# spdiags

Extract nonzero diagonals and create sparse band and diagonal matrices

## Description

example

Bout = spdiags(A) extracts the nonzero diagonals from m-by-n matrix A and returns them as the columns in min(m,n)-by-p matrix Bout, where p is the number of nonzero diagonals.

example

[Bout,id] = spdiags(A) also returns the diagonal numbers id for the nonzero diagonals in A. The size of Bout is min(m,n)-by-length(id).

example

Bout = spdiags(A,d) extracts the diagonals in A specified by d and returns them as the columns of min(m,n)-by-length(d) matrix Bout.

example

S = spdiags(Bin,d,m,n) creates an m-by-n sparse matrix S by taking the columns of Bin and placing them along the diagonals specified by d.

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S = spdiags(Bin,d,A) replaces the diagonals in A specified by d with the columns of Bin.

## Examples

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Create a tridiagonal matrix using three vectors, change some of the matrix diagonals, and then extract the diagonals.

Create a 9-by-1 vector of ones, and then create a tridiagonal matrix using the vector. View the matrix elements.

n = 9;
e = ones(n,1);
A = spdiags([e -2*e e],-1:1,n,n);
full(A)
ans = 9×9

-2     1     0     0     0     0     0     0     0
1    -2     1     0     0     0     0     0     0
0     1    -2     1     0     0     0     0     0
0     0     1    -2     1     0     0     0     0
0     0     0     1    -2     1     0     0     0
0     0     0     0     1    -2     1     0     0
0     0     0     0     0     1    -2     1     0
0     0     0     0     0     0     1    -2     1
0     0     0     0     0     0     0     1    -2

Change the values on the main (d = 0) diagonal of A.

Bin = abs(-(n-1)/2:(n-1)/2)';
d = 0;
A = spdiags(Bin,d,A);
full(A)
ans = 9×9

4     1     0     0     0     0     0     0     0
1     3     1     0     0     0     0     0     0
0     1     2     1     0     0     0     0     0
0     0     1     1     1     0     0     0     0
0     0     0     1     0     1     0     0     0
0     0     0     0     1     1     1     0     0
0     0     0     0     0     1     2     1     0
0     0     0     0     0     0     1     3     1
0     0     0     0     0     0     0     1     4

Finally, recover the diagonals of A as the columns in a matrix.

Bout = spdiags(A);
full(Bout)
ans = 9×3

1     4     0
1     3     1
1     2     1
1     1     1
1     0     1
1     1     1
1     2     1
1     3     1
0     4     1

Extract the nonzero diagonals of a matrix and examine the output format of spdiags.

Create a matrix containing a mix of nonzero and zero diagonals.

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];

Extract the nonzero diagonals from the matrix. Specify two outputs to return the diagonal numbers.

[Bout,d] = spdiags(A)
Bout = 5×4

0     0     5    10
0     0     6    11
0     3     7    12
1     4     8     0
2     5     9     0

d = 4×1

-3
-2
1
3

The columns of the first output Bout contain the nonzero diagonals of A. The second output d lists the indices of the nonzero diagonals of A. The longest nonzero diagonal in A is in column 3 of Bout. To give all columns of Bout the same length, the other nonzero diagonals of A have extra zeros added to their corresponding columns in Bout. For m-by-n matrices with m < n, the rules are:

• For nonzero diagonals below the main diagonal of A, extra zeros are added at the tops of columns (as in the first two columns of Bout).

• For nonzero diagonals above the main diagonal of A, extra zeros are added at the bottoms of columns (as in the last column of Bout).

spdiags pads Bout with zeros in this manner even if the longest diagonal is not returned in Bout.

Create a 5-by-5 random matrix.

A = randi(10,5,5)
A = 5×5

9     1     2     2     7
10     3    10     5     1
2     6    10    10     9
10    10     5     8    10
7    10     9    10     7

Extract the main diagonal, and the first diagonals above and below it.

d = [-1 0 1];
Bout = spdiags(A,d)
Bout = 5×3

10     9     0
6     3     1
5    10    10
10     8    10
0     7    10

Try to extract the fifth super-diagonal (d = 5). Because A has only four super-diagonals, spdiags returns the diagonal as all zeros of the same length as the main (d = 0) diagonal.

B5 = spdiags(A,5)
B5 = 5×1

0
0
0
0
0

Examine how spdiags creates diagonals when the columns of the input matrix are longer than the diagonals they are replacing.

Create a 6-by-7 matrix of the numbers 1 through 6.

Bin = repmat((1:6)',[1 7])
Bin = 6×7

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

Use spdiags to create a square 6-by-6 matrix with several of the columns of Bin as diagonals. Because some of the diagonals only have one or two elements, there is a mismatch in sizes between the columns in Bin and diagonals in A.

d = [-4 -2 -1 0 3 4 5];
A = spdiags(Bin,d,6,6);
full(A)
ans = 6×6

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

Each of the columns in Bin has six elements, but only the main diagonal in A has six elements. Therefore, all the other diagonals in A truncate the elements in the columns of Bin so that they fit on the selected diagonals:

The way spdiags truncates the diagonals depends on the size of m-by-n matrix A. When $\mathit{m}\ge \mathit{n}$, the behavior is as pictured above:

• Diagonals below the main diagonal take elements from the tops of the columns first.

• Diagonals above the main diagonal take elements from the bottoms of columns first.

This behavior reverses when $\mathit{m}<\mathit{n}$:

A = spdiags(Bin,d,5,6);
full(A)
ans = 5×6

1     0     0     1     1     1
2     2     0     0     2     2
3     3     3     0     0     3
0     4     4     4     0     0
5     0     5     5     5     0

• Diagonals above the main diagonal take elements from the tops of the columns first.

• Diagonals below the main diagonal take elements from the bottoms of columns first.

## Input Arguments

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Input matrix. This matrix is typically (but not necessarily) sparse.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

Diagonal numbers, specified as a scalar or vector of positive integers. The diagonal numbers follow the same conventions as diag:

• d < 0 is below the main diagonal, and satisfies d >= -(m-1).

• d = 0 is the main diagonal.

• d > 0 is above the main diagonal, and satisfies d <= (n-1).

An m-by-n matrix A has (m + n - 1) diagonals. These diagonals 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 diagonal numbering.

If you specify a diagonal that lies outside of A (such as d = 7 in the example above), then spdiags returns that diagonal as all zeros.

Example: spdiags(A,[3 5]) extracts the third and fifth diagonals from A.

Diagonal elements, specified as a matrix. This matrix is typically (but not necessarily) full. spdiags uses the columns of Bin to replace specified diagonals in A. If the requested size of the output is m-by-n, then Bin must have min(m,n) columns.

With the syntax S = spdiags(Bin,d,m,n), if a column of Bin has more elements than the diagonal it is replacing, and m >= n, then spdiags takes elements of super-diagonals from the lower part of the column of Bin, and elements of sub-diagonals from the upper part of the column of Bin. However, if m < n , then super-diagonals are from the upper part of the column of Bin, and sub-diagonals from the lower part. For an example of this behavior, see Columns and Diagonals of Different Sizes.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | logical
Complex Number Support: Yes

Dimension sizes, specified as nonnegative scalar integers. spdiags uses these inputs to determine how large a matrix to create.

Example: spdiags(Bin,d,300,400) creates a 300-by-400 matrix with the columns of B placed along the specified diagonals d.

## Output Arguments

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Diagonal elements, returned as a full matrix. The columns of Bout contain diagonals extracted from A. Any elements of Bout corresponding to positions outside of A are set to zero.

Diagonal numbers, returned as a column vector. See d for a description of the diagonal numbering.

Output matrix. S takes one of two forms:

• With S = spdiags(Bin,d,A), the specified diagonals in A are replaced with the columns in Bin to create S.

• With S = spdiags(Bin,d,m,n), the m-by-n sparse matrix S is formed by taking the columns of Bin and placing them along the diagonals specified by d.

## Version History

Introduced before R2006a