DM41X Matrix

Matrix functions in the Advantage Pac

The matrix functionality isn't a standalone program, with a menu, like CFIT and PLY.

It's a collection of individual programs that can be used for many matrix tasks. Each one has to be called individually as needed.

There are many commands, the most used are those for

• Dimensioning a new matrix MATDIM
• Editing a matrix to enter values - there are 2 versions
  • Real Matrix MEDIT
  • Complex Matrix CMEDIT
• Inverting a matrix MINV
• Transposing a matrix TRNPS
• Finding the determinant MDET
• Multiplying two matrices together M*M
• Solving a system of equations defined by 2 matrices MSYS

To make things easier it's best to use the DM41X CST custom command shortcut mechanism.

Start with a blank CST and add the commands you need. Then save this CST so that it's easy to switch to other CST setups, and then return to the Matrix setup later.

I have the following

Matrices are saved in Extended Memory, and I put commands in CST to find and remove them once I'm finished: EMDIR, EMDIRX and PURFL and I have ZK?YN to quickly access DM41X 41z Module complex number functions.

All matrix addressing is done via the ALPHA register

• Put the name of the matrix in the ALPHA register.
  • Use single letters A, B, C etc. - these will create the matrix in Extended Memory
  • ALPHA A ALPHA
• Put the dimensions in the X register
  • this is in the form r.ccc with 3 decimal places for the column
  • for a 3×3 : 3.003 (3 rows, 3 columns)
  • for a 4×2 : 4.002 (4 rows, 2 columns)
  • 3.003
    • puts the dimensions in the X register ready for MATDIM to assign to the matrix named in ALPHA
• Execute the MATDIM command from the CST menu A
  • CST A
  • This creates matrix A in extended memory

• Populate the matrix with values using MEDIT from the CST menu B

For a 3×3 matrix

| 1 1 -1 | | 4 6 6 | | 1 8 9 |

• CST B
• 1:1= 1R/S
• 1:2= 1R/S
• 1:3= 1CHSR/S
• 2:1= 4R/S
• 2:2= 6R/S
• 2:3= 6R/S
• 3:1= 1R/S
• 3:2= 8R/S
• 3:3= 9R/S

With A still in the ALPHA register use the MDET command from the CST menu F

• CST F
• -50.000

• Matrix A will now be in LU Decomposition form.
• It can be used in this form for some other functions, but it's possible to get it back to normal form by Inverting twice using MINV from CST menu D
  • CST D
  • CST D
  • You can check the element values by using MEDIT again, but just cycling through each with R/S and not changing any values
    • it's possible to set flag 08 which prevents any changes to values

Using M*M

For matrix multiplication you need 3 matrix definitions:

• Matrix A and Matrix B are the two matrices to be multiplied
  • Matrix A and Matrix B are dimensioned with MATDIM and populated with MEDIT
• Matrix C is the matrix to hold the result
  • Matrix C is simply dimensioned with MATDIM and will be filled automatically by the function M*M

To multiply two matrices together they must have the correct dimensions

• Columns in A must equal Rows in B
• Results matrix C will have the dimensions Rows of A x Columns of B

Follow the same procedure as above.

Make sure ALPHA has only the letter A

A different numerical example

A will be a 3×2 matrix

• use MATDIM via CST menu
  • 3.002 CSTA

| 2 3 | | 4 5 | | 7 6 |

• Use MEDIT via CST menu
  • CST B
  • 1:1= 2R/S
  • 1:2= 3R/S
  • 2:1= 4R/S
  • 2:2= 5R/S
  • 3:1= 7R/S
  • 3:2= 6R/S

B will be a 2×2 matrix

Make sure ALPHA only has B and then dimension B with MATDIM from CST menu A

• ALPHA B ALPHA
• 2.002 CSTA

| 2 3 | | 4 5 |

* Use MEDIT via CST menu

• CST B
• 1:1= 2R/S
• 1:2= 3R/S
• 2:1= 4R/S
• 2:2= 5R/S

We now have defined the matrices to be multiplied together.

We need a results matrix C

The dimensions of C will be rows of A and columns of B which is 3 rows and 2 columns : 3.002

Make sure ALPHA only has C and dimension C with MATDIM from CST menu A

• ALPHA C ALPHA
• 3.002 CSTA

That's enough to create the results matrix

• Put the matrix names in ALPHA in the form Matrix 1,Matrix2,Results Matrix
  • ALPHA = A,B,C
• ALPHA A , B ,CALPHA
• Execute M*M from the CST menu
  • CST E
• C will now hold the answer

• Put C in the ALPHA register
• ALPHA C ALPHA
• Use MEDIT from the CST menu and cycle through the elements (it's a 3×2 matrix)
• CST B
  • 1:1= 16.000? R/S
  • 1:2= 21.000? R/S
  • 2:1= 28.000? R/S
  • 2:2= 37.000? R/S
  • 3:1= 38.000? R/S
  • 3:2= 51.000? R/S

The result in C is

| 16 21 | | 28 37 | | 38 51 |

A simple example first

3 unknowns

3x + 4y + 6z = 39.7 2x - 4y - 3z = -7.0 y + 8z = 63.4

| 3 4 6 | | 2 -4 -3 | | 0 1 8 |

• Dimensions 3.003

| 39.7 | | -7.0 | | 63.4 |

• Dimensions 3.001

Simultaneous Equations represented in Matrix Form

|A| x |C| = |B| | 3 4 6 | | x | | 39.7 | | 2 -4 -3 | x | y | = | -7.0 | | 0 1 8 | | z | | 63.4 |

• ALPHA A ALPHA
• 3.003 CST A (MATDIM)
• CST B (MEDIT)
  • 1:1= 3 R/S
  • 1:2= 4 R/S
  • 1:3= 6 R/S
  • 2:1= 2 R/S
  • 2:2= 4 CHS R/S
  • 2:3= 3 CHS R/S
  • 3:1= 0R/S
  • 3:2= 1R/S
  • 3:3= 8R/S

• ALPHA B ALPHA
• 3.001 CST A (MATDIM)
• CST B (MEDIT)
  • 1:1= 39.7 R/S
  • 2:1= 7 CHS R/S
  • 3:1= 63.4 R/S

• Put A,B in ALPHA
  • ALPHA A , B ALPHA
• Run MSYS - the solution matrix for x, y and z will be placed in Matrix B
  • CST H

• Put B in ALPHA
  • ALPHA B ALPHA
• Run MEDIT from CST menu
  • CST B (MEDIT)
    • 1:1= 1.500 'R/S/
    • 2:1= -3.800 R/S
    • 3:1= 8.400 'R/S

x = 1.5 y = -3.8 z = 8.4

• A is now in LU-decomposed format
• It can be restored to normal by Inverting Twice
• Put A in ALPHA
• Run MINV via CST menu twice
  • CST D CST D
• A is now back to normal and can be inspected and edited by MEDIT via the CST menu

Instead of the MSYS function you could do it manually by entering Matrix A and Matrix B, and define Matrix C such that

A X C = B

C = B / A

C = A* X B where A* is Inverse A

Create Matrix C with dimensions 3.001 in the usual way then and do:

• Put A in ALPHA
• Invert A using MINV from CST menu (A is replaced by its inverse)
• Put A,B,C in ALPHA
• Multiply using M*M
  • this multiplies B by the inverse of A which is effectively dividing B by the original A
• C now contains the solutions for x, y and z
• Put C in ALPHA
• Examine C with MEDIT
  • 1:1= 1.500 R/S
  • 2:1= -3.800 R/S
  • 3:1= 8.400 R/s
• This preserves B unlike the MSYS method.
• The Inverse of A can be reversed by repeating MINV with A in ALPHA
• We can even run M*M on A and C to re-create B
  • Put A,C,B in ALPHA
  • Run M*M from CST menu
  • B will be re-created

It is possible to define, edit and carry out operations with matrices containing Complex Numbers.

The matrix is defined with twice as many rows and columns as appears necessary, and edited with the function CMEDIT, in CST menu C

| 1+j2 5+j6 | | 3+j4 7+j8 |

Two rows and two columns 2×1 containing complex numbers, so the definition/dimension is 4.004

The numbers are actually saved as if in a larger real matrix as

| 1 -2 5 -6 | | 2 1 6 5 | | 3 -4 7 -8 | | 4 3 8 7 |

The entry is via CMEDIT via CST menu C, which asks for each Real and Imaginary part in turn

• ALPHA A ALPHA
• 4.004 CST A
• CST C
  • RE. 1:1= 1 R/S
  • IM. 1:1= 2 R/S
  • RE. 1:2= 5 R/S
  • IM. 1:2= 6 R/S
  • RE. 2:1= 3 R/S
  • IM. 2:1= 4 R/S
  • RE. 2:2= 7 R/S
  • IM. 2:2= 8 R/S

• Matrix B of size 2×1 which can be multiplied by A as A x B

| 3+j4 | | 2+j6 |

This is dimensioned as double the basic size of 2×1 : 4.002 and then populated with CMEDIT

• ALPHA B ALPHA
• 4.002 CST A
• CST C
  • RE. 1:1= 3 R/S
  • IM. 1:1= 4 R/S
  • RE. 2:1= 2 R/S
  • IM. 2:1= 6 R/S

Matrix 'C' which will be also be a 2×1 complex with dimension :4.002

• ALPHA C ALPHA
• 4.002 CST A

• ALPHA A , B , C ALPHA
• CST E

• ALPHA C
• CST C
  • RE. 1:1= -31.000 R/S
  • IM. 1:1= 52.000 R/S
  • RE. 2:1= -41.000 R/S
  • IM. 2:1= 82.000 R/S

The result matrix is therefore

| -31+j52 | | -41+j82 |

Which agrees with my long-hand calculation (thanks to DM41X 41z Module )

|(1+j2).(3+j4) + (5+j6).(2+j6)| |(3+J4).(3+J4) + (7+J8).(2+J6)|

| (-5+j10) + (-26+j42) | | (-7+j24) + (-34+j58) |

| -31+j52 | | -41+j82 |

Once you can enter and manipulate Complex Matrices you can also use MSYS to solve systems with complex matrices in the same way as usual - the only difference is entering the matrices as Complex.

After creating and manipulating (and some finger trouble) there will be a bunch of files in Extended Memory.

Most will be simple matrix definitions with obvious names A, B, C etc. but occasionally you'll accidentally have run MATDIM with some nonsense in the ALPHA register and created a nonsensical filename in extended memory, which it's tricky to spell out correctly in ALPHA in order to run PURLF - this is where EMDIRX comes in.

To simplify clearing up I include EMDIR, EMDIRX and PURFL in the CST menu.

Assuming you have other files in Extended Memory, the quickest way is to find the number of files in EM with EMDIR.

• Run EMDIR from CST menu
  • CST N
• Count the files to find the number of the first file you've created using MATDIM - i.e. the first one after your other EM files
• Put this number in x
• Run EMDIRX from CST menu
  • CST O
  • this puts the name of the specified-numbered file in ALPHA and means you don't have to type it in yourself
• Run PURFL from CST menu to delete this filename
  • CST P
• Run EMDIRX again (with the same number in X)
  • If X returns as non-zero we've found another junk-matrix file in EM and ALPHA will have its name
  • Run PURFL again
• Keep doing this until EMDIRX returns 0.00 in X which means there are no more files to delete
  • all your original files are still safe, as they are numbered below the number we were searching for