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Extract and create sparse band and diagonal matrices

`B = spdiags(A)`

[B,d] = spdiags(A)

B = spdiags(A,d)

A = spdiags(B,d,A)

A = spdiags(B,d,m,n)

The `spdiags`

function generalizes the function `diag`

.
Four different operations, distinguished by the number of input arguments,
are possible.

`B = spdiags(A)`

extracts
all nonzero diagonals from the `m`

-by-`n`

matrix `A`

. `B`

is
a `min(m,n)`

-by-`p`

matrix whose
columns are the `p`

nonzero diagonals of `A`

.

`[B,d] = spdiags(A)`

returns
a vector `d`

of length `p`

, whose
integer components specify the diagonals in `A`

.

`B = spdiags(A,d)`

extracts
the diagonals specified by `d`

.

`A = spdiags(B,d,A)`

replaces
the diagonals specified by `d`

with the columns of `B`

.
The output is sparse.

`A = spdiags(B,d,m,n)`

creates
an `m`

-by-`n`

sparse matrix by taking
the columns of `B`

and placing them along the diagonals
specified by `d`

.

In this syntax, if a column of `B`

is longer
than the diagonal it is replacing, and `m >= n`

, `spdiags`

takes
elements of super-diagonals from the lower part of the column of `B`

,
and elements of sub-diagonals from the upper part of the column of `B`

.
However, if `m < n`

, then super-diagonals are
from the upper part of the column of `B`

, and sub-diagonals
from the lower part. (See Example 5A and Example 5B, below).

The `spdiags`

function deals with three matrices,
in various combinations, as both input and output.

A | An |

B | A |

d | A vector of length |

Roughly, `A`

, `B`

, and `d`

are
related by

for k = 1:p B(:,k) = diag(A,d(k)) end

Some elements of `B`

, corresponding to positions
outside of `A`

, are not defined by these loops. They
are not referenced when `B`

is input and are set
to zero when `B`

is output.

An m-by-n matrix `A`

has m+n-1diagonals. These
are specified in the vector `d`

using indices from
-m+1 to n-1. For example, if `A`

is 5-by-6, it has
10 diagonals, which are specified in the vector `d`

using
the indices -4, -3 , ... 4, 5. The following diagram illustrates this
for a vector of all ones.

For the following matrix,

A=[0 5 0 10 0 0;... 0 0 6 0 11 0;... 3 0 0 7 0 12;... 1 4 0 0 8 0;... 0 2 5 0 0 9] A = 0 5 0 10 0 0 0 0 6 0 11 0 3 0 0 7 0 12 1 4 0 0 8 0 0 2 5 0 0 9

the command

[B, d] =spdiags(A)

returns

B = 0 0 5 10 0 0 6 11 0 3 7 12 1 4 8 0 2 5 9 0 d = -3 -2 1 3

The columns of the first output `B`

contain
the nonzero diagonals of `A`

. The second output `d`

lists
the indices of the nonzero diagonals of `A`

, as shown
in the following diagram. See How the Diagonals of A are Listed in the Vector d.

Note that the longest nonzero diagonal in `A`

is
contained in column 3 of `B`

. The other nonzero diagonals
of `A`

have extra zeros added to their corresponding
columns in `B`

, to give all columns of `B`

the
same length. For the nonzero diagonals below the main diagonal of `A`

,
extra zeros are added at the tops of columns. For the nonzero diagonals
above the main diagonal of `A`

, extra zeros are added
at the bottoms of columns. This is illustrated by the following diagram.

This example generates a sparse tridiagonal representation of
the classic second difference operator on `n`

points.

e = ones(n,1); A = spdiags([e -2*e e], -1:1, n, n)

Turn it into Wilkinson's test matrix (see `gallery`

):

A = spdiags(abs(-(n-1)/2:(n-1)/2)',0,A)

Finally, recover the three diagonals:

B = spdiags(A)

The second example is not square.

A = [11 0 13 0 0 22 0 24 0 0 33 0 41 0 0 44 0 52 0 0 0 0 63 0 0 0 0 74]

Here `m =7`

, `n = 4`

, and ```
p
= 3
```

.

The statement `[B,d] = spdiags(A)`

produces ```
d
= [-3 0 2]'
```

and

B = [41 11 0 52 22 0 63 33 13 74 44 24]

Conversely, with the above `B`

and `d`

,
the expression `spdiags(B,d,7,4)`

reproduces the
original `A`

.

This example shows how `spdiags`

creates
the diagonals when the columns of `B`

are longer
than the diagonals they are replacing.

B = repmat((1:6)',[1 7]) B = 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 4 4 4 5 5 5 5 5 5 5 6 6 6 6 6 6 6 d = [-4 -2 -1 0 3 4 5]; A = spdiags(B,d,6,6); full(A) ans = 1 0 0 4 5 6 1 2 0 0 5 6 1 2 3 0 0 6 0 2 3 4 0 0 1 0 3 4 5 0 0 2 0 4 5 6

This example illustrates the use of the syntax ```
A
= spdiags(B,d,m,n)
```

, under three conditions:

`m`

is equal to`n`

`m`

is greater than`n`

`m`

is less than`n`

The command used in this example is

A = full(spdiags(B, [-2 0 2], m, n))

where `B`

is the 5-by-3 matrix shown below.
The resulting matrix `A`

has dimensions `m`

-by-`n`

,
and has nonzero diagonals at `[-2 0 2]`

(a sub-diagonal
at `-2`

, the main diagonal, and a super-diagonal
at `2`

).

B = [1 6 11 2 7 12 3 8 13 4 9 14 5 10 15]

The first and third columns of matrix `B`

are
used to create the sub- and super-diagonals of `A`

respectively.
In all three cases though, these two outer columns of `B`

are
longer than the resulting diagonals of `A`

. Because
of this, only a part of the columns are used in `A`

.

When `m == n`

or `m > n`

, `spdiags`

takes
elements of the super-diagonal in `A`

from the lower
part of the corresponding column of `B`

, and elements
of the sub-diagonal in `A`

from the upper part of
the corresponding column of `B`

.

A = full(spdiags(B, [-2 0 2], 5, 5)) Matrix B Matrix A 1 6 11 6 0 13 0 0 2 7 12 0 7 0 14 0 3 8 13 == spdiags => 1 0 8 0 15 4 9 14 0 2 0 9 0 5 10 15 0 0 3 0 10

`A(3,1)`

, `A(4,2)`

, and `A(5,3)`

are
taken from the upper part of `B(:,1)`

.

`A(1,3)`

, `A(2,4)`

, and `A(3,5)`

are
taken from the lower part of `B(:,3)`

.

A = full(spdiags(B, [-2 0 2], 5, 4)) Matrix B Matrix A 1 6 11 6 0 13 0 2 7 12 0 7 0 14 3 8 13 == spdiags => 1 0 8 0 4 9 14 0 2 0 9 5 10 15 0 0 3 0

Same as in Part A.

When `m < n`

, `spdiags`

does
the opposite, taking elements of the super-diagonal in `A`

from
the upper part of the corresponding column of `B`

,
and elements of the sub-diagonal in `A`

from the
lower part of the corresponding column of `B`

.

A = full(spdiags(B, [-2 0 2], 4, 5)) Matrix B Matrix A 1 6 11 6 0 11 0 0 2 7 12 0 7 0 12 0 3 8 13 == spdiags => 3 0 8 0 13 4 9 14 0 4 0 9 0 5 10 15

`A(3,1)`

and `A(4,2)`

are
taken from the lower part of `B(:,1)`

.

`A(1,3)`

, `A(2,4)`

, and `A(3,5)`

are
taken from the upper part of `B(:,3)`

.

Extract the diagonals from the first part of this example back into a column format using the command

B = spdiags(A)

You can see that in each case the original columns are restored
(minus those elements that had overflowed the super- and sub-diagonals
of matrix `A`

).

Matrix A Matrix B 6 0 13 0 0 1 6 0 0 7 0 14 0 2 7 0 1 0 8 0 15 == spdiags => 3 8 13 0 2 0 9 0 0 9 14 0 0 3 0 10 0 10 15

Matrix A Matrix B 6 0 13 0 1 6 0 0 7 0 14 2 7 0 1 0 8 0 == spdiags => 3 8 13 0 2 0 9 0 9 14 0 0 3 0

Matrix A Matrix B 6 0 11 0 0 0 6 11 0 7 0 12 0 0 7 12 3 0 8 0 13 == spdiags => 3 8 13 0 4 0 9 0 4 9 0

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