmr

both have full bandwidth (

kl

=

m

- 1

and

ku

=

n

-1

), and differ only in the

pivtng

and

pack

parameters, then the matrices generated differs only in the order of the rows and columns, and otherwise contain the same data. This consistency cannot be maintained with less than full bandwidth.

Array, size (

n

or

m

) This array specifies the permutation used. After the basic matrix is generated, the rows, columns, or both are permuted.

If row pivoting is selected,

?latmr

starts with the last row and interchanges row

m

and row

ipivot

[

m

- 1]

, then moves to the next-to-last row, interchanging rows

[

m

- 2]

and row

ipivot

[

m

- 2]

, and so on. In terms of "2-cycles", the permutation is

(1

ipivot

[0]) (2

ipivot

[1]) ... (

m

ipivot

[

m

- 1])

where the rightmost cycle is applied first. This is the inverse of the effect of pivoting in LINPACK. The idea is that factoring (with pivoting) an identity matrix which has been inverse-pivoted in this way should result in a pivot vector identical to

ipivot

. Not referenced if

pivtng

=

'N'

.

On entry, specifies the sparsity of the matrix if a sparse matrix is to be generated.

sparse

should lie between 0 and 1. To generate a sparse matrix, for each matrix entry a uniform ( 0, 1 ) random number

x

is generated and compared to sparse; if

x

is larger the matrix entry is unchanged and if

x

is smaller the entry is set to zero. Thus on the average a fraction

sparse

of the entries is set to zero.

On entry, specifies the lower bandwidth of the matrix. For example,

kl

= 0

implies upper triangular,

kl

= 1

implies upper Hessenberg, and

kl

at least

m

-1 implies the matrix is not banded. Must equal

ku

<